High precision structural measurements on thin epitaxial layers by means of ion-channeling

Nuclear Instruments North-Holland

and Methods

in Physics

HIGH PRECISION S~~~~~ OF ION-~~NNELrNG A. CARNERA Qipartimento Received

6 February

B44 (1990) 357-366

~AS~~~S

and A.V. DRIGO

di Fisica dell’ Universit2

Research

ON THIN EPITA~AL

357

LAYERS BY MEANS

*

- CISM,

Via Marzolo 8, 35131 Padova, Italy

1989 and in revised form 2 June 1989

The possibility of obtaining a detailed description of the crystalline structure of single-crystal samples is intrinsic to several applications of the ion-channeling technique. This paper reports a new approach to the use of channeling measurements which allows a precise characterization of the crystallography of very thin crystalline layers. The use of a precise mathematical description of the sample rotations which are involved in a typical channeling experiment gives the possibility of having a direct correlation between the sample lattice structure and the anguhu coordinates where the axial and planar channeling minima are located. The model is fully tested and the precision of the measurements obtained by this technique is compared to the results of double crystal X-ray diffraction measurements on the same systems. This technique is particularly well adapted to the measurement of lattice strain in heteroepitaxiai structures

1. Introduction Since its discovery nearly twenty years ago, ion channeling - the influence of a crystal lattice on the trajectories of energetic particles - has become a well established technique in materials science for the investigation of the c~st~lograp~c structure of the near surface region of solids [l]. Ion channeling essentially consists of the steering of the trajectories of particles for a well collimated beam entering a crystal parallel to low index axes or planes. This steering ability gives rise to a number of phenomena useful for the investigation of the structure. The main use of channeling is in connection with Rutherford Backscattering Spectrometry (RBS), since RBS-channeling is a mass-dispersive and depth-resolved structure probe. Because of its flexibility the RBS-channeling technique has been widely applied to the study of surface c~st~lograp~c structures [2], lattice damage and damage recovery, lattice location of impurities and dopants, and the formation of surface crystalline, polycrystalline or amorphous phases [3]. In general a very high accuracy in the measurement of the rotation angles is not required because the bulk or the host lattice channeling dips supply the reference direction. Therefore, no accurate description of the geometrical relationships which are involved in a typical channeling experiment is available in the literature up * Present address: Dipartimento di Scienza dei Materiali Universita - CISM, Via Amesano, 73100 Lecce, Italy. 0168-583X/90,/$03.50 (North-Holland)

0 Elsevier Science Publishers

B.V.

dell’

to now. Only recently, by applying the channeling technique to the study of epitaxial relationships [4] or the strain associated to epitaxial heterostructures [5], it became clear that a very accurate measure of the rotation angles is required. For instance, the measurement of strain relies on the measurement of the angular deviation, A+, for lattice directions inclined to the surface normal with respect to the angle C$of the corresponding direction of the unstrained lattice. The strain induced A+‘s are maximized at + = 45 Q and are of the order of a few tenths of a degree so that a 10% precision in the measured strain value requires a relative precision of the order of 10m4 in the measurement of the rotation angles. In this paper we report the basic mathematics which is required to make precise measurements of the directions of the axes and of the poles of the planes of surface crystalline layers by ion-channeling. A description of a standard experimental setup for channeling studies is given in section 2; the basic equations are reported in section 3; the most important parameters which are involved in the pm-alignment of the goniometric sample holder are discussed in section 4; the tests we performed in order to have an actual estimate of the precision which can be obtained by means of this method are shown in section 5; finally in section 6 we outline some possible applications and a few results on the measurements of epitaxial layer tetragonal distortion. Some details on the equations describing the angular coordinates of planar channeling minima are reported in the appendix.

