CHAPTER 1 AN INTRODUCTION TO SEMICONDUCTOR MATERIALS

Transcription

1 CHAPTER 1 AN INTRODUCTION TO SEMICONDUCTOR MATERIALS This hapter disusses the intentions of the book, the general properties and generation of X-rays, and the strutural and hemial information that an be extrated through diffration. Muh of the emphasis is on omposite layer strutures, beause of their tehnologial importane. These layer strutures reate interesting hallenges beause of the stresses generated, whih ause strains that an be very anisotropi. A detailed explanation is presented of the elastiity for all ombinations of orientation and symmetry. Some of the basis of epitaxy are disussed and how with ertain ombinations of materials, omplex mirostrutures an exist. 1

2 2 X-RAY SCATTERING FROM SEMICONDUCTORS 1.1. General outline Semiondutor materials exist in many strutural forms and therefore require a large range of experimental tehniques for their analysis. The advantage of onentrating on this lass of materials is that the amount of detailed strutural information is impressive, however the appliation of the methods desribed is not exlusive to these material types. Sattering from powder samples will be overed theoretially and how a greater understanding of the sattering an give rise to extra information. The overall intention of this book though, is to allow the reader to obtain a good working knowledge of X-ray diffration theory, tehniques and analysis, so that he or she is fully aware of the possibilities, assumptions and limitations with this form of strutural investigation. The moleular struture of semiondutors is in general well known. Most materials of interest have been manufatured in some way; therefore an approximate knowledge of the elements and layer thiknesses and sequene is assumed, and is the starting point for many of the approahes used. Most samples of interest, however, are not of a simple moleular form but are omposite strutures, ommonly onsisting of multiple thin layers with different ompositional phases. There are many important strutural parameters that an modify semiondutor devie performane. These parameters inlude phase omposition, miro-strutural or layer dimensions and imperfetions, et. A desription of how the properties are ategorised and material form is given, sine this largely determines the X-ray sattering experiment for the analysis, Fewster (1996). The theory of X-ray sattering is presented from a physial basis and therefore naturally starts with dynamial theory and its extensions before desribing the more approximate kinematial theory. The theories are largely onsidered at the single photon level. In reality the experiment ollets many photons that are divergent and oupy a range of energies. These effets influene the experimental results and are overed in the

3 Chapter 1 An Introdution to Semiondutor Materials 3 desription of the instruments. In general the sattering is threedimensional beause the sample is three-dimensional and this is born in mind throughout. The mapping of the sattering in three-dimensions is the most general experiment and all other approahes are obvious projetions. The assumptions on moving to two-dimensional reiproal spae mapping and ultimately one-dimensional X-ray sattering, e.g. roking urves will then appear more obvious and understandable. The assumptions assoiated with interpreting data olleted in various ways will be disussed. This will then allow the reader to understand the subjet oneptually and extend the tehniques to his or her partiular problem. The onentration is on near-perfet semiondutor materials, whose elements are defined by their valene in the Periodi Table, that is group IV, III-V, II-VI, et., sine the available information is large as well as being ommerially very important. The onentration on near-perfet materials is to ontain the range of tehniques and available information. Although the tehniques will be onsidered in general terms so as not to limit the book to speifi materials. These near-perfet materials are essentially extended single rystals, however as the perfetion delines they beome more mosai until eventually polyrystalline. Sine this edition is more omprehensive and onentrates on the whole proess of olleting data, inluding the diffration proess and the instrumental aberrations, this is extended to inlude polyrystalline material analysis and powder diffratometers. Interpretation requires a good understanding of the instrument aberrations, the assumptions in the methods and presumed details onerning the sample. Aspets of quality ontrol with X-ray diffration methods based on omparative measurements requires an understanding of the sensitivity of the onditions to the parameters of interest, these aspets will also be overed.

4 4 X-RAY SCATTERING FROM SEMICONDUCTORS 1.2. Semiondutors Semiondutors range from the most perfet rystals available to amorphous materials. The sample dimensions an range from 12 inh (although 18 inh is possible) diameter ingots of bulk Si, down to layers of partial-atomi overage with nanometre sale lateral dimensions embedded in a multi-layer struture. The moleular struture of most ommon semiondutors is of a high symmetry extended lattie, for example Si has the spae group (this represents the relative relationships between atoms) of Fd3 m and the spae group for GaAs and InP is F 43m, whereas GaN in hexagonal form has the spae group P6 3 m. The two former spae groups are simple fae-entred ubi latties, figure 1.2.1, whih leads to relatively simple rystallographi relationships, whereas the latter spae group is more omplex and requires a deeper understanding of the rystallography. Figure The arrangement of atoms in typial III-V semiondutor strutures (e.g. GaAs, InP). The degree of omplexity in semiondutor samples is inreasing with the manufature of laterally patterned or phase separated strutures having dimensions suffiiently small to reate zero dimensional quantum size effets. The list of strutural parameters required to define these materials is inreasing, although the basi definitions an be

