A method for the determination of atomic displacements in compound crystals by means of RBS-PIXE-channeling experiments

Nuclear Instruments North-Holland

and Methods

in Physics Research

Nuclear Instruments 8 Methods in Physics Research

B63 (1992) 451-461

secmnI3

A method for the determination of atomic displacements in compound crystals by means of RBS-PIXE-channeling experiments D. Comedi 1 and R. Kalish Solid State Institute and Physics Department, Technion, Israel Institute of Technology, 32000 Technion City, Haifa, Israel

John H. Barrett Solid State DiGsion, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6032, Received

3 September

1991 and in revised

form 15 October

USA

1991

A method which combines proton-induced X-ray emission (PIXE), Rutherford backscattering (RBS) and ion channeling, which enables the determination of static or dynamic root-mean-square displacements of constituent atoms in compound crystals is described. Full PIXE and RBS channeling angular scan curves are measured simultaneously by scanning through a major axis of the crystal under study. The experimental data are interpreted by using two different methods: A) a simple semi-analytical model which takes advantage of the depth information contained in RBS spectra to estimate depth correction factors needed to analyze the PIXE angular scan data, and B) a Monte Carlo simulation program of channeling, which takes into account the special structural features of the materials studied and allows for a direct comparison of simulated results with experimental PIXE data. The use of the technique is demonstrated by deducing rms displacements of constituent atoms of some II-VI (binary and ternary) semiconductors analyzed by both methods. The present results agree well with vibrational amplitude data obtained from X-ray diffraction measurements available from the literature.

1. Introduction The knowledge of root mean square (rms) displacements of constituent atoms in compound crystals can give important information about the local lattice structure. In addition, the study of the temperature dependence of the atomic vibrational amplitudes provides a way of learning about the potential well in which each particular atom vibrates, and through it, about properties of the local bonding. The conventional ways of extracting atomic rms displacements are X-ray and neutron diffraction analysis, and Mijssbauer spectroscopy. However, for most mixed crystals the availability of such data is rather limited due to difficulties encountered in separating the contributions of different atoms to the detected intensities in diffraction experiments and the limited number of existing suitable isotopes for MGssbauer experiments. The ion channeling technique offers an alternative method for the determination of rms atomic displace-

’ Present address: Centre for Electrophotonic Materials and Devices, McMaster University, Hamilton, Ontario, Canada L8S 4Ml. 0168-583X/92/$05.00

0 1992 - Elsevier

Science

Publishers

ments in crystalline solids. The dependence of channeling angular scan curves (the so-called channeling dips) and their features such as half width (I)~,~> and minimum yield (x,,,~,,) on the atomic rms vibrational amplitudes are well established for monoatomic crystals [1,2]. For compound crystals, on the other hand, theoretical calculations of channeling dips within the binary collision approximation [3] suggest that, for projectiles moving along a channel surrounded by strings containing different atomic species having different displacements, the collision probability of the projectiles with a given atom is mainly determined by the rms displacement of that atom. Thus, if the interaction yields of channeled particles with the different atomic species in a mixed crystal can be resolved and measured (i.e. by measuring the corresponding different channeling dips), information about the individual rms displacement magnitude of each constituent atom can be deduced. The most straightforward way of obtaining channeling data is by measuring the backscattering yield of the channeled particles (Rutherford backscattering (RBS)) as a function of the angle between one of the target major crystallographic axes and the incident beam. Unfortunately, the elemental resolution of RBS is limited and for many interesting polyatomic targets

B.V. All rights reserved

D. Comedi et al. /Atomic

452

displacements in compound crystals

the discrimination of particles backscattered by different constituent atoms is difficult when using ion beams and particle detectors commonly available in the laboratory. In such cases the use of a complementary close encounter scattering process in the channeling experiments, such as the particle induced X-ray emission (PIXE), becomes necessary. The characteristic X-rays emitted by different atoms can, in general, be easily resolved by ordinary Li-drifted Si detectors, even for X-rays originating from atoms which are not far apart in the periodic table. Unfortunately, the intrinsic poor depth resolution of PIXE presents an obstacle for its use as a scattering process for channeling measurements. The above limitation is probably the main reason for the fact that many PIXE-channeling studies of crystalline solids reported so far have been only semiquantitative in character. The present work reports on a method which combines Rutherford backscattering spectrometry (RBS), proton-induced X-ray emission (PIXE) and channeling while overcoming the depth resolution problem thus enabling the determination of rms displacements of constituent atoms in mixed crystals. Full PIXE and RBS channeling angular scan curves are --- measured simultaneously by scanning through the [ill] channels of the crystalline samples. The experimental data are compared with the results of a simple semi-analytical model, as well to those of a Monte Carlo simulation program of channeling, which includes the special structural features of the systems studied.