358

A. Camera, A. V. Drigo / Channeling analysis of epilaxial layers

2, Experimental

Scattering chambers for solid state physics measurements are installed both at the 2 MV and at the 7 MV Van de Graaff accelerators of the Laboratori Nazionali di Legnaro (Padova, Italy). Two silicon surface barrier detectors can be moved indep~ndentIy, controlled from outside the chamber, to any angle with respect to the beam and there are provisions for other detectors at fixed positions for charged particles, X-ray and y-ray detection. The beam is collimated by two sets of diaphragms whose distance is about 3 m and of variable size. In the most used arrangement for channeling measurements the diaphragm diameter is kept at 1 mm giving a measured beam divergence (HWHM) of less than 0.01” , while the current at the target is maintained at several tens of nanoamps. The goniometer is the model originally developed at the Atomic Energy Research Establishment in Harwell and manufactured and marketed by McLean Research Engineering Company Ltd. Three rotation axes allow complete freedom of orientation of the sample with respect to the beam. The rotations are operated by three independent stepping motors coupled by antibacklash gear boxes so that one step corresponds to O.Ol” for all axes. The guaranteed repeatability and overall precision are 0.01’ and 0.03” respectively, the x and z axes (see fig. 1) being operated through 360° and the y axis through f 30 O.

This apparatus has been working satisfactorily for several years in our laboratory for channeling measurements on a wide variety of single crystals. However only in the last three years it became necessary to have high precision angular scans. For this purpose a series of experiments on different materials have been performed in order to test the overall accuracy and all the possible applications of the system. Silicon samples were used in order to test the overall performances because Si single crystals of high crystalline quality are easily available. Strained single layer Ga, _,In,As/GaAs samples having low (< lo-*) strain values have been measured to determine the minimum detectable strain and to compare the results obtained by using this channeling-based technique with double crystal X-ray diffraction (DCD) measurements.

3. Angular meas~emen~ The ion-channeling experiments require the alignment of a crystallographic direction parallel to the incident beam. This is performed by rotating the crystal and by looking for the minima of the reaction yield as a function of the rotation coordinates. By assuming, for the sake of simplicity, that during the alignment procedure no rotation around the z axis (see fig. 1) is necessary, each direction is identified by the angles 6, and BY, corresponding to the rotations around the x and y axes, respectively, required to put that direction

Beam

a

b

Fig. 1. Schematics of the goniometric sample holder for channeling measurements: a) front view (beam and I axis normal to the plane of the figure); b) side view. The tilt angle eY around the y axis is shown together with the y angle which accounts For the non-orthogona~ty between the go~ometcr x axis and the beam direction.

A. Camera, A. V. Drigo / Channeling analysis of epiiaxial layers

parallel to the ion beam. These two coordinates are usually measured with respect to the normal to the sample holder, that is both 19, and By are assumed to be zero when the normal to the sample holder is parallel to the beam. A suitable fixed frame of reference to describe the rotations of the sample with respect to the beam is the frame whose X axis coincides with the goniometer x axis (which remains unchanged during the rotations of the crystal), the Y axis is chosen orthogonal to both the X axis and the beam direction (the y axis of the goniometer coincides with the Y axis when y is orthogonal to the beam) and the 2 axis is consequently defined as the third orthogonal axis. We will refer to this system as fixed frame of reference (FFR). As a consequence of this definition the X axis is not necessarily orthogonal to the beam and therefore the 2 axis generally does not coincide with the beam direction although it is usually very close. We will discuss later the effects of this misalignment. By using this fixed frame of reference, whose unit vectors will be indicated as f, Y, 2, we can easily describe the effect of the rotations of the goniometer on unit vectors (directions) which are fixed to the sample. Let P = (V,, I’,, Vz) be such a unit vector written as a function of its coordinates in the FFR and R,( 0,), R,,( 8,) the matrices describing the rotations around the x and y axes of the goniometer respectively. These two matrices are defined as

R,(B,)=

t 0 i

co:‘% sin 19,

0 - sin 0, cos e,

, 1

e,

0

_ sin e,

1

0

0

sin 8,

cos Ry(@,)=

0

i

359

sin ey

. 1

As a consequence of the rotations t is transformed into a unit vector ci= Rx(ex)R,(ey)l?

