BB-33

DYNAMICAL X-RAY DIFFRACTION FROM NONUNIFORM CRYSTALLINE FILMS: APPLICATION TO X-RAY ROCKING CURVE ANALYSIS*

C. R. WIE**, T. A. TOMBRELLO, and T. VREELAND, JR. Divisions of Physics, Mathematics, and Astronomy, and

Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125

ABSTRACT

A dynamical model for the general case of Bragg x-ray diffraction from arbitrarily thick nonuniform crystalline films is presented. The model incor­porates depth-dependent strain and a spherically symmetric Gaussian distribution of randomly displaced atoms and can be applied to the rocking curve analysis of ion damaged single crystals and strained layer superlattices. The analysis of x-ray rocking curves using this model provides detailed strain and damage depth distributions for ion implanted or MeV ion bombarded crystals and layer thickness, and lattice strain distributions for epitaxial layers and superlattices. The computation time using the dynamical model is comparable to that using a kine­matical model. We also present detailed strain and damage depth distributions in MeV ion bombarded GaAs (100) crystals. The perpendicular strain at the sample surface, measured as a function of ion beam dose (D), nuclear stopping power (S ), and electronic stopping power (S ), is shown to vary according to (1- kS )DS an~ saturate at high doses. e e n

Submitted to the Journal of Applied Physics

*Supported in part by the National Science Foundation [DMR83-18274]. **Permanent address: Dept. of Electrical & Computer Eng., SUNY at Buffalo, Amherst, NY 14260.

ONE OF THE BROWN BAG PREPRINT SERIES IN BASIC ANV APPLIEV SCIENCE

September 1985

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I. Introduction

The double crystal x-ray diffraction (or x-ray rocking curve) technique is

being widely used in the study of ion implanted crystals [1.6]. superlattices [2].

MeV ion bombarded crystals [3), epitaxial layers [ 4]. and thermally diffused cry­

stals [5]. The x-ray rocking curve technique. which gives the sample retlecting

power as a function of the angle between the sample surface and the incident x­

ray beam. provides information on depth protlles of strain and damage for ion

implanted [ 1] or MeV ion bombarded [3] crystals. on layer thickness, composi­

tion and strain tor superLattices [2] and epitaxial Layers [ 4]. and on diffusion

profiles for thermally diffused crystals [5].

To extract information. the x-ray rocking curve is analyzed using a

kinematical [7] or a dynamical [6] diffraction theory. The kinematical theory,

which is an approximation to the dynamical theory that neglects the extinction

(i.e., multiple scatteri~). serves as an adequate model when the maximum

re!lecting power is less than about 6% [7]. When the retlecting power exceeds

6%, the dynamical theory should be used in the rocking curve analysis. The

dynamical theory. however. has the shortcoming of taking too much computa­

tion time in the analysis [6). In this paper, we present the dynamical theory in

the Layer approximation; this model requires computation time which is com­

parable to that by a kinematical model. This dynamical model is applied to the

analysis of rocking curves taken from MeV ion bombarded GaAs(lOO) crystals to

get a detailed strain and damage depth distribution.

D. Theory

The general theory of x-ray diffraction, which properly accounts for normal

absorption (photoeLectn c process and Compton scattering) and extmction

(coherent scattering or diffraction) of wave fields in a crystal medium. is called

the dynamical x-ray diffraction theory. It is a first order theory in that it takes

the deviation of the x-ray refractive index from unity up to first order, and the

variation of the complex amplitudes over one x-ray wavelength to first order [B) .

When the homogeneity of the crystal specimen varies in only one direction, i.e ..

in depth with no lateral variation, the change in the x-ray complex amplitudes in

- 3 -

depth is well described by the Takagi-Taupin equation which was originally

developed by S. Takagi [9] and independently by D. Taupin [ 10]. The Takagi­

Taupin equation is:

where

Do.H(r) = complex amplitudes of the incident and diffracted waves,

Po.H = wave vectors of the incident and diffracted waves ,

e2 ")....2 Fo.H 1/lo.H = -me 2 rr -v-'

Fo.H =structure factors for the incident and diffracted waves,

V = unit cell volume,

(1 )

The above equation expresses the spatial variation or the complex ampli­

tudes or the incident and d.it!racted waves along the incident and the diffracted

directions. respectively. The Takagi-Taupin equation has been used in the study

of diffraction from curved crystals by Kl.ar and Rustichelli [ 11]. They have

presented the theory in a form useful for the present study, writing the depth

dependent scattering amplitude in the following form.

i~ = (l+ik).x2- 2(y+ig)X + (1+ik). (2)

where

Dn(r) I -y I X = = scattering amplitude, b = "'Ho , ...tbDa(r) ,

?'o.H = direction cosines of the mcident and diffracted waves Wlth respect to

the inward surface normal.