2. General considerations For the specific application of channeling in the present work, it is important to note that the velocities of channeled protons with energies exceeding = 0.1 MeV are such that the time of interaction with a particular atom on the atomic row is very short as compared to the period of the thermal oscillations of the atom. The particles thus interact with a momentarily “frozen” lattice, the atoms being displaced from their normal lattice sites with a spatial distribution P, which is generally assumed to be described by a Gaussian function with a standard (one dimensional) deviation u,, the thermal vibrational amplitude. The interaction probability x between the flux of channeled particles F and the target atoms can be expressed, in the language of the continuum model of channeling, as t41 x(+1=

/F(p,

@Y(P)

dp.

(I)

Here p is the position vector relative to the atomic row in the plane perpendicular to the channel and Q the angle of the incident beam with respect to the channel

direction. The flux of channeled particles F(p, Q) is determined mainly by the transverse continuum potential due to the atomic rows bordering the channels. When different atomic species are present in the rows, the fact that channeled trajectories have typical wavelengths of the order of a few hundred atomic spacings suggests that it is the average atomic charge which mainly determines the effective transverse potential and hence the flux F in such a case. The same argument applies when different interatomic spacings are present along the atomic rows [5], as is the case for the (111) strings in the zinc blende structure. However, it is the individual displacement distribution of each constituent atom P(p) which will determine the degree of interaction between the channeled projectile and the different atoms. Atomic species having different P(p) (and different rms displacements) will therefore yield different measured channeling angular scan curves. This is the basic idea of the present experimental method for the determination of individual rms atomic displacements in compound and mixed crystals. In many RBS studies of compound crystals a situation is encountered in which the available accelerator beams (ion mass and energy) do not provide sufficient mass resolution to permit a clear discrimination between particles backscattered from different constituent atoms in the crystal. In such cases, RBS continuum spectra show small steps, whose quantitative interpretation involve sophisticated background subtraction schemes and hence relatively low accuracies. In the present method the ion mass and energy are deliberately chosen so that the backscattering signals originating from different atoms at the same depth are practically indistinguishable, i.e., the differences between the different projectile kinematic factors for collisions with different atoms are comparable to, or below, the detector energy resolution. For the study of Zn,Cd, _,Te and Hg,_,Cd,Te this can, in general, be accomplished by the use of sub-MeV (i.e. 200 keV) protons. This particular ion beam-target combination enables a simple interpretation of the RBS spectra measured under channeling conditions; namely, the RBS channeling angular scans under these circumstances give a weighted RBS average of the individual channeling responses of all constituent atoms in the mixed crystal at various depths. The discrimination of different constituent atoms is provided by the detection of the particle induced characteristic X-rays (PIXE). The individual X-ray peaks emitted by the different atoms are usually easily resolved when using common Si(Li) detectors (see fig 1). The use of low energy (200 keV) protons as exciting particles diminishes the continuum background in the spectra which is commonly seen at higher beam energies. Protons moving along the crystal channels with energy _t 0.1 MeV typically approach the atomic rows

D. Comedi et al. /Atomic

displacements

to a minimum distance which is about 10-l A. Since the impact parameters required for L-shell ionization at the present proton energy are of the order of 10m2 A, PIXE spectra show a strong reduction in yield under channeling conditions and PIXE angular dips can be readily obtained for different constituent atoms. However, a known disadvantage of PIXE is its intrinsic limited depth resolution. As the projectiles penetrate into the solid they gradually lose their energy with depth, therefore reducing the probability for target atom inner shell excitation. Furthermore, after the X-ray photons were created, they are partially self absorbed in the crystal on their way out to the detector. As a result, measured PIXE channeling angular dips for different constituent atoms are depth averages of the channeling responses of each atom over broad depth bins. The weighting factors for these averages are determined by the product of the depth dependences of both the inner shell excitation probability [6] and the X-rays self absorption [7], which are different for the different emitting atomic constituents of the crystal. Fig. 2 shows the depth dependence of the above products for the excitation of Zn, Cd and Te L shells, as calculated for 200 keV protons incident in Zn,,,Cd,,,Te. Since the characteristics of the channeled beam is also depth dependent, the interpretation of PIXE channeling dips is not straightforward. In view of this, we have developed two methods of analysis which enable the deduction of the rms displacements of constituent atoms in polyatomic crystals. The first is based on a simple model which makes use of the depth information contained in RBS spectra to compute depth correction factors for the measured PIXE channeling angular scan parameters, and the second is a Monte Carlo simulation, which permits a direct interpretation of the PIXE channeling data. As will be shown below, an advantage of the Monte Carlo technique for the analysis of complex crystals is the possi-

400L ; .’

300 t

zn L,p .