5 Chapter 1 An Introdution to Semiondutor Materials 5 desribed quite simply, Table Sine these strutures are manufatured the analysis has a good starting point in that an approximate understanding of the struture will exist. The hallenge therefore is to determine these parameters more preisely, not only to aid the manufaturing proess but also to analyse them for defets and strutural quality, et. All physial properties, be they eletroni or optial, rely in some way on the strutural properties of the material. As the degree of devie sophistiation inreases the strutural toleranes are redued. Therefore the neessity for aurate determination of the omposition, thikness and defets, et., beomes very important. X-ray analysis tehniques are very well developed for obtaining this information and mature enough to reognise its weaknesses. Of ourse no tehnique should be used in isolation but ompared and omplimented with other methods. Most tehniques measure something that does not ompare diretly with any other; this in itself gives valuable information not only on the assumptions in deriving the information, but also on the sensitivity of the tehnique. Amorphous semiondutors are important in the areas of large area eletronis, although beause of the low free-arrier mobility some of the assoiated iruitry requires re-rystallisation. This hange of strutural form improves the mobility and devie speed. The sizes of the rystallites, the loation of rystallite boundaries with respet to the devie ative region are all-important parameters. As we inrease the degree of perfetion the long-range strutural order gives rise to welldefined eletroni band strutures. This leads into the possibilities of band struture engineering. The variables that the devie engineer has are omposition, shape and orientation to reate strutures with the required optial or eletroni responses. Control over the growth of these strutural parameters at the required level, sometimes at the atomi level, is not a trivial task. The growth of these strutures is dominated by epitaxy either from the liquid phase (LPE), hemial vapour deposition (CVD), moleular beam epitaxy (MBE) or extended forms of these; metal organi CVD (MOCVD), metal organi MBE (MOMBE), et.

6 6 X-RAY SCATTERING FROM SEMICONDUCTORS These growth methods require very areful ontrol and therefore areful analysis (in-situ and ex-situ) to ensure the strutural parameters are those wanted. Also different ompositional phases have different interatomi spaings and therefore these must be aommodated either by elasti strains or plasti deformation. Plasti deformation exists in the form of raks and disloations, whih an at as harge arrier reombination entres and alter the devie performane. It is therefore very neessary to have knowledge of the defets in the ative region of the devie that ontrols their behaviour. Again very areful strutural analysis is required. All these properties an depend strongly on the quality of the substrate material, i.e. its defet density, orientation and surfae strains, et. It should be lear that these semiondutors annot be grown by pressing a few buttons and ahieving the ultimate performane. The growth mahine has to be haraterized for growth-rate that ould relate to temperature stability, whih an influene the alloy omposition in a layer, et. A very good and thorough understanding of the materials and the growth method are required. In-situ analysis methods used to monitor the growth are developing but generally the most thorough analyses are performed ex-situ that an be very exhaustive. The in-situ methods generally rely on a detailed understanding from post-growth analysis. X-ray diffration methods are sometimes used in-situ but in general ontribute to improving yield by analysing material at various stages in manufature, help in ontrolling the growth proess and for detailed materials analysis ex-situ. The X-ray analysis tehnique to apply depends on the material quality, the level of detail and the preision required. This book will desribe all the levels of preision and assumptions made to arry out ertain types of analysis. Beause the rystalline quality of many semiondutors is very high the diffration proess annot be treated in a simple way. Most analyses require the appliation of the dynamial diffration theory and therefore an understanding of this and the assumptions involved are important and desribed in Chapter 2. The development of instrumentation for olleting the sattered X-rays has

7 Chapter 1 An Introdution to Semiondutor Materials 7 also reated new possibilities in analysis that make X-ray methods a very versatile tool in probing the struture of materials. Table Definition of the strutural properties of materials. Type of property General property Speifi property Marosopi Shape Layer thikness Lateral dimensions Composition Strutural phase Elements present Phase extent Form Amorphous Polyrystalline Single Crystal Orientation General preferred texture Layer tilt Distortion Layer strain tensor Lattie relaxation Warping Homogeneity Between analysed regions Interfaes Interfae spreading Density Porosity Coverage Mirosopi Shape Average rystallite size Crystallite size distribution Composition Loal hemistry Orientation Crystallite tilt distribution Distortion Crystallite inter-strain distribution Crystallite intra-strain distribution Disloation strain fields Point defets Craks Strain from preipitates Interfae Roughness laterally Homogeneity Distribution within region of sample studied 1.3. Method In this setion a brief desription of the aessible information to X-ray diffration tehniques will be given. How and why this following

8 8 X-RAY SCATTERING FROM SEMICONDUCTORS information is possible to extrat will beome lear in later hapters. Table presents the definition of various strutural parameters used to define a material. The first subdivision of the strutural properties is into marosopi and mirosopi. These are X-ray definitions and an be onsidered respetively as aspets that define the major features of the diffration pattern (peak position and intensity) and those that alter the pattern in a more subtle way (peak shape and weak diffuse sattering). Table Definition for strutural types. Strutural type Nearly perfet epitaxial Textured epitaxial Textured polyrystalline Nearly perfet polyrystalline Amorphous extended lattie Random moleules Definition A single extended rystal having near perfet registry with the same orientation as the underlayer, whih is also nearly perfet. The layer orientation is lose to registry with the underlayer, both normal and parallel to the surfae plane. The layer is omposed of mosai bloks. Crystallites preferentially orientated normal to the surfae, but random in the plane. They have a distribution in sizes. Random orientated rystallites of similar size and shape. Similar strength interatomi bonds but no length sale orrelation greater than this. Essentially amorphous struture with weak interlinking between moleules, possibly giving some ordering. X-ray diffration is a very sensitive strutural analysis tool and the extent to whih detailed information an be obtained depends on the sample itself. Suppose that the sample is poorly defined and ontains numerous rystallites, with a distribution of strutural phases, sizes, orientations and strains, then, separating the various ontributions is not trivial. However if ertain properties an be determined rather preisely then others an be determined by extending the range of experiments. Clearly therefore, the initial assumptions onerning the sample will define whih strutural details an be obtained readily. At this stage we should define the sample sine these will define the likely information that an be determined by X-ray methods, and the appliable information to use, Table