-4-

rr I '!f'n I A= ~:v' l .>.o.>..n l z,'!fo.H ='¢-'oJI +i'¢'"0.n.z =depth,

(1 +b )'¢'"0

g = 2 1'¢-'n I v'O "//"n

absorption, k = "¢/'n .

( 1 +b )"/1'0 -b an . . y =

2 1 "/!' H I vo = devtat10n from the Bragg angle .

The damage reduces the structure factor to Fn.

where

FIJ = structure factor tor the undamaged crystal,

u = average atomic displacement [7].

The strain is taken into account through an.

where

(3)

(4)

upper sign= for the incident angle of @B-rp with respect to the sample sur­

face,

lower sign = for the incident angle of G)B +rp with respect to the sample sur­

face,

rp = angle between the surface and the retlecting lattice plane,

e 1 = strain perpendicular to the surface,

e2 = strain parallel to the surface [2].

Larson and Barhorst have used equation (2) to obtain the s train depth dis­

tribution in an implanted and laser annealed silicon crystal [6]. They decom­

posed the complex differential equation (2) i.nto two coupled real equations by

putting X = X1 + i)(2, and integrated numerically for each equation to obtain the

rocking curve. We take, however, a different approach which takes less compu­

tational time.

- 5 -

Equation (2) can be integrated analytically [ 12],

( ) _ sXa + i(B+CXa)tan [s(A-Ao)] X A - s - i(C+BX0}tan [s(A-Ao)] '

where

X(Ao) = Xo. B = -(1+ik),

c = y +ig 0 s = vf'l-82, and

I X(O) 12 = reflecting power at the sample surface.

(5}

In the analysis of rocking curves for ion damaged crystals, the surface

strained and/or damaged layer is divided into an arbitrary number of parallel

laminae (layer approximation), and an average strain and/ or damage is

assumed in each layer (see Fig. la). For a known scattering amplitude X at the

interface of the bottom strained layer "1" and the undamaged and unstrained

substrate crystal (i.e., the initial condition), one can calculate the scattering

amplitude at the sample surface (i.e., at the top of layer "n") by repetitively

applying the formula (5) from layer "1" to layer "n." In the Bragg case, the tnitial

condition on the scattering amplitude, which is given at the interface of the

strained layer and the substrate, is an infinite crystal solution because the

thickness of the substrate undamaged crystal is much larger than the x-ray

absorption length in most cases. The initial condition is most easily obtained

from equation (5) by the following consideration.

Imagine a layer of thickness A 0 on an infinitely thick crystal. The layer and

substrate crystals are assumed to be identical with no damage or strain. Then,

"Xo" in (5) is the scattering amplitude of the substrate which is the i.n..fulite cry­

stal solution, and ''X(O}" at the layer surface will again be the in.tlni.te crystal

solut10n because the layer crystal is identical with the substrate with no damage

or strain. Thus, by putting X(A) = Xo in (5) and solving for Xo. we get the infinite

crystal solution [14] , which is

(6a)

The lower sign in (6a) gives the correct infinite crystal solution. That ts, the ini­

tial condition is

- 6 -

(6b)

The rocking curve is calculated using equations (5) and (6b) for given strain

and damage in each layer, and the best fit of the calculated rocking curve to the

experimental curve gives the strain and damage distribution in the layer

approximation. Since the correct evaluation of "S" (which is a square root of

complex variables) in (5) is important for the rocking curve calculation, we pro­

vide the formula in the appendix. Equations (5) and (6b) can also be used for

virgin or ion damaged epitaxial layers or superlattices with suitable structure

factors tor each composition.

For virgin superlattices, the computation can be much simpler because the

strain and structure factor is identical for all periods. The scattering amplitude

X at the top of a layer which is the jth period from the substrate is given by

where

_ (P-iQ)JC;- 1 + (T-iR) X; - (T+iR)X;-1 + (P+iQ) '

P = (BtB2-CtC2)tan(s tAt)+s 1s2,

Q = C1s 2tan(s 1A1) + C~ 1tan(s2A2),

R = s 1B2tan(s2A2) + s2B1tan(s 1A1), and

T = (BtC2-B2G)tan(stAt)ta:n(s2A2),

(7)

Xo is the infinite crystal solution, and A1.A2 .B 1, etc. are for composition "1 " and

"2" (see Fig . l b) and are defined in (2) and (5) . The parameters, P, Q,R , and T

are the same for all periods, and thus (7) can be used for a very fast calculation

of rocking curves for superlattices.

Finally, we present in Fig . 2 an experimental rocking curve (das hed)

obtained from a virgin GaAs(lOO) crystal for FeKa1(400) reflection, and a c alc u­

lated rocking curve (solid) from the formula (6b) with a 10 arcsec half width

Gaussian convolution.