(lb)

R,( 0,) and Rv(fJ,,), CJ given by (2)

The order of the product has been chosen in order to account for the fact that a y rotation does not affect the orientation of the x axis but the opposite is not true. R,(B,) and Ry(By) being orthogonal matrices the following relation holds RR = I (where R is the transpose of R and I is the unit matrix) and the eq. (2) can be easily inverted giving P= A,(e,)B,(e,)O. By imposing

(3)

the condition

O=b,

(4)

(where & is the unit vector of the beam direction) eq. (2) or (3) becomes the mathematical description of the alignment of the direction P to the beam. As a consequence of the definition of the FFR the vector 6 does not necessarily coincide with 2 but it certainly lies in the (2, b) plane. If y is the angle between k and 2, the beam direction can be written as

(Ia) B = (-sin

y, 0, cos y).

mo1

b Fig. 2. Representation

of a diamond lattice: a) in the goniometer angular coordinates (8,, 8,) in the case of y = 0 and b) in the standard stereographic projection.

360

A. Carncra,

When condition (4) is imposed, the following set of equations

A. I;‘. Drip

eq. (3) corresponds

/ Channeling

analysts uf’eptluxial

buyers

to

and t';>is given by

j,~,rccos(P,*

Q

(6)

and the dot product 9,

l

can be written

by using eq, (5) as:

VJ= cos( ep - 0Y2)(sin2y + cos’y Cos OX1Cos tlx2) -Csin( O,, - 0Y2) sin

+cn&

y cos y (WE; Oil - Cos t?,,)

sin #x1 sin 8*X

(71

whwe f$,, f&, and &> rt,, ale the jptiom&x coos& nrttes of the two unit uectars VI and ff* respe&v& on the basis oE q. (7) and of the set of ttrcpticit Cartesian eoor&~ates given in eq. (5) it is p&bfe to predict the angular coordinates f& and gY of any crystal direction. The result of such a calcuIation can be summarized by of the pXotting in a (OX, 6),) plane the coordinates crystallographic axes, and of the tra~e,s of the planes. WC will refer to these plots as “goniomctric projections”. Fig. 2a shows such a projection far a diamond structure having its [OOl] axis at 8, = 0, 8, = Q and its (010) plane orthogonal to the x axis. In this case the angle y is assumed to be zero. Fig. 2b shows the standard stereo-

gfaphk projection ctf the SiLme struialre son.

4. Gontorneter

alignment

for compati-

procedure

Fig. 3.

Goniometric projectionof a diamond fat&e in the (&, $) plane assuming y = - 0.3 O.

It must be pointed out that any systematic error in the “zero” of the Sy coordinate does not influence the calculation of the angle between two directions bscause the dot pradxxct of two vectors Ieq. (7)] depends on the difference of &be 8, coordinates. However this cnlculation requires the knmkdge CS y and eq. (9) &OWS that there is no prtssibiiity of an j~~e~e~~e~t ~&&ion of the 8, md of the y parameter of the ex~e~men~~ setup by means of this simple alignment procedzmz For this reason it is necessag to develop a method for measuring independently the amount of misalignment between

In order to use these formulae to analyze channeling experimental results it is necessary to exactly determine the ‘zeroes’ of the 8, and t$ coordinates, i.e. to find the actual relations between the FFR and the axes of the goniometer.

-SO

By using eq. (5) these angles turn out to bo 8, w_* 0

“60

-30

0 30 0 x fdegreesk

60

80

Fig. 4, Trace of a nearly horizontal plane ($I =i -O.CKBO, I= -O.l” ) as it &ppeassassuming different values for the y angle: 1, y-Oa; 2, y= -O.l”; 3, y=-O.2O; 4, ym -0.3’; 5, y= -0.49. The experimental data points (i- ) me aXso reported showing perfect agreement with the trace obtained by assting y = -0.3”. For the defk&imz d i& and f see eq.

A. Carnwa, A. V. Drigo / Channeling ana(vsis

the Z axis and the beam direction ii that is the angle y. If y # 0 the trace of a plane being perpendicular to the x axis does not appear as a straight line in the goniometric projections and it can be demonstrated (see appendix) that its curvature depends only upon the actual value of y. Fig. 3 shows the same goniometric projection as in fig. 2a but assuming y = -0.3O which is the actual value for one of our experimental apparatus. The major feature of this projection is the curvature of the “horizontal” planes which is clearly evident at large 6, values even for this small value of y. We use this effect for measuring the angle y. Fig. 4 shows the experimental points where the channeling minima from a Si(Ol0) plane have been located in the (f?,, 8,) plane together with the traces of a plane having fixed orientation with respect of the FFR calculated by assuming different values for y. The best fit has been obtained for a value y = -0.3” with a precision of about k 0.03”. After this evaluation the orientation of the goniometer axes with respect to the FFR is unequivocally defined and any calculation of the angular distance between two directions can be exactly performed.