9 Chapter 1 An Introdution to Semiondutor Materials 9 Orientation in this ontext refers to the alignment of low index atomi planes (these are planes separated by distanes of about one unit ell spaing) to some other referene, e.g. the surfae. These definitions onentrate on samples with laterally extended homogeneity and therefore an be expanded to inlude patterned strutures and random strutural variations in the lateral plane by onsidering them as olumns. Figure The main marosopi parameters that haraterise a layered struture. Sine any strutural probe will determine an average of a region or analyse an unrepresentative region of an inhomogeneous sample, it is lear to see that the useful information may be limited to some average parameter and its variation. For strutural types that are highly inhomogeneous, e.g. random moleules and textured polyrystalline materials, then X-ray diffration will average some long-range order, orientation distribution and their variations. This is where it is important to link the physial or hemial property of the material to the strutural property, for example is it the marosopi average or the mirosopi details that determines the property of interest? The next question may well be the sale of the variation; is it homogeneous at the miron or nanometre sale? X-ray diffration averages in several ways, within a

10 10 X-RAY SCATTERING FROM SEMICONDUCTORS oherently diffrating volume and the X-ray beam dimensions on the sample. Controlling the beam divergene an modify the former and the latter an be subdivided by analysing the sattered beam with an area detetor as in X-ray topography. The range of X-ray analysis tehniques therefore annot be simply ategorised into finite bounds of appliability but depend upon the material, the property of the material of interest, the versatility of the diffratometer, the X-ray wavelength, et. Understanding the details of diffration proess, the nature of X-rays and assumptions onerning the sample are all-important to making a good and reliable analysis. Figure The main mirosopi parameters that haraterise a layered struture. Some typial marosopi and mirosopi properties are given in figures and respetively. The important aspet here is the X-ray probe dimension with respet to the properties. Clearly the probe is not simply defined two dimensionally but also has some depth into the figures, onsequently we must be aware how this probe brings all this information together to reate a signal whih is then interpreted. Having defined some basis onerning the sample we shall onsider some basi information about the X-rays used to extrat this information.

11 1.4. Properties of X-rays Chapter 1 An Introdution to Semiondutor Materials 11 X-ray wavelengths ompare with the energy transitions of inner eletron orbitals in atoms. It is this property that is used to reate laboratory monohromati X-rays. High energy deelerating eletrons will also emit X-radiation and this is the reason for the ontinuum of radiation from laboratory soures. A laboratory soure is shown diagrammatially in figure with an aompanying spetrum. The radiation from a laboratory soure is not very uniformly distributed and in general muh of the radiation is not used. However the intense harateristi lines at as a good internal standard and it is these lines that are used in the majority of laboratory experiments. Figure The interior of a modern sealed soure X-ray tube and an indiation of the spetral variation with intensity for a typial anode material Synhrotron radiation soures work on a very different priniple. A synhrotron is really a storage ring for eletrons, whih are ontained by magneti fields to prevent exessive divergene and onsequent energy loss. When the eletrons are deviated from a straight line using so-alled bending magnets, wigglers and undulators, the onsequent aeleration

12 12 X-RAY SCATTERING FROM SEMICONDUCTORS towards the entre of the urve reates an energy orbital jump thus produing eletromagneti radiation. If this energy hange is large (i.e. high speed eletrons and small bending radius from intense magneti fields) then X-rays an be produed. The X-rays from the synhrotron are emitted tangentially from the radius and are onentrated into a narrow one with the eletri field vetor predominately onfined to the plane of the orbit; i.e. the beam is horizontally polarised. However it is possible to rotate this plane of polarisation but in general this aspet does restrit most experiments to sattering in the vertial plane. This an lead to extra toleranes required for mehanial movements of diffratometers beause of the gravitational pull. Another aspet to onsider is the wavelength alibration; this has to be done before any experiment sine the emission is smooth and there are few referene lines exept at absorption edges. Laboratory soures are rather less effiient at produing X-rays. The emerging X-rays are randomly polarised and almost radially symmetri, yet only a small perentage of this divergent soure an be used. Beause of the method of injeting eletrons into a synhrotron they are arranged in bunhes and onsequently the X-ray emission will have a time struture. This an prove useful for some experiments espeially when only a single bunh is injeted, if the X-ray pulse an be synhronised with some dynami experiment, (Barrington-Leigh and Rosenbaum, 1976, Whatmore, Goddard, Tanner and Clark, 1982). The third generation soures have a very high brilliane level, giving rise to very small soure sizes that an reate some phase oherene aross the whole soure. This phase oherene an lead to observable interferene effets when the beam travels along different optial paths (phase ontrast topography and tomography, Cloetens et al, 1999). This oherene over the soure an also reate routes to reonstruting the sattering objet (Miao, Charalambous, Kirz and Sayre, 1999). Laboratory X-rays from similar sized soures have very low power output. However there are methods of effetively moving the soure lose to infinity with rystal optis: the phase front formed from an objet ontaining several different optial paths an then be separated with an analyser rystal (Davis, Gao, Gureyev, Stevenson and Wilkins,