- 7 -

m. Strain/Damage Distribution in lleV Ion Bombarded GaAs(lOO) Single Cry­

stals

Formulae (5) and (6b) were used for the analysis of rocking curves taken

for GaAs single crystals bombarded with 15 MeV Cl ions to various doses at room

temperature. The experimental rocking curve was taken by measuring the

(400) reflection of FeKa1 radiation from ion bombarded (100) cut GaAs at each

angle which is varied in 0.0015 degree s teps.

Figure 3 shows the experimental (dashed) and calculated (solid) rocking

curves in the left column, and the depth distributions in the layer approximation

ot the perpendicular strain (solid) and average random atomic displacements

(dashed) on the right column. The figure shows that the strain and damage in

the surface layer saturate in a hlgh dose irradiation.

The strain at the sample surface, as obtained by the dynamical theory

analysis, was measured as a function of beam does (D), and nuclear (Sn) and

electronic (S.) stopping powers by bombarding GaAs( 100) crystals with various

MeV ions (see Fig. 4) . Figure 4 shows that the surface strain varies with (1-

kS. )DS" with k = 0.09 (appropriate units are given in the figure), and saturates

to -0.4% after a high dose bombardment.

This result illustrates that the strain produced by nuclear collisions is par­

tially annealed by the electronic collision processes. The saturation and stop­

ping power dependence of strain has been explained in a separate paper by an

ion-lattice single collision model in which the point defect production and

saturation is discussed for MeV ion bombarded GaAs single crystals [ 13].

- 8 -

.Appendix

Formula for Evaluating s = vC'La2

s = s 1 +is 2 for ys-q 1 ,

s = -s 1 +is2 for y >q 1 ,

where

and

s2= vf9-. q 1 = Y l -k2+g2,

rl = IY2-g2 +k 2 - ll

r 2 =2(yg-k) ,

-9-

References

[ 1] V. S. Speriosu, B. M. Paine, M-A. Nicolet and H. L. Glass. Appl. Phys . Lett. 40,

604 (1982).

[2] V. S. Speriosu and Thad Vreeland. Jr .. J. Appl. Phys. 56, 1591 (1984) .

[3] C. R. Wie. T. Vreeland, Jr. and T. A. Tornbrello. Nucl. Instr. Meth. B9, 25

( 1985) .

[ 4] A. Fukuhara andY. Takano, Acta Cryst. A33, 137 (1977) .

[5] A. Fukuhara andY. Takano, J. Appl. Cryst. 10, 387 (1977).

(6] B. C. Larson and J. F . Barhorst, J . Appl. Phys. 51, 3181 (1980) .

[7] V. S . Speriosu, J. AppL Phys. 52, 6094 (1981) .

[8) W. H. Zachariasen, "Theory ot X-Ray Diffraction in Crystals" (Wiley, New York

1945).

[9] S. Takagi, Acta Cryst. 15, 1311 (1962) .

[10] D. Taupin. Bull. Soc. Franc. Miner. Cryst. 87, 469 (1964) .

[11] B. Klar and F. Rustichelli, Nuovo Cimento 138, 249 (1973) .

[12] M. A. G. Halliwell, M. H. Lyons and M. J . Hill. J. Cryst. Growth 68, 523 (1984).

[ 13] C. R. Wie , T. A. Tombrello and T. Vreeland. Jr .. subm1tted to Phys. Rev. B

( 1985) .

[ 14] C. R. Wie, Ph.D. Thesis, California Institute of Technology ( 1985) .

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Ftgure Captions

1(a). The surface strained/ damaged region of an ion-irradiated crystal is divided

into parallel laminae, in each of which the average strain and damage is

assumed in the layer approximation.

1(b).Undamaged superlattice for which the formula(?) is used. The parameters

P, Q, R and Tare the same in all periods.

2. · Dynamical diffraction theory calculation of rocking curve. Experimental

(dashed) and calculated (solid) curves are shown for an Fe.Ka1 ( 400)

reflection from a virgin GaAs( 100) crystal. The formula (2.6b) was used for

the calculation with 10 arcsec halfwidth Gaussian convolution.

3. Dynamical x-ray diffraction theory analyses of the GaAs rocking curves.

Left: experimental (dashed) and calculated (solid) rocking curves. Right:

strain (solid) and damage (dashed) depth profile . Note that the layer at 5

J.Lm depth has become amorphous at 5 x 101!5 cm-2 .

4. The perpendicular strain at the surface of GaAs( 100) crystals obtained by

the dynamical theory analysis of the experimental rocking curves t aken for

( 400) symmetric reflections of Fe.Ka1 radiation from the GaAs( 100) cr ystals

bombarded with various MeV ions as shown in the figure .

"0 0 ..... c <1)

a... .....,.

c .Q

~

r

I (j')

0 a. 1-1 "' ·I E 0 u

J...

..) )

c: 0 <l . + 0

...... "'

0

l

.