5. Accuracy and reliability

tests

In order to check the overall precision of the goniometer, the accuracy in the calibration parameters and the reliability of the procedure used in order to find the orientation of the crystal lattice with respect to the goniometer axes, a set of tests were performed using [OOl] Si crystals. First we checked the accuracy of the goniometer in measuring the angles between different axial directions, namely between the [ill), [OOlJ and [lil] axes in a Si[OOl] crystal along a (110) plane placed orthogonal to the x axis of the goniometer. In order to have a good measure of the channeling minimum the centroid of the data points is computed. The precision of such an estimate depends both on the number of data points and on the statistics of the integrated counts; for routine measurements it is of the order of 0.02O. The measured angular distances between axial directions in Si are shown in table 1 and compared to the theoretical values. As it can be seen the maximum discrepancy is of kO.025 o for the angle between the

Table I Experimental and theoretical

angles

between

different

direc-

ofepitaxial layers

361

.’ -90

-60

-30

0 0,

30

60

Fig. 5. Experimental data points from the same plane of the previous figure but after different rotations around the z axis: squares Art, = 0 O; open dots A& = 0.1”; crosses AI?, = 0.3O; triangles A.8, = OS”. The traces of the plane have been calculated using the best fit + values reported in table 2.

11111 and the [OOl] axes. This discrepancy is of the order of the inaccuracy in the determination of the minimum of the channeling dip. On the other hand the error in the angle between the two [ill] axes is undetectable so that we can conclude that the goniometer could be affected by a maximum nonlinearity of less than 5 X 10-4* One of the most important features of our calculational scheme is its ability to account for the misalignment, y, between the Z axis and the beam direction. Even though y may have a small value, as in our case, its influence on the goniomet~c projection of the crystal structure is important since it affects the 19~coordinate of a given direction. This effect can be rather strong for high 8, values as shown in fig. 4. The distortion of the trace of a plane is also influenced by its azimuthal coordinate # (for the definition of cp see appendix). This is illustrated in fig. 5 where we report the position of the planar channeling minima. taken at different 6, positions, for the Si (010) plane with different azimuthal orientations. The ~ntinuous lines are the traces of the plane computed by fitting the data points taking the azimuthal angle as the fitting parameter. The azimuthal angles resulting from this fitting are reported in table 2 and are compared to the A@, rotations (i.e. the rotation

Table 2 Fitted azimuthal angle (+), rotation of the goniometer around the z axis (A&) and difference between the fitted values (A+)

tions in a [OOl] Si crystal Angle between

axes

Exper.

Theor.

and [OOI]

54.71 0

54.736 o

[OOIJ and [it 11 [lil] and [ill]

- 54.76 0

- 54.736 D

109.47 D

109.471 0

[lit]

9

(degrees)

-0.038” 0.062O 0.262 o O+WJO

0” 0.10 0 0.30 o 0.50 o

_ 0.10 o 0.30 o 0.48O

B,

(degrees)

Fig. 6. Goniometric projection of a Si lattice having its (010) plane ~muth~ly tilted by 5.0~5~. The traces of the planes have been computed by assuming this tilt and by imposing the experimental coordinates of the [OOl] axis. The experimental coordinates of the planar channeling minima for the (010) plane (open dots) and the (ial) plane (triangles) are reported. The accuracy of the lattice description is shown in the insert reporting the experimental and calculated angular distances between the [Ool] and the [l&l) axes,

angles

around

the

z axis

of

the

goniometer)