13 Chapter 1 An Introdution to Semiondutor Materials ). It is lear that the developments and possibilities ontinue and this is far from a stati subjet. These developments will then lead to new possibilities in analysis. Another soure that onentrates on the phase oherene is the Free Eletron Laser. This soure utilizes the alternating magnets of an undulator that deviate the eletrons from side to side, produing X-rays. Now if the X-rays produed at eah sideways movement are in phase with eah other, i.e. being separated by an integral number of wavelengths, then the output will be very strongly oherent. This makes the oherent sattering more useful, rather then the synhrotron that is not fully oherent. The disadvantage though, is that this neessitates a pulsed output rather than a more ontinuous soure, beause this is a linear devie and not a storage ring as in the ase with a synhrotron. The early free eletron lasers are predominately soft X-rays (long wavelengths or low energy), although harder X-ray soures with wavelengths loser to those reated at synhrotrons and sealed laboratory soures are lose to ompletion. The highly diretional aspets of the synhrotron generated X-rays leads to very intense soures ompared with laboratory soures. However the onveniene and improvements in intensity output makes the laboratory soures suitable for most experiments. One of the earliest methods for inreasing the intensity in the laboratory was ahieved by rapidly rotating the anode (rotating anode soure) to distribute the heat. This has lead to inreases in intensities by almost an order of magnitude for 15 kw soures; 60 kw soures are also available. However there are many other ways of improving the intensity and whatever method is used it has to be related to the problem to be solved, sine the intensity output should be qualified with flux, divergene, wavelength distribution, et. Sine X-rays are primarily generated from inner atom ore transitions the photon wavelengths are in the region of 0.1 nm, whih is of the order of the interatomi spaings in materials. Bragg s equation (derived in Chapter 2) indiates that the differene in the sattering angle of two interatomi spaings of 0.14 and 0.15 nm determined with a 0.15 nm

14 14 X-RAY SCATTERING FROM SEMICONDUCTORS X-ray wavelength is ~ As will be seen later the peak widths of diffration maxima an be loated within about This gives X-rays the high strain sensitivity at the part per million level and sensitivity to atomi sale spatial resolutions Instrumentation There have been onsiderable developments in new instrumentation. The power of laboratory X-ray soures have inreased and various fousing mirrors and X-ray lenses an reover the divergene of laboratory X-ray soures with onsiderable intensity enhanements. The degree of sophistiation is inreasing with the various omponents reognising eah other (i.e. exhanging the X-ray tube will be reognised by the system, thus limiting the power delivered, et.). Computer automation has made very signifiant improvements in time and freed user involvement and this will ontinue. This has onsiderably helped in the thinking to doing ratio. The mehanial stability has also improved with optial enoding on the axes, allowing fast movement to very high preision. Interhangeable omponents (monohromators, X-ray mirrors, slits, et.) inrease the versatility of diffratometers and an be pre-aligned so that several very different experiments an be performed on one instrument with a simple hange. The experiment an now be fitted to the sample and property of interest instead of the former more established approah of having an array of instruments for eah experimental tehnique. The hoie of instrumental onfiguration and its onsequential influene on the information required from the sample will be overed in Chapter Sample definition The various sample types have been desribed in setion 1.2 and 1.3, but here the desription will be defined more losely to that required for X-ray diffration. These definitions also indiate the information of importane in analysis, Table Basially any struture will be an

15 Chapter 1 An Introdution to Semiondutor Materials 15 arrangement of atoms. However a rystal is defined as any struture having essentially a disrete diffration pattern. This is the aepted definition (Ata Crystallographia A48 928, 1992). To have a diffration pattern that is observable with X-rays in the simplest ase requires some form of periodiity or repeat unit ell. A semiondutor, for example GaAs, Si, GaN, onsists of an extended periodi array and would fit into the above ategories of perfet epitaxy, textured epitaxy and possibly textured polyrystalline in thin layer form. Although material with no disloations (missing lines of atoms) an be grown, most do have disloations threading through them. The generation of onvetion urrents during growth an reate mosai bloks (rystallites surrounded by defets) that an be tilted with respet to eah other. These are all fairly typial features found in bulk material and thin films. One of the most fundamental problems in thin films is that the atomi spaing of the layer differs from that of the underlying material. This will ause either elasti distortion or, if the internal stress exeeds that whih an be aommodated by elasti strains, plasti deformation ours and misfit disloations are generated. Misfit disloations an be formed from the high stress levels and imperfetions at the growing surfae nuleating disloation loops that glide to the interfae, or by turning a threading disloation to lie in the interfae plane. Knowledge of the state of strain, the number of defets, et., an be very important for devie performane and X-ray diffration methods are very sensitive to these effets. Figure gives a three dimensional view of the strutural properties of a thin film. Basially we have a unit ell repeat that an vary laterally and in depth, having parameters a, b,, α, β and γ. Within this there are relative rotations between regions and layers, defets (disloations and point defets (atomi site errors, e.g. interstitials, vaanies and impurity atoms)). These features all influene the diffration pattern of X-rays. The reation of the sattering pattern from X-rays is one thing but to interpret the features is quite another and a reasonable understanding of the sample in question is neessary. The important aspets will now be onsidered here.