. .

,__

.

.

.

-1 1-

(\J

"l r ~

I I 1 I I I I <1)

I -0

I ..... -(j') .0

I .0

I ::) (j')

I I I

.I . - 1 N -1 "' J .l I

- .__

"0 <1)

"0 ..... 0 .0

E 0 .0 c 0 c

-0

II "0

"'-<1)

"0 <1)~ "'- - <1)

8' 0 0

E 0 .0 "'-

f<i <l E - 0 . (/) "'-+ + 0 .0 0 0 0 .0 :::1 a.

I (/)

0

c <1) 0 (.) ·- ·---0 --

"0 <1)

-o .... 0 .0

. 0 0 E 0 .0 c 0 c __ ..__ I ..__.-

Fig. 1

-12-

~----------.-----------~----------,------------r-----------,

0 1.(")

--o--oo qO --

Q) . 0.(.)

>< a wu

0 (T')

0 N

C%) ~3~0d ~NilJ3l33~

Fig . 2

0

1.(")

1.(") 0

0

1.(")

0

1.(") -

< 1-­w :t: I-

< 1-­.....J UJ CJ

-1 3-

J5 . 6 . 6

15 MeV CIH JD dose=txt 013 ern~ . ~

~ .. zo

. 3

I !>

.2 . 2

10 I

/l I . I ~ . I

~

I ~_, .. :·~ -

0 0 ........ -.

0 -. 35 -. ) -- ~ -. 2 -. 15 -. I - . 05 0 . 05 0 10 20 )(l • O !>() 60 70

)5

dose=4x 1013 em-~ 6

......... .........

lO - ~ 5 U)

N ~ 0 - ......... ex:

25 E-< N .. .. U)

ex: c.:J - ~ ~ 2C

(J 0 . 3 , ) -Q..

15 z H

I ~ .2 I :··· ;I ~ . 2 E-< ,.J

t.:) 10 U) w :z: V' :~ c.:J H I ~ t; .I .: i . I ~ t&J 0 ,..J

' ,-- ;......, ""' t&J .. .. .. ex: 0 0 J

- . )5 -. 3 -. 25 -. 2 ·. I~ -. I - 05 0 . 05 0 10 20 30 •o 5C 60 70

35 . 6 . 6

dose=1 .5x1014 em-2 I I JD . 5 l

1' 5

~ .. [ ~: 1 .•

zo I I . 3

I 1 . 3

15 I

. 2 L . 2

10 I , .. , ... ... .

5 . I , ..

0 0 •. J -25 -. 2 •. 15 ·.I •. 05 0 . 05 0 10 20 )() • O !>() 60 ·c

DELTA THETA (DEGREES) DEPTH (1000 A)

Fig . 3

-14-

)'5 . 6 . II

dose=6x 1014 cm-z ]I)

1 . 5 1 . ~ I I

l5 j I

. . j .. I I

20

1 I . 3 i . 3

15

. 2 , ....

1 ' 10

.----------.. . I . 1

~ 0 0 -. )'5 -. 3 -. 15 -. 2 -. 1~ -.I - - ~ 0 - ~ 0 10 20 30 •O 50 eo 70

15 . II . II - -dose= I 2xto1='>cm-z ~ (/) - ]I) ::;:: . ~ - ~ 0

0::: E-o

l5 - (/) .. . . c..:> 0::: ~ ~ ~ 20 -0 -p., . 3 . 3

15 ... ...

c..:> ;::; . 2 ,--· • 2

z 10 ~

: ~ w E-o E-o c..:> u (/)

........... . I .. ----------' 'I . I

~ w 5 ..... '-'--'

.-. w \.. --~

0 0::: 0 0 0

- . 15 - . 3 -.15 -. 2 ·.I ~ ·. I -. D:l 0 - ~ 0 10 20 ]I) •o 50 eo 70

:1!1 . II . II

dose=5x 101:; cm-z ]I)

. 5 - ~

l5 ~----~ .. 20 I . 3 . 3

15 I :I

. 2 ,., . 2

10 I; ...................

. I . I ~ I

. .. . .) I

·--.. ----- ----- ___ .j ; 0 0 0 ·. le -. l -. 15 -. 2 ·. I~ · . I -. D:l 0 . 05 0 10 20 ]I) .a 50 10 70

DELTA THETA (DEGREES) DEPTH (1000 A)

Fig . 4

X

~ X

oou >>> Q)Q)Q)

~~~ r0 a:)c;;:tl()

<l • •

l()

0

-15-

(\J

u

0

---uuu >>> Q) Q) Q)

~~~ C\.1

l()l()r<)

X 0 0

Fig. 5

0 0

I I ----> so cxr -o ...._____ (J)

(f)

(j)

0 .

---->lo<! ~

c (J)

(\J I

E u

f'()

0 -0