. .--Random

Channel

Fig. 1. Characteristic X-ray spectra induced by 200 keV protons incident parallel to the [ill] channel and in a random direction in Zn,,,,Cd,,,,9Te.

453

in compound crystals I”“‘/J”,““‘,1,‘,““‘I”‘,““““I,“““’II

300, g

;

1

-.A A

c .E

q

A

200- . 'A

9

qA qA

.

$ FG, loo-

.

DA .AACd Te",AA o A 0 'A

Z”..

.

i?

0: 0

.

qo

l.

0.2

l*.

A~ 00 AAA 00

****a. 0.4

q,'AAA pm . 0.6

0.8

1.0

Depth (11m)

Fig. 2. Product of the L,, inner shell ionization cross section and the emitted X-ray photon absorption probability as function of depth, calculated for (A) Cd (0) Te, and (0) Zn in Zn,,,Cd,,,Te, for 200 keV protons.

bility of taking into account tures of the systems studied

3. Experimental

the special structural feain a rather natural way.

procedure

The channeling experiments were carried out in the ion implantation laboratory of the Solid State Institute at the Technion. Proton beams accelerated to energies of 200 keV on the Danfysik-HVEE accelerator were collimated to an angular definition better than 0.02”. The beam current was kept lower than 1 nA/mm2 in order to avoid sample heating. After a preset charge density had been accumulated ( _ 30 JLC) the beam was moved to a fresh spot in order to minimize damage production by the probing beam. Below this dose, no beam damage effects were noticeable on RBS and PIXE spectra. The samples were mounted on a 3-axis goniometer having an angular resolution of 0.01”. The target holder could be cooled by a liquid nitrogen dewar, and the temperature could be continuously varied (100 < T < 450 K) by heating the cooling finger. The samples studied were (111) oriented Zn,Cd, _,Te (x = 0, 0.077, 0.114, 0.2, 0.31) and Hg,_,Cd,Te (x = 0, 0.24, 0.4, 0.7) grown by the modified Bridgman method and the traveling heater method, respectively. The crystallographic polarity characteristic of the {ill) faces of zinc blende crystals, which turned out to be of importance in the present experiments, was determined by the etching procedures outlined by Fewster et al. [8,9]. Backscattered protons were detected with a Si surface barrier detector set at 165”, while characteristic X-rays were collected in a Si(Li) detector at an angle of 135”. Both RBS and PIXE spectra were recorded simultaneously --_ for each crystal tilt angle, scanning through the [ill] axis, typically at 0.1”/0.2 increments. The angular scan curves for each element

D. Comedi et al. /Atomic

454

dirplacements

in compound crystals

weighted average of the individual a, which are given by [2] a = 0.4683( Z;‘3 + Z;13) - “’

$ $

e A

F

0.0

2

0.6

Tilt Angle (Deg.)

--Fig. 3. PIXE angular scans through [ill] of Hg M+ (o), Cd L,, (A) and Te L,, (Cl) induced by 200 keV protons in at room temperature. The solid lines are Hg o,76Cd,,z,Te results from Monte Carlo computer simulations (see text).

in the crystal were extracted from the PIXE spectra by integrating over the L lines of Zn, Cd and Te, and the M lines of Hg. Raw data of such PIXE angular scans for Hg, Cd and Te in Hg,,,,Cd,,,,Te are shown in fig.

4. Methods of analysis 4.1. Method

and results

A

In the first method the half width and minimum yield of the channeling dips are measured at various depths by RBS and are extrapolated to the crystal surface. The $p,, and x,$, at zero depth so obtained are introduced into Barrett’s [l] equations

(2)

screening

distances

A.

(4)

Note that in eq. (2), Lindhard’s analytical continuum potential has been used [2] instead of the Moliere potential used previously [l], due to the greater simplicity of the former. The individual displacements of different atomic constituents U: are discriminated from the average u, by the use of PIXE in the following way. Channeling half widths $~~~“~’ and minimum yields xPrXE*’ for IIll” every element i are determined from the experimental PIXE angular scans. Since the characteristic X-rays emitted by different atoms originate at different depths which are of the order of 0.5 urn, corrections for these depths must be applied to the measured $~~$“~’ and x~~~~,’ values before introducing them to eqs. (2) and (3). In the present work, depth correction factors have been approximated as follows. The depth information contained in the RBS spectra together with the known inner shell ionization cross sections and the X-ray absorption coefficients are used to “simulate” the PIXE process for every excited atom. For a proton beam incident at a small angle IJ with respect to a channel direction, the numbers of protons which have undergone close collisions at a depth z can be expressed as N(z,