used

to

obtain the change in the azimuth. Here again the precision of the computation scheme appears to be excellent. As stated in the previous section, once the orientation of a pair of directions in the crystal have been determined (for instance an axis and the pole of a plane crossing that axis), the angular coordinates of any direction of the crystal lattice can be calculated and the goniometric projection of the crystal can be drawn. The experimental data are shown in fig. 6 witb superimposed the projection of the crystal structure obtained by irn~os~~~ the angular coordinates of the [OOl] axis and the azimuthal angle (+ = 5.05 ’ > of the (010) plane. The angular distances A@, and A6, between the [OOl] and [10X] axes have been calculated and compared to the experimental values: the results are shown in the insert of fig. 6. The agreement is surprisingly excellent because the discrepancies are well below the exp~r~ental uncertainties. However the results of other less accurate measurements lead us to state that the error in locating a given direction with such a method is at most of some hundredths of a degree. Another important feature resulting from the results of fig. 6 is that our computation scheme allows a very precise absolute estimate of the angle between two crystalline directions, which is very important for channeling-based angular measurements. In fact when two directions do not he in a plane perpendicular to the x axis, the A@, angle is in no way a measure of the angular distance between two directions. For instance the angle between the [OOl] and the flOl]

in a cubic structure, which is used for measuring the strain effects in superlattice structures because it maximizes the strain effects, in the case of fig. 6 is Ati, i= 44.79 * and not 45 O, i.e. the effects of the nonorthogonality with respect to the x axis of the plane defined by the two lattice axis is 0.2”, which is of the same order of ma~~tude as the strain effect. This fact is particularly important for goniometric sample holders having onIy two rotation (x. y) axes because they do not allow the plane to be positioned exactly perpendicular to the x axis. The correction which must be applied to the direct measure of A@, as a function of the azimuthal angle + of the plane strongly increases with increasing the + angle so that it is small (I 0.05 * ) only for + < 2O, but it amounts to about 0.5O already at (p=7”.

6. Application

As a first example of the problems which can be solved by this technique we present the channeling analysis of a SLS made of 4 periods each formed by a 15 nm thick InGaAs and a 30 nm thick GaAs layers It;]. Fig. 7 reports the RBS-channehng result of an angular scan around the [llO] direction of this sample. It appears that the channeling dip of the top layer shows the usual symmetric shape while the dips for the successive

O.O4W

46

u, (DEGREES) Fig. 7. Angular scans around the fllO] axis in a 4 period inGaAs/GaAs strained layers superlattice. Different symbols refer to different InGaAs iayers (I: topmost layer; 3 and 5 buried layers) and to the GaAs buffer layer. A+ is the distance between the channeling minimum of the topmost strained layer and the [IlO] axis of a cubic lattice.

A. Camera, A. V. Drigo / Channeling analysis of epiiaxial layers

8

363

(degrees) Y

Fig. 8. Goniometric projection of the full set of axial (full dots) and planar (X) minima measured in a single strained InGaAs layer grown on GaAs in order to determine the tetragonal distortion of the structure. The traces of the planes are computed on the basis of

the best fit value of the tetragonal distortion.

layers and even the dip of the buffer/substrate layer are strongly asymmetric. This effect is due to the “kinks” of the atomic rows at the interface and to the steering power [7] which induces re-channeling of part of the beam which was de-channeled in the preceding layer. The main point here is that the minimum in the buffer channeling dip is in no way a measure of the angular position of the channeling direction in the unstrained lattice. The tetragonal distortion measured by this technique without taking into account such an effect would be systematically underestimated. On the contrary our measurement based on absolute rotation angles gives A$ = (0.70 + 0.04) o in perfect agreement with the value computed from the continuum elasticity theory and from the measured In composition (x = 0.180 &-0.005, AJi = 0.71” + 0.02O). 6.2. Strained single layers In the case of SLS the structure may be investigated by means of X-ray diffraction. This technique gives better and better results the higher is the number of periods of the structure. Therefore this latter method is usually preferable to the channeling technique. However the opposite holds in the case of single strained layers. In fact the diffraction peak from the epilayer broadens and weakens with decreasing layer thicknesses. For this reason the errors associated to these measurements when