16 16 X-RAY SCATTERING FROM SEMICONDUCTORS Figure The range of strutural parameters aessible to X-ray methods for imperfet (real) samples. The very high strain sensitivity an allow measurement of x <1% (absolute) omposition variations in Al x Ga 1-x As alloys, <0.1% (absolute) omposition variations in In x Ga 1-x As, et., for peak shifts of o or 3.6 seonds of ar. However to ahieve this, the strain has to be related to the omposition using some assumptions. As disussed above a thin layer grown epitaxially on a substrate will distort either elastially or plastially. In both ases we need to determine the unit ell parameters of the layer of interest and alulate how this would hange if it was free standing. Clearly we have to inlude the influene of elasti parameters.

17 Chapter 1 An Introdution to Semiondutor Materials 17 Table A broad overview of the strutural parameters that haraterise various material types. Those parameters that have meaning in the various materials are given with filled diamonds, those that ould have meaning are given by open squares. Thikness Composition Relaxation Distortion Crystallite size Orientation Defets Perfet Epitaxy Nearly perfet epitaxy Textured epitaxy Textured polyrystalline Perfet polyrystalline Amorphous layers The influene of elasti distortions The arrangement of atoms in silion is similar to that given in figure 1.2.1, exept that all the atoms are idential. If we try and ompress the struture along the bonds, then the [111] type diretions will be muh more diffiult to ompress than along a [100] diretion, where we would distort angles, for example. So although the struture is of high symmetry, its elasti properties are very anisotropi. Compression along one diretion will neessitate an expansion in another. This an be haraterised by examining the relationship between stress and strain. Initially we an suppose that the strain is elasti until the internal stress is too large and plasti deformation ours. The plasti deformation will our as raks or disloations, however the strain parallel to any interfae will be related to the degree of alignment of the atoms in a layer with that underneath. Hooke s law gives the relationship of stress to strain, but beause we are onsidering an anisotropi medium we have to

18 18 X-RAY SCATTERING FROM SEMICONDUCTORS generalise the problem and the elasti stiffness to fourth rank tensors, Nye (1985). σ = a a a a T ε ij im jn ko lp mnop ij The a ij are the diretion osines of the diretion assoiated with i to that of j, σ ij and ε ij are the stresses and strains along the various diretions related by i and j. The parameters σ zz (=σ 33 ) and ε zz (=ε 33 ) are the stress and strain diretions normal to the surfae. T mnop is a fourth rank tensor with 81 oeffiients, where m, n, o and p run from 1 to 3. When equivalent oeffiients are onsidered, this an be simplified to a 6 6 matrix, ij, where i and j run from 1 to 6. The onversion from one to another is obtained by ombining the suffixes m and n, and o and p, using the following rules: Table The rules for onverting the 4 th rank stiffness tensor to a more onvenient matrix form. m,o=1, m,o=2, m,o=3, m,o=2, m,o=1, m,o=1, n,p=1 n,p=2 n,p=3 n,p=3 n,p=3 n,p=2 i,j=1 i,j=2 i,j=3 i,j=4 i,j=5 i,j=6 T ij represents the stiffness oeffiients, whereas the ij that are tabulated for most semiondutor materials. This simplified form is given by σ xx 11 σ yy 21 σ zz 31 = σ yz 41 σ xz 51 σ xy ε xx ε yy ε zz ε yz ε xz ε xy 1.6.2

19 Chapter 1 An Introdution to Semiondutor Materials 19 σ ij and ε ij represent the stresses and strains along various diretions. The onvention is that; 11 is the stiffness oeffiient for the a axis [100] diretion for ubi system, whereas for other symmetries this an vary, for example the hexagonal symmetries have 11 along [2-1-10], 22 along [01-10] and 33 along [0001]. This full array of oeffiients is the general ase for trilini strutures, although there are further equivalenes, ij = ji, hene for the trilini ase there are 21 unique stiffness oeffiients required. For higher symmetry many of these oeffiients beome zero and some beome equivalent. Suppose we onsider the growth along the <001> diretion then this orresponds to the zz indies and the layer will be onstrained in the plane of the interfae and the stress will be zero normal to this diretion, i.e. the top surfae is unonstrained then from equation σ zz 0 = 31ε xx + 32ε yy + 33ε zz + 34ε yz + 35ε xz + 36 = ε xy Now if we take the example of a ubi system, the high symmetry leads to the following equivalents 31 = 32 = 13 = 23 = 12 = 21, 22 = 33 = 11 and 34 = 35 = 36 = 0, therefore, equation an be rearranged to yield the strains along three orthogonal diretions to be related rather simply: ε zz = ε xx ε yy { ε xx + ε yy} Hornstra and Bartels (1978) have solved the ondition for several typial orientations of ubi systems, giving some examples for III-V ompounds. Any solutions of the ubi system will just inlude ombinations of the oeffiients 11, 12 and 44. However it is most useful with the inreasing range of materials and orientations that a ompletely general solution is required. So working with equation 1.6.1, the oeffiients need to be transformed bak into their 4 th rank form, e.g. 46 beomes T 2312, et., following Table In this tensor form the rystal an be rotated by invoking the relevant diretion osines and then related to sample stresses and strains. 11