4) =cYnns(z>

IL)E,‘(zt

+I),

(5)

where Y,,, is the RBS yield measured at a depth z at the angle of incidence + and c is a constant. The depth dependence of the particle energy prior to scattering squared Ej appears as a normalization factor due the quadratic dependence of the Rutherford scattering cross section. It is calculated by using the proton stopping power for amorphous materials [lo] and Bragg’s rule [ 111, multiplied by a factor which expresses the reduction in stopping power due to channeling: p = Pmin for $ = 0, and p = 1 for a random angle of incidence. Pmin is a free parameter of the model to be determined later, while the variation of the function /?(I,+) between p = Pmin and p = 1 is assumed to scale with the measured PIXE channeling angular scan curves. The calculated normalized PIXE yield expected for a constituent atom i is given by

(3)

_/N( z, +)ui[ E,( z, $)I

e-W~z’cOs 8x dz

xi = to deduce a self-consistent RBS-compositional average of the atomic root mean square displacements u,Here Z, is the atomic number of the projectile while Z, and d are the weighted average of the atomic numbers of the crystal constituent atoms and the average interatomic spacing along the rows; E is the beam energy, N is the crystal atomic concentration, and a is the

(6) IN~(z)~;[EF(z)]

e-“~r/cosex dz

’

where CL, is the absorption coefficient of the X-ray photons emitted by the atom i in the crystal [7], ox is the X-ray detector angle, gi is the ionization cross section of the inner shell considered [6] and the index R stands for a random angle of incidence. It should be

D. Comedi et al. /Atomic

455

displacements in compound crystals

pointed out here that for eq. (6) to yield accurate results, the depth range included in the measured RBS spectra should be larger than (or at least equal to) the PIXE effective depths (see fig. 2). In the present work this has been accomplished by choosing the particular proton energy used (200 keV), for which the stopping power is about its maximum value, combined with the particle detector being set at a backscattering angle (0 = 165”). Eq. (6) gives RBS simulated channeling dips for every PIXE line. These may be compared with the measured actual PIXE angular scan curves. The parameter Pmin is adjusted by requiring

Table 1 Rms displacements of Cd, Te and A = Zn or Hg, as determined by methods A and B (see text) for various of the II-VI semiconductor crystals studied. Also shown are similar data available from the literature for CdTe and HgTe. The uncerin tainties in the present U, values (due to uncertainties experimental data only) are estimated to be = 3%

CdTe

method A method B ref. [24]

HgTe

method

A

ref. [25]

where x~~“~’ are the PIXE normalized minimum yields and x!,,~, are the corresponding minimum yields computed from eq. (6) by setting $ = 0. The average value for Pmin so obtained was pmi, = 0.85 i 0.05, in agreement with the value quoted in the literature [12] (0.825) for channeling in similar materials. Since eq. (6) is based on RBS spectra which monitor the average channeling response of the crystal as a whole, any difference between a given PIXE angular scan curve and the corresponding curve calculated from eq. 6 may be attributed to a difference between the behaviour of a particular atomic species and the average. The measured PIXE channeling dips half widths IJ~~~“~’ can now be corrected for the depth effects by multiplying them by the correction factors

(8) where I/J!,, is the half width of the curve computed by the use of eq. (6) for the element i and I,!J:,, is the half width measured at the crystal surface (by RBS). The corrected PIXE half width values for every element i can now be introduced into eq. (2) to obtain the individual root mean square displacements. This is the method used by us to obtain the vibrational amplitudes of various II-VI semiconductors previously reported [13,14]. Some of these “old” results are listed in table 1 for a comparison with results obtained by analysis using the Monte Carlo simulation method to be described below. Also listed in the table are some relevant results on thermal vibrational amplitudes deduced from diffraction methods by other authors.

4.2. Method B: Monte Carlo simulations The Monte Carlo simulation method makes use of a modified version of the channeling simulation program LAROSE, which was originally written for monoatomic cubic crystals [l]. Two major alterations of this pro-

u ,(Cd)

u ,(Te)

u ,(A)