the layer thickness is lower than some tens of nm are unacceptably high. On the other hand, when the thickness of a strained layer increases above a critical value, misfit dislocations are generated and the diffracted peak broadens again as an effect of the lattice distortions around the dislocation line. These disadvantages are not present in the RBS-channeling technique which allows measurements for film thicknesses ranging from 10 to 1000 nm. In the following we present a method allowing very high precision and the results are compared with those obtained from DCD. We have seen that the mathematical description of the rotations allows one to compute exactly the goniometer angular coordinates of any direction, and conversely that it allows one to compute the angular distance between any couple of directions whose goniometer coordinates are measured by locating the minimum in the channeling dips. Axial channeling dips are generally flat-bottomed so that the location of the minimum is performed by computing the centroid of the dip data points. The error is then reduced but it is still of the order of a few hundredths of a degree. In order to further reduce this error many channeling directions are located (mainly planes exhibiting narrower channeling dips). The tetragonal distortion of the strained structure is then obtained by a least-squares fitting of all the data points as a function of the fitting parameter a i/n,, (the ratio

364

A. Carnero, A. f/ Drigo / Channeling analysis of epituxial layers

-10 52

53

54

ex

55

56

(degrees)

Fig. 9. Enlarged view of the go~omet~c projection around the [Ill] axis showing the sensitivity of the technique in the case of very small tetragonal ~stortion [ ft i / a,,= (1.~47~0.~3)]. Dashed lines show the traces of the planes of the undistorted (cubic) structure for comparison.

between the lattice parameter perpendicular and parallel to the interface). The procedure used is the following: first the orientation of the sample with respect to the goniometer is found by locating a couple of (orthogonal) planes around the surface normal. One of these planes is then put “horizontal” (i.e. orthogonal to the n axis of the goniometer) and the trace of this plane is determined in a few points far from the surface normal so that the x axis ~sahgnment angle, y, is determined. This latter parameter usually does not vary much from run to run but, since it depends both on the mechanical mounting of the goniometer on the chamber and on the beam alignment it is routinely checked. The coordinates of the appropriate crystallographic axes and of the planes around them are then determined as shown in fig. 8 (zinc-blend structure (001) cut, (110) “horizontal” plane, [ill] axis and (110) and (112) planes around it). In order to further improve the precision of the measurement the same procedure can be repeated for opposite @, rotations as shown in fig. 8. Fig. 8 refers to the case of an In,Ga, _, As layer grown by MBE on a GaAs substrate. The in mole fraction is x = 0.104 i 0.006, as determined by RBS and AES [8], and the epilayer thickness is (68 + 4) nm, i.e. very close to the critical thickness for this system so as to optimize the sample for DCD analysis. The results of the channeling analysis is a I/a,, = (I .0132 f 0.0004)

in perfect agreement with the DCD results of a ,~‘a,, = (1.0127 + 0.0004). It must be observed that the precision of the channeling estimates is comparable to that of DCD when both these techniques can be used [9]. The precision of the channeling technique allows the measure of strains of the order of lop3 as shown in fig. 9 where we report the channeling data around the [ - 111) axis relative to a single In,Ga, _,As layer on GaAs having x = 0.035. The fitting procedure gives a i/a,, = (1.0047 -+ 0.~3~ ~rresponding to a parallel strain F,, = ( - 0.0025 + 0.0002) which perfectly corresponds to the misfit of the structure computed on the basis of the measured In concentration.

7. Conclusions The intrinsic possibility of the ion-channeling technique to measure the angles between axes and planes of a crystalline structure is fully exploited by means of a detailed mathematical description of the typical experimental geometry. Such a mathemati~l description allows to directly relate the angular coordinates at which axes and planes are located in a channeling experiment to the directions of these symmetry elements in the crystal lattice. Fundamental parameters of the crystalline structure under investigation like the ratio between