20 20 X-RAY SCATTERING FROM SEMICONDUCTORS For the ase of an extended thin layer, there will be no overall shear for the probed volume, and again it is possible to extrat the σ zz line, but now the number of terms inreases (depending on the orientation). So the general relationship for an extended thin layer with no shear is given by where F F2 ε zz = ε yy F F ε xx m= 1 n= 1 o= 1 p= 1 3 F = a a a a T i The T mnop is now the full fourth rank tensor representation and an be onverted aording to Table to relate them to the more familiar tabulated form, ij. Eah summation has 81 terms, however this is redued by symmetry sine some of the T mnop / ij values are zero. If we now onsider the ubi system for a sample, with a surfae diretion that is not isotropi in the surfae plane, e.g. <110>, then F 1 F 2. Equation then beomes: ε zz = ε xx + ε yy This formula is based on the alulation of the strains, ε xx and ε yy along <110> and <001>, and therefore these parameters F 1 and F 2 are only orret if the strains in these diretion are known. Clearly the elasti parameters and therefore the distortion is a funtion of the diretion. This leads to signifiant asymmetry in the elasti parameters in GaAs for example: zz xx 3m 3n ε ε ε io ip yy mnop = So provided the layer symmetry (ubi in this example) mathes that of the substrate and both have the same orientation and the layer is strained to math that of the substrate then equation is valid for obtaining the state of strain in this <110> layer. However if there is some strain

21 Chapter 1 An Introdution to Semiondutor Materials 21 relaxation, the strains need to be measured along the diretions hosen for ε xx and ε yy. Similarly if the layer and substrate are of a different symmetry or orientation (e.g. GaN on sapphire not grown on 0001, i.e. non isotropi in the basal plane) then the elasti parameters need to be alulated along the diretions of measurement. Layer growth is sometimes onduted on viinal planes to improve surfae morphology or to grow quantum wires, or the substrate is not perfetly on orientation. This will also influene the parameters F 1 and F 2. By way of example a ubi GaAs layer grown on a ubi substrate that has a surfae orientation of 0 0, , 4 0 and 10 0 away from the <001> towards <101> will have differing parameters F 1 and F 2 given in Table 1.6.3: Table The variation in the prinipal elasti parameters as a funtion of misorientation with respet to the 001 diretion, the % error in ε zz is independent of strain magnitude. Viinal angle F 1 /F 3 F 2 /F 3 % error in ε zz introdued % % % % From Table 1.6.3, it is lear that minor misorientations, i.e. those within typial nominal values, have little effet on the strain estimations, however for larger misorientations this an be signifiant. The X-ray diffration experiment will give an approximate one to one relationship of omposition to the strain and therefore this should be onsidered for any preision measurement. To further illustrate the variation in the relationship between the stiffness oeffiients and the strain, a few examples for hexagonal GaN are given, sine several growth surfaes are used, e.g. -plane (0001), a-plane (2-1-10), m-plane (01-10) and r-plane (10-12). These planes have relevane as an orientation option sine the a- and m-planes are nonpolar surfaes and r-plane is semi-polar. The a, m and r orientations will

22 22 X-RAY SCATTERING FROM SEMICONDUCTORS remove or redue the piezo-eletri effet brought about by the strain of epitaxy, whih an be quite a signifiant effet in -plane material. For -plane GaN, equation simplifies to ε zz = ε yy ε xx + ε yy = ε xx From symmetry 23 = 13 and hene any stress applied in the surfae will have a similar strain response, i.e. it is isotropi in the surfae plane. However for a-plane GaN ε zz = ε xx + ε yy = ε xx ε yy Also in this ase of m-plane GaN ε zz = ε yy ε xx + ε yy = ε xx Beause the stiffness oeffiients 11 = 22 for this symmetry the response to in-plane stress for these two orientations is idential. Whereas for r-plane GaN ε ε ε zz = xx The ombination of oeffients runs to 8 non-zero terms to evalute F 1 and F 3, and 4 non-zero terms to evalute F 2, equations and The orientations within the surfae plane, for -plane, a-plane and m-plane are simple low index planes, whereas for the r-plane surfae (10-12), the two orthogonal diretions for determining these oeffiients are [-0.568,0,0.568,1] and [1-210], sine there are no simple integer ombinations for this orientation and the oeffiients are funtions of the lattie parameters. This non-integer index of diretion is determined from the vetor produt. Clearly deposition on any surfae with different lattie parameters, other than (0001), will not distort evenly in the plane, and onsequently the layer symmetry will no longer be hexagonal. These detailed alulations give a range of distortion fators within the bounds of A reasonable approximation for lower yy

23 Chapter 1 An Introdution to Semiondutor Materials 23 symmetry materials an be made within these bounds if the stiffness oeffiients are unknown, assuming there is only a small diretional anisotropy or the symmetry is isotropi in the interfae plane. Alternatively if an engineering Poisson ratio, ν, is known then this an give a distortion oeffiient given by: ν ε zz = ( εxx + εyy) ν For rigid materials, ν takes on a low value <0.33 and for flexible materials, e.g. rubber ν an be as high as 0.5, whih is also the maximum value it an take. A Poisson ratio of 0.5 indiates that onstant volume is maintained during distortion, however the strutural form for ommon semiondutors of this Poisson ratio an vary onsiderably with struture and orientation, beause of the nature of the highly diretional ovalent bonding. Strutures of similar form and orientation however do not vary too exessively, so approximate estimates of a Poisson ratio an be assumed from similar strutures if no values are available. Clearly as the moleular form beomes more ompliated and the bond diretions more random this anisotropy will derease, making a more general distortion oeffiient / Poisson ratio a reasonable approximation. Additional shear ations ome into effet along diretions of lower symmetry, however these may be of less onern in a homogeneous thin layer of large lateral dimensions. For example, ubi materials with a surfae orientation of [001], from equation , the Poisson ratio is given by + 12 ν = For patterned wafers, whih have small lateral dimensions that are omparable to the layer thikness, the stresses are not simply relieved at the top surfae, and the full matrix, equation should be used, e.g. as in finite element analysis, to estimate the distortions in more than one diretion. This is important for analyzing surfae quantum dots, freestanding quantum wires, et. 12