Gl

[Al

iA1

0.150 0.155 0.154

0.136 0.135 0.135

-

-

0.148

0.165

-

0.141

0.170

Zn ,,.,, ,Cd,,,,,Te

method method

A B

0.103 0.107

0.075 0.077

0.096 0.095

Zn0.&do.6YTe

method method

A B

0.142 0.150

0.133 0.127

0.138 0.148

Hg “.&d”.24Te

method method

A B

0.154 0.155

0.144 0.145

0.168 0.175

gram were developed, one is to include calculations for the (111) channels of ideal zinc blende binary compound crystals, and the other to simulate the channeling process in ternary alloys of the above materials. In addition, features which are particularly important in determining PIXE channeling data such as the energy dependence of the projectile electronic energy loss and multiple scattering, and the inner-shell ionization mean radii of constituent atoms, have also been included. As will be shown, the simulated results reproduce the experimental RBS and PIXE channeling angular scan data accurately and enable the determination of the rms displacements of the atomic constituents of the crystal. As in the original computer code [l], the zinc blende version describes the ion-atom interaction by Moligre potentials. The particles trajectories and their nuclear energy loss are calculated within the binary collision model using the impulse approximation. The special structure of the atomic rows parallel to the (111) channels (i.e. atomic species and atomic spacing alternation, see fig. 4) and their distribution in the plane perpendicular to the channel characteristic of the zinc blende lattice are taken into account. Each atom is assumed to be isotropically displaced from its lattice site due to its vibrations, with a distribution described by a Gaussian function with standard deviation u,, which is the parameter to be determined for every element i. At each collision, a contribution to the nuclear encounter probability [l] (NEP) is calculated

D. Comedi et al. /Atomic displacements in compound crystals

4.56

Zinc Blende Lattice

.

. a

.

l

.

.

0

0

.

\

(110) Plane l

. a

l

.

.

.

l

0

.

.

0

\ \

l

h

l

.

.

. a

l

WI l

\

.

‘-Sublattice

l-Sublattice

l

1 (Hg, Cd or Zn) 2 (Te)

Fig. 4. Projection of the ideal zinc blende lattice on the (li0) --plane. The channeling directions [ill] and [ 1111 are indicated.

and stored by the program. For the calculation of PIXE channeling dips, the NEP is estimated from the expression

where m, and M, are the electron and projectile masses, A E the electronic energy loss, E the projectile energy and s is a parameter which ranges between l/4 and l/2 and describes the fact that not all collisions with electrons causing energy loss involve angular scattering [16]. In the calculations the stopping power for channeled particles has been assumed to be reduced with respect to that of particles in amorphous material by the factor Pmin determined previously by method A. Fig. 5 shows the influence of the electronic multiple scattering parameter s on the calculated NEP as function of depth (number of atomic layers traversed), for two angles of incidence, 0.0” and 0.8” with respect to the [ill] axis. It can be seen that the slope of the curve changes considerably with s for angles of incidence which are almost parallel to the channel direction (fig. 5a). However, for angles of incidence near to the critical angle (fig. 5b) variations in s have almost no effects on the NEP. This is because at higher angles the nuclear deflections become much more important than the electronic scattering. Good agreement between the experimental xmin as function of depth (as determined from RBS spectra, solid line in fig. 6) and the depth dependence of the NEP calculated for parallel angle of incidence was obtained by assuming s =

cos *

P; = 2?r[(Q2+

x,z,

(ri)2]NdNc

exp -(‘~+Y~)/2[(U{)2+(ri)z]7

(9)

where ri is the mean impact parameter for the inner shell ionization of element i [15]. xi and yj are the particle coordinates in the plane perpendicular to the channel and NC is the total number of calculated collisions. Eq. (9) is an approximation to be used for a convolution of the Gaussian distribution of atomic displacements with the ionization probability density, which is also assumed to be of Gaussian shape. The above approximations are expected to hold provided ui > ri, which turns to be the case for the present experimental conditions even for the lowest measurement temperature (100 K). A table of the electronic stopping power of the projectile in each target as function of projectile energy is calculated separately using the available polynomial fits to experimental data in amorphous monoelemental targets [lo] and Bragg’s rule [ll]. The simulation program uses this table to compute the mean energy of the channeled particles at each depth, and the electronic multiple scattering angles, which are assumed to have a Gaussian distribution with standard deviation _ (L!2Y2=

m, SZE

AE

) 1

(10)

t

0.4 -88

0.2 -

Fig. 5. Influence of the electronic multiple scattering parameter s on the calculated NEP as a function of the number of atomic layers traversed for (a) the [ill] channel direction, and (b) at an angle of 0.8” from [ill]. The simulations were done for 200 keV protons incident on CdTe for (A) s = 0 t@) s=0.3,and(~1)s=O.S.

D. Comedi et al. /Atomic

4.57

displacements in compound crystals

& 0.7 g f) 2 0.6 -

.baAfi*A 0.3t .A *A -$ 0.2 A 2) g 0.1 0.4

5c

0.00~

1200

Atomic Layers

in all cases studied,

200 keV protons

channeled

as can be seen in fig. 6 for

along the [iii]

0.7

w 4

0.6

=

0.5 0.4 0.3

Fig. 6. Comparison between the experimental xrnin (solid line) and the calculated NEP as function of depth. The simulations were --- done for the case of 200 keV protons incident along the [Ill] channel of Hg o,,6Cdo,,,Te for (0) s = 0.35, and (A) 5 = 0.30.