A. Camera, A. V. Drigo / Channeling analysis of epitaxinl layers

the crystal lattice parameters and the angles between them can be therefore precisely determined. Moreover when in-depth resolving techniques like Rutherford Backscattering are used as probes in the channeling experiment the crystallographic structure of thin surface layers can be analyzed without any significant decrease in the precision of the measurements even when dealing with surface layers as thin as few nanometers. This ability of keeping the precision of the measurement on thin layers is the main advantage of this approach with respect to more conventional double crystal X-ray diffraction techniques whose precision gets poorer with decreasing layer thickness. The accuracy of the measurements can be further increased by determining the angles between several crystalline directions and by looking for the crystalline structure which gives the best fit to the experimental results. For instance, for measuring the tetragonal distortion of thin mismatched epitaxial layers, parallel strain values as small as 2.5 X lop3 were measured with a precision of 2 x lob4 by locating 12 planar channeling minima and three axial minima. The comparison between the experimental angular coordinates and the directions expected assuming a strained lattice gave an average discrepancy of less than 0.01’ with maximum differences of the order of 0.02”. The SLS and single strained layers were MBE grown by F. Genova and C. Rigo at CSELT (Torino) and the DCD measurements were performed by P. Franzosi and C. Ferrari at MASPEC-CNR (Parma). The electronic control of the goniometer was developed by R. Storti, G. Bovo and 0. Piovan. The technical assistance of G. Egeni, G. Gonella, I. Motti and V. Rude110 for the installation and alignment of the whole apparatus is kindly acknowledged. This work was partly supported by the CNR Finalized Project “Materials and Devices for Solid State Electronics”.

Appendix: Equations of the planes

and therefore 3.

365

if P lies in the plane it must be

P=o.

(A.3)

From eq. (A.l) and (5) the implicit plane therefore is - sin 5 cos + (sin y cos +sin

of the

e, + cos y sin ey cos ex)

5 sin $ cos y sin

+cos [(cos

equation

e,

y cos e, cos 8, - sin y sin e,) = 0.

(A.4)

Although the general explicit solution 6$ = I$( 0,) of eq. (A.4) is somewhat complicated, simple and useful solutions can be obtained in some particular cases. In particular it can be easily demonstrated that around the 0, = 0 direction the slope of the trace of a plane can be written in the simple form

de, i-ide, /I,=0

= tan $

(A.51

(5 = a/2)

thus allowing an immediate measure of the azimuthal angle $. Moreover the trace of a ‘horizontal’ plane (i.e. a plane having $ = 0) shows a characteristic curvature which can be obtained from the derivative

PBY-

sin y sin e,

de, - - sin y tan y + cos y

c0s2ex

(@=O),

(‘4.6) which shows that the curvature depends only upon y giving the possibility of an independent estimate of the y parameter. As a final remark it must be pointed out that the angle that a given direction fi = fi( 0,, By) forms with a “horizontal” plane, as defined by eq. (A.2) is sin $=

-sin

y sin(i3,+[)+cos

ycOs(ey+t)

cos e,

(G=O>. Even for the simple case where .$ = 71/2 the goniometer angle 0, does not correspond to the angle $ and at least the cos 0, factor must be taken into account. This is the reason why planar dips taken at different 0, values apparently do not give the same width G,,,.

Let us define the pole vector of a plane k as B= (sin .$ cos $,

sin 5 sin $,

cos [)

(A.1)

thus giving the definition of the angles [ and + as the declination and azimuthal angles of the pole vector in the FFR. Using these angles it is evident that a plane having $ = 0 is parallel to the Y axis (and it appears as ‘horizontal’ in the (0,) fly) projections), whereas a plane having 5 = 7r/2 is parallel to the Z axis. We define as usual a general direction as P = fi( 0,, 0,). The angle 4 between the vector P and the plane having @ as its pole is simply given by sin $=B+

P

(A.2)

References [l] L.C. Feldman, J.W. Mayer and S.T. Picraux, Material Analysis by Ion Channeling (Academic Press, New York, 1982). [2] J.F. Van der Veen, Surf. Sci. Rep. 5 (1985) 199. [3] ST. Picraux, in: New Uses of Ion Accelerators, ed. J.F. Ziegler (Plenum, New York, 1975) p. 229. [4] A. Camera, G.C. Celotti and A.V. Drigo, Proc. Int. Symp. Three day in-depth Review on the Nuclear Accelerator Impact in the Interdisciplinary Field, eds. P. Mazzoldi and G. Moschini (Laboratori Nazionali di Legnaro, 1984) p. 295.

366 [5] ST. Picraux,

A. Camera, A. V. Drigo / Channeling analysis of epitaxial layers

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