24 24 X-RAY SCATTERING FROM SEMICONDUCTORS The onentration here is for planar strutures, whih are the most ommon appliation for these analyses, where the strains in the plane of the interfae an be expressed as ε L d x L d L d x y L d 0 0 y xx =,ε yy = L0 d x L0 d y and in the diretion normal to the surfae L d z L0d z ε = εzz = d Ld x, et., are the atual atomi plane spaings along x, and L0d x are the unstrained or free standing atomi plane spaings along x, et. L0 z The epitaxial relationship The atoms of material deposited on a substrate will try and bond to those in the substrate and the lowest energy onfiguration will result in the most likely arrangement. The energy is a ombination of the diret bond energy (related to the differene in bond length to the ideal), the torsion energy (effetive next-nearest neighbour atom distanes from the ideal), deviations from the ideal dihedral angle (effetively the next-next-nearest neighbours) and so on, plus the elasti ompliane of how the whole struture responds to this interation and the influene of any strutural faults, e.g. grain boundaries. Examples of a simplified energy alulation will be given later, but initially simpler systems will be onsidered, that have similar atom arrangements and small differenes in atom spaings from layer to substrate. Let us firstly onsider a ubi (001) GaAs substrate with a thin film of ubi AlAs on top. Both strutures have the same spae group, the same arrangement of atoms, but slightly different lattie parameters and elasti parameters. If a thin layer (~0.2 μm) is deposited then the atoms will align with those of the substrate and the struture will appear as a ontinuous lattie with an abrupt hange in lattie parameter and omposition at the substrate interfae. The alignment of the atoms in the interfae plane will define the lattie parameter of the layer in the

25 Chapter 1 An Introdution to Semiondutor Materials 25 interfae plane and through the appropriate elasti stiffness ombinations will define the lattie parameter normal to the interfae plane, figure Figure The undistorted (before deposition) and a distorted (after deposition) unit ell for a simple ubi layer on a ubi substrate; both are orientated along a ubi edge diretion. Figure The problems that our when the elasti parameters are inapable of aommodating the distortions neessary for perfet epitaxy.

26 26 X-RAY SCATTERING FROM SEMICONDUCTORS As the thikness of the layer inreases the layer will beome progressively more relutant to distort. Eventually the elasti limit will be reahed and only partial registry will exist. When this situation arises there will be more rows of atoms in the substrate than in the layer (the lattie parameter of GaAs is less than that of AlAs). The average lattie parameter in the plane of the interfae will therefore differ and some distortion will extend into the layer and substrate with possible tilting between the two, figure Clearly the struture beomes quite omplex even for this simple system. For perfet epitaxy we are trying to math the layer interatomi spaings to that of the layer or substrate below, S d x, et., however if the layer has partially relaxed bak to its strain-free state then we an rewrite equation as S d x L0d S d y L0d x y ε xx = [1-Rx ],ε yy = [1-Ry ] L0 d x L0 d y where R x is the relaxation in the misfit along x, et. The relaxation is then zero for perfet mathing and unity when the layer relaxes to its unonstrained shape. Substituting equation into will therefore give the perpendiular strain from knowledge of the original lattie parameters of the omponent layers and the degree of relaxation in two orthogonal diretions. S d z L0d z F1 S d x L0d x F2 S d y L0d y ε = = [1-Rx ] + [1-Ry ] d F d F d L0 z 3 L0 x When the unit ells of the two materials differ signifiantly then the registry of the atoms beomes very omplex. Consider for example (0001) GaN on (0001) sapphire, both are hexagonal strutures but the lattie parameters differ quite onsiderably. The atom arrangement of both materials is given in figure Now the orientation of the GaN on the sapphire (0001) surfae is determined by the interfae free energy, whih depends on the thermodynamis of the system, this is overed in onsiderable detail in Sutton and Balluffi (1995). For the purposes of illustration here a simple approximation is made by assuming the bond energy is at a minimum at some equilibrium distane, but inreases as the square of this separation. Similarly a seond order effet of the 3 L0 y

27 Chapter 1 An Introdution to Semiondutor Materials 27 torsion angle an be treated in a similar manner. Further interations next-next-nearest neighbours assoiated with a dihedral angle are ignored to simplify the proess, so it then beomes possible to suggest possible orientation arrangements. The saling between ontributions is a potential soure of error and the validity of this simple square-rule relationship at large distanes is dubious, although a non-bonding ut-off is imposed. However it is not the purpose here of defining anything with preision, but rather suggest likely onfigurations of a layer on a substrate. It must be remembered that these poorly mathed strutures an have a omplex mirostruture and differing response to elasti distortion, et., i.e. there are many unertainties. The energy to be minimized is given by: 2 ( l l ) + E ( Φ ) 2 E E B 0 T Φ0 where l and l 0 are the losest approah and ideal bond length, Φ and Φ 0 are the torsion angles for the next-nearest neighbour and that for the ideal onfiguration and E B and E T are sale-fators (E B >> E T ). The alulation is based on determining the ideal bond length and torsion angle for an atom within the layer, and then determining the energy E for various onfigurations for the atoms at the interfae. The best atomi math appears when the two latties are aligned along different diretions, i.e. the x-diretion is rotated through 30 0 with respet to the other. However we an see that this gives approximate alignment of the Al in the sapphire to the Ga in the GaN. The mismath is so large however that the GaN is very heavily relaxed towards its unstrained state and will be full of defets assoiated with this poor math.