0.3-0.35

0.8

a g

axis in

Hg0.&d0.2.+TeThe influence of the ion incidence direction into the (111) channels (i.e. either [iii] or 11111 directions) of the zinc blende structure was also investigated. Since two alternating different atomic spacings are present along the (111) rows in this structure (see fig. 4) the [ 11l] and [ll l] directions are not equivalent. The depth dependence of the NEP for Cd and Te as calculated by the simulation program for 200 keV protons channeled along the [iii1 and [llll directions of --CdTe are displayed in fig. 7. It can be seen that for [ 11 I] incidence the close encounter probability for Cd is considerably higher than that of Te, as one would expect since Cd has been assumed to have a larger rms displacement fas will be shown below). However, when the channel is “reversed” (i.e. when the the protons move along the [ill] direction) the NEP for the Cd is no longer higher and is even slightly lower than that of --- Te. This can be understood by noting that for the [ill] direction the Cd atoms are always ahead of Te following the larger atomic spacing and vice versa for the [ill] direction. As a result, in the first case the Cd atoms have a higher probability of interacting with the protons, as they partially shadow the Te atoms behind (fig. 7a). The opposite is true for the [ill] incidence; however, since the Cd atoms have been assumed to have higher rms displacements the differences between the NEP of Cd and Te are reduced in this case (fig. 7b). Experimental evidence for the above effect is shown in fig. 8 where the measured PIXE angular scans for Cd and Te induced by 200 keV protons channeled in the [iii] and

0.2

-I

0.1 t

Oc,,,,,,,,,1 0

500

Atomic Layers

1000

1500

Fig. 7. Nuclear encounter probability as function of depth as calculated for 200 keV protons incident at an angle of 0.8 with respect to the (a) [iii], and (b) 11111directions of CdTe. (0) NEP for Cd (A ) NEP for Te.

[ill] directions of Zno,o,,Cdo~,,Te are displayed. It can be noted that, as anticipated by the simulations, the differences between the PIXE normalized yields

0.4 -

.Cd A Tt

p

Q2-

'S

%f

0.2O-5

I -3

I

I -I

Tilt

I 0 Angle

I I

I

I. 3

I 5

(Deg.)

Fig. 8. Measured Cd (0) and Te (a > PIXE channeling dips as obtained with --- 200 keV protons incident on Zn,.,,,Cd,,,Te at the (a) [ill], and (b) [llll directions. The scanning plane was almost parallel to the (2ii) plane in both cases.

458

D. Comedi et al. /Atomic

0.0

ZnTe

Fig. [ill] keV mal

0.1

0.2

0.3

0.4

0.5

0.6

x in Zn,.,Hg,Te

0.7

0.6

0.9

displacements in compound crystals

1.0 HgTe

9. Half widths of simulated channeling dips for Te near in Zn, _,Hg,Te of various compositions (dots), for 500 protons, assuming constant lattice parameter and theramplitudes. The solid line represents the behaviour expected by the use of eq. (2).

for Cd and Te are increased

considerably when meas--ured at different angles close to the [ill] direction (fig. 8a). Since the above effect occurs in an axial channeling configuration and does not strongly depend on the scanning plane angle [17], we believe that it is different than (but related to) the previously reported asymmetries [l&19] of channeling dips due to row shadowing near a (110) channel scanned within the (110) planes in the zinc blende structure. The program LAROSE was also generalized to simulate the channeling process in A,_,B,C ternary zinc blende alloys, where x is the molar concentration. For this purpose, a fraction x of atoms B are assumed to be randomly distributed on sites of the A sublattice. At each collision the atom with which the collision occurs is identified and the corresponding atomic data are used in the calculation of the scattering parameters. The way alloying affects the half widths of channeling dips was simulated for the case of 500 keV protons channeled near the [ill] axis in Zn,_,Hg,Te. The results are shown in fig. 9, where the calculated half widths for Te are plotted as function of the composition x. Also shown is the theoretical curve according to eq. (2), in which a weighted average of the atomic numbers of the constituent atoms for 2, was assumed. For simplicity, the lattice parameter and atomic rms displacements were assumed to remain constant with x in both calculations. It can be seen that the replacement of Zn (Z, = 30) by Hg (Z, = 80) leads to the increase of the effective repulsive Coulomb force on the channeled particles, leading to a broadening of the angular dips. Furthermore, within the present calculational accuracy, there is a reasonable agreement between the simulated data and the results obtained by the use of eq. (2). It should be pointed out, however, that it should be in principle possible to achieve a better agreement between both calculations

by using the Moliere potential instead of Lindhard’s potential in eq. (2), as it is done in the simulation program calculations. Static deviations from the zinc blende structure, which are known to exist [20,21] in most tetrahedrally coordinated ternary alloys A, _,B,C due to differences in bond lengths between the constituent binaries AC and BC, are taken into account in the simulation program by assuming a simple structural model. The model assumes a perfect (A,B) FCC sublattice but allows the C sublattice to be distorted in such a way as to reproduce the values of nearest neighbour distances as measured by EXAFS 1211over the entire composition range in the ternary materials under study. For the case of Zn,Cd, _xTe, the static displacements deduced from the model for the Te atoms are of the order of - 0.1 A, and the agreement with EXAFS data [21] is better than 0.3% [22]. Details on the structural model used in the simulations will be described elsewhere

La. In the simulations a number (typically 2000) of proton trajectories were followed through a total depth of about 2000 atomic layers and the NEP for all constituent elements of the crystal were calculated and stored as function of depth. The above procedure was then repeated for different angles of incidence, above and below the channel direction. The beam was assumed to have an initial angular spread of Gaussian shape with 0.26” standard deviation, which is the value expected to be caused by the nuclear scattering of the projectiles in a layer of tellurium oxide 10 A thick. Such a layer has been observed by optical Raman measurements to be present always on the surface of etched CdTe crystals [23]. The PIXE channeling angular dips for the element i were calculated using the expression Pb,,,=

/

P;(z, $)u;[E(z,

$)I e-cr~z/c0s8~ dz

(11)

where Pi is the generalized NEP for element i as function of depth provided by the simulation program [eq. (9>1. In the fitting procedure the atoms were initially assumed to have identical rms displacements, equal to the average value obtained from the RBS channeling angular scan data and eqs. (2) and (3). The NEPs were then calculated at different depths for parallel channeling incidence and were compared with the measured RBS normalized minimum yield as a function of depth. By doing so, the final value for the initial angular spread parameter could be determined; it was always found to be of the order of the expected value 0.26”. Simulations for angles below and above the channel direction were then performed for each element, thus producing full computer generated angular scans for different sets of rms thermal vibrations. The individual

D. Comedi et al. /Atomic

displacements

in compound crystals

459

between calculated and experimental values can be noted. Simultaneous agreement of all three PIXE calculated angular scan curves and surface channeling half width with the corresponding experimental values could be obtained after typically 5-7 iterations. The final rms displacements obtained this way for the different constituent atoms are listed in table 1 for some of the crystals studied. The physical meaning of these results has already been briefly described elsewere [13,14] and will be further discussed in a forthcomming publication [22]. z

_ 0.6 -

5. Discussion

0.4 0.2 (b) O.O-"""""""',"',"""l'l"'L -6 -6 -4 -2

0

2

4

6

6

Tilt Angle (Deg.)

Fig. 10. Fit of the simulated data (solid lines) mental PIXE channeling dips (dots) of (a) Cd, [iii] Zn a,&d,,,,Te at room temperature. The was 200 keV and the scanning plane angle respect to the (2ii) plane.

to the experiand (b) Te in proton energy was 11” with

rms displacements were determined by fitting each of the calculated PIXE channeling dips to the measured ones. Typical simulated results are shown in fig. 10 for Cd and Te in Zno.04Cdo,96Te and are compared with the experiment. The deviation of the calculated results from the experiment in the region of the shoulders of the channeling dips has been observed before [2] and is pressumably due to the higher sensitivity of the channeled particles incident at the corresponding angles to inaccuracies in the description of the ion-atom potential and the nuclear scattering process [2]. In order to obtain an additional check on the internal consistency of the method, the simulated channeling half widths extrapolated to the crystal surface were compared to the corresponding experimental values as determined from the RBS measurements. These are shown in table 2 for some of the materials studied, where, within uncertainties, an excellent agreement Table 2 Channeling half widths extrapolated to the crystal surface for some of the crystals studied, as determined experimentally from RBS spectra (I/J$:), and as calculated by the simulation program ($lois2)using the best fit parameters for each case, as described in the text

CdTe Zna.,&do.ss~Tc Zno.&do.,,Te Hgo.&d o.z4Te

fi$$ (deg.) ( f 0.02”)