28 28 X-RAY SCATTERING FROM SEMICONDUCTORS Figure The basal plane view of GaN and sapphire; this indiates the rotation neessary to aommodate the alignment of atoms for epitaxy. Al and Ga are oloured blak. Pashley (1956) has given a very full aount of the possibilities in epitaxy and enompasses the early theories. A full all-enompassing theory explaining the nuleation and orientation dependene is still elusive, but there are some general guidelines that an be given. Generally an orientation dependene ours when the mismath (the frational differene in the lattie plane spaing in the plane of the interfae) between the overlayer and underlayer is less than ~14%. Theoretial models and experimental evidene on a wide range of systems support this. The orientation depends on the relative alignment of atoms in the overlayer and underlayer and is not governed by integer relationships of atomi plane spaings of the two latties. The thikness of the layer influenes the extent to whih the elasti distortion an be aommodated, the greater the misfit the thinner the layer should be to maintain good epitaxy. One epitaxial growth is established in the fabriation of a struture then a full understanding of these nuleation proesses may seem irrelevant, however some strutures make use of some of the nuleation properties of ertain materials.

29 Chapter 1 An Introdution to Semiondutor Materials 29 We an onsider growth to our in three priniple ways. The first is a two-dimensional mehanism, i.e. the layer is built up atomi layer by atomi layer, and this relies on the atoms migrating aross the surfae and preferring to loate at atomi layer steps. A omplete atomi layer overage will reate a very smooth surfae, whereas for intermediate overage the surfae is atomially rough. This osillation in smooth and rough surfaes explains the osillating speular refletivity observed in Refletion High Energy Eletron Diffration (RHEED) during growth by Moleular Beam Epitaxy (MBE), Neave, Joye, Dobson and Norton (1983). When the surfae does not wet easily with the deposited atoms of the overlayer, the growth an our in distint islands that gradually enlarge and eventually oalese. This is termed three-dimensional growth and an lead to mosai or olumnar growth with defets onentrated at the boundaries. Another interesting growth mode is a mixture of both three- and two-dimensional growth first desribed by Stranski and Krastanow (1938). This mehanism is haraterised by the initial formation of a wetting layer (two-dimensional growth) that is very thin (no more than a few atomi layers) and the subsequent growth of islands. Examples of these mehanisms an be seen in semiondutor materials. InGaAs deposited on GaAs at low In ompositions, <10%, will grow two-dimensionally up to about 70 nm before misfit disloations are formed at the interfae, whereas InAs will grow by the Stranski-Krastanow mehanism. These differing growth modes may appear troublesome but an be used to advantage in reating strutures defined in all three dimensions by optimising the growth method. These an have very speial properties and offer another hallenge to analytial methods. Determining the growth mode an only be aomplished with preise surfae diffusion data and bond strengths, et. It is then possible to onstrut a surfae by modelling the whole proess and extrating a statistial signifiane. These approahes have been very suessful at prediting some of the general observations of surfae topography, Itoh, Bell, Avery, Jones, Joye and Vvedensky (1998).

30 30 X-RAY SCATTERING FROM SEMICONDUCTORS Prediting the situation when defets form at the interfae between two materials, i.e. when the elasti limit has been exeeded, has been the subjet of many studies. This of ourse is a very important parameter beause defets in general are detrimental to semiondutor devies. Knowledge of the bounds of lattie parameter misfit and thikness define whether a devie is possible to fabriate. Hull and Bean (1992) have reviewed the mehanisms of disloation generation and propagation and disussed the definitions and derivation of the ritial thikness defining their onset. Of ourse there are many experimental studies that have questioned the theoretially derived values. Dunstan, Kidd, Howard and Dixon (1991) have taken a very pragmati approah to the evaluation of ritial thikness and ompared the residual strain as a funtion of thikness. The resulting urve is remarkably preditable for a large range of material systems and offers a very quik proedure for prediting the onset of relaxation. Understanding and ontrolling the influene of surfae misorientations to redue defets has been the subjet of many studies, e.g. LeGoues, Mooney and Tersoff (1993) who studied the distribution and arrangement of disloations and related these parameters to the proximity of glide planes. Stringfellow (1982) has reviewed the relationship of small misorientations ompared with low-index planes and how these promote step-edge growth, and Shukin and Bimberg (1999) have used this growth ontrol to influene the growth of quantum wires. If the mismath is large or the substrate is amorphous then the orientation dependene of the layer an be governed by very different riteria and the layer an beome essentially textured polyrystalline or even random polyrystalline. Knowledge of the likely form of these materials will define the type of experiment neessary to obtain detailed strutural information. Chapter 2 will desribe the theoretial basis of sattering from various strutures typially enountered in the field of semiondutor physis as well as more general material forms, e.g. powders. The onentration on semiondutors is of partiular interest, beause these materials represent the ase when a large amount of information an be