@pi; (deg.) (+0.0153

1.05 1.48 1.18 1.09

1.06 1.44 1.14 1.09

As can be seen in table 1, the results obtained by the semi-analytical (A) and simulation (B) methods for the rms atomic displacements of Cd and Te in CdTe and by method A for Hg and Te in HgTe are found to be in good agreement with thermal vibrational amplitude values determined by diffraction techniques [24,25]. This result, together with temperature dependent channeling data obtained for the crystals studied [13,14,22], show that the displacements presently deduced are mainly vibrational in character, and that any possible static contribution (i.e. defects) is negligible [13,14,22]. Interestingly, the results obtained by method A agree with those determined by the more detailed Monte Carlo analysis (method B), despite the simplifying assumptions that were made in the first case. This has to do with the fact that for the present case most of the additional features included in the simulations and neglected in method A (such as the deviations from the ideal zinc blende structure or the electronic multiple scattering) affect mainly the minimum yields, while their influence on the calculated yields for angles of incidence close to the half angle is small. Furthermore, the relatively high atomic vibrational amplitudes existing at room temperature partially diminish the contributions to the calculated yields by the additional details included in the simulations. However it should be noted that for crystals having lower atomic rms displacements, or for crystals at lower temperatures, the simplifications implicit in method A may lead to misleading interpretations of the experimental channeling results. Further discrepancies between the methods are apparent for the case of Zn,,,,Cd,,,,Te where a significant difference between the rms displacements of Cd and Te is revealed when analyzing the data by the simulation program, but not by using method A (see table 1). The deviations from the zinc blende structure due to the significant difference between the CdTe and ZnTe bond lengths [21], which has been taken into account in the simulations but not in method A, is responsible for the disagreement between the methods

460

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displacements in compound crystals

in this particular case. For the case of Hg,.,,Cd,,,,Te, on the other hand, the constituent binaries CdTe and HgTe have almost equal lattice parameters and it is therefore justified to neglect the small deviations from the ideal zinc blende structure [26]. Indeed less disagreement between the two methods employed in the analysis is present in this case. The previously reported [13] strong reduction of the vibrational amplitudes in the Zn,~,,,Cd,,,,,Te crystal as compared to pure CdTe (see table 1) is confirmed by the more accurate simulation analysis of the experimental results. The inclusion of alloy disorder in the analysis produces almost no effect on the results, presumably due to the relatively low concentration of Zn in this case.

6. Conclusion The new experimental technique of RBS-PIXEchanneling measurements and the methods of its analysis have been described in this work. It has been shown that this technique offers a powerful tool for the determination of individual rms displacements in mixed crystals. The validity of the method is supported by the good agreement between calculated channeling angular scans and the experimental ones, by the agreement of the rms vibrational amplitudes presented in this work with similar data obtained by more conventional methods, and by the correlations found between microscopic vibrations and some mechanical properties of the studied semiconductor materials [22]. The simultaneous fitting of the measured PIXE channeling angular dips for all the elements and the crystal surface RBS half width by the simulated results provides a systematic way of determining a unique solution for the unknown rms displacements of the constituent atoms of the crystal under study. The influence of other adjustable parameters such as the initial angular spread of the beam or the electronic multiple scattering parameter have been shown to have significant influence on the calculated RBS and PIXE minimum yields, but yet little effect on the simulated yields for angles of incidence near the half width of the channeling dips. As a result their effect on the rms displacements, which are determined by fitting the entire angular scan curves, turns out to be negligible. The accuracy of the technique, however, relies on the knowledge of the ion-atom potential, bonding effects on the potential, the energy and impact parameter dependence of the inner shell ionization cross sections, the X-ray attenuation coefficients, and the exact crystal structure. Which one of these factors is dominant is a question that requires further detailed studies. However, the method already at its present stage

provides a reliable tool for a systematic study of the rms displacements of constituent atoms in a series of mixed crystals having similar characteristics. In contrast to diffraction techniques (in which all the elements in the crystal contribute collectively to the detected diffraction pattern), in the RBS-PIXE-channeling technique different characteristic X-rays are emitted by the different atoms individually and they are clearly discriminated in the measurements. Therefore, no initial assumptions have to be made about the dynamic behaviour of different atoms sharing a given sublattice, as it is usually done in X-ray diffraction analysis experiments [27]. Furthermore, in contrast to EXAFS measurements for the determination of vibrational amplitudes, where only relative one-dimensional rms displacements in the direction of atomic pairs can be determined [28], the ion channeling technique offers the possibility of obtaining the projections of the rms displacements in the planes perpendicular to, at least, the three major crystallographic directions. The above could be used to study anharmonicity and/or strain induced anisotropies of lattice vibrations which may be present in complex mixed crystals. Optimization of the sensitivity of channeling dips to small changes in rms displacements can be achieved by reducing the vibrational amplitudes by cooling the crystal such that u, 2 a [see eq. (2)]. By doing so, both the absolute and relative changes in half widths induced by small differences in ui are significantly increased. Furthermore, the determination of rms displacements at different temperatures provide a way of discriminating between vibrational and possible static contributions [13,14].

Acknowledgements The authors would like to thank Dr. V. Richter and Mr. Barak Philosoff for technical assistance, and Prof. Alex Zunger for stimulating and fruitful discussions. The financial support of the US-Israel Binational Science Foundation (BSF contract No. 88-00295) is gratefully acknowledged. The work in Oak Ridge was supported by the Division of Material Sciences, US Department of Energy, under contract No. DE-ACS84OR21400 with Martin Marietta Energy Systems, Inc.

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