particular temperature is taken from experiments /4/,. kT. D(C1) = (0.071 to 2.4)exp k Boltzmann constant and T absolute temperature. Equation (4) is of the form.
Short Notes phys. stat. sol. ( b ) 160, K85 (1990) Subject classification: 6 6 . 3 0 ; 58.13 School of Physics, Madurai Kamaraj University )
Chlorine Diffusion in CdTe BY K. SADAIYANDI and K. RAMACHANDRAN Introduction
Chlorine is widely used as a donor impurity to produce high
resistivity material in melt-grown CdTe. However, the information with regard to its solubility and its diffusivity is very limited. The diffusion study in CdTe is taking a momentum in the recent past and there are reviews on diffusion ir. this system 111. The self-diffusion study in 11-VI compounds prominently takes place through a single mechanism as suggested by Shaw 111, which has also been verified
theoretically
12,
31.
Because of
the technological applications of
C1
diffused CdTe, Shaw and Watson 141 have studied experimentally the diffusivity of C1 in CdTe by the radiotracer method. This experimental study leads to the conclusion that C1 diffuses via the neutral defect pair (VTe
- Vc,).
It would be
interesting to study this aspect as much works are not available. It is known that C1 is generally taken to be a substitutional shallow donor on the Te sublattice, so that the defect diffusion mechanism must involve VTe at some stage. As a first step, since C1 migration in the Te sublattice is via VTe, the theoretical investigation for the vacancy mechanism for diffusion is attempted and reported here. Theory The jump frequency and isotope effect for the diffusion of C1 in CdTe are calculated following the theory of the scattering matrix formalism and the
.
reaction coordinate approach / 5, 6 / Diffusion of an atom in a crystal takes place through discrete jumps, which will be obstructed by a ring of neighbouring atoms. The actual construction of the ring which obstructs the jumping atom and the expression for the amplitude of vibration of the diffusing atom and the neighbours are reported in 12, 3 1 . The jump frequency r can be calculated in the quasi-harmonic approximation,
where
I)
Madurai 625 021, India.
7 physica ( b )
K86
physica status solidi ( b ) 160
Here
ro
attempt frequency,
reaction coordinate due to the
X ( 4 , A ) the (6, A) mode,
contribution to the fluctuation in the and Xc the critical value of fluctuation
necessary for the jump to be certain. The isotope effect, a measure of the diffusion is given b y
where
rl
and
r2
are the jump frequencies for the isotopic masses M1 and M2,
respectively. Results and discussion
The
diffusion
coefficient
for
C1 in
CdTe for
a
particular temperature is taken from experiments / 4 / , D(C1) = (0.071 to 2.4)exp
kT
(4)
k Boltzmann constant and T absolute temperature. Equation (4) is of the form
D = D~ exp[-
21.
Also H D
o
=
ro
2nn
d2
,
H Havan's ratio, d the jump distance, and n the number of identical jumps. From the values of D and Do the activation energy Q is deduced and is given in Table 1 along with the experimental values. Q and X can be related as C
Using (7), Xc is found. Knowing XC, the jump frequency is estimated from (1). This is repeated for the isotopes 35Cl, 36Cl, 37Cl and for temperatures of 300, 800, 1000, 1200, and 1400 K. The results are reported in Table 1. Here the diffusion is assumed to take place through jumps between vacancy and diffusing C1 atom. Charge compensation is not taken into account explicitly, whereas it has been indirectly incorporated through the force constant change. The isotope effect, which is a measure of the tracer diffusion rate, would be generally above 50% / 3 / for the mechanism assumed
Short Notes T a b l e
1
The attempt frequency ( ro), jump frequency for C1 diffusion in CdTe temperature
14 sotope 10
ro
1200 1400 300 800 1000 1200 1400 300 800
1000 1200 1400
isotope effect (Ak)
I r
(calc.)
(exper.)
0.15313
1.47
1.60
1.38
0.15175 0.15167 0.15162 0.15160
1.14 1.01 D.88 0.75
1.60 1.60 1.60 1.60
1.20 1.13 1.06 3.98
.O. 15311
1.47 1.14 1.01 0.88
1.60
0.15173 0.15165 0.15160
1.34 1.17 1.1
0.15158
0.75
1.60
0.95
0.15390 0.15172 0.15163 0.15159 0.15156
1.47 1.14 1.01 0.88 0.75
1.60 1.60 1.60 1.60 1.60
1.31 1.15 1.08 1.00 0.92
(K) 300 800 1000
( r ), and
1.60 1.60 1.60
2.698~10-l~ 3.366~10~ 1. 192x108 3.017~10~ 3.034~10~~ 2.698~10-~~ 3.366~10~ 8 1.192~10 3.017~10~ 3.034~10~~
1.03
2.698~10-~' 5 9.366~10
-
1.192~10~ 3.017~10~ 3.034~10~~
The activation energy (Q) is in eV and the critical amplitude Xc is in % of atomic radius of C1. The attempt frequency (To) and jump frequency
( r ) are
in units of rad s ' .
to work. The theoretical investigations on C1 diffusion in CdTe give an isotope effect very much less than 1%,which indicates that the single vacancy mechanism (which is assumed) is not the way, the diffusion can take place. The next possible mechanism is b y assuming a vacancy and interstitial (pair defect mechanism) which is more stable than two vacancies (VTe
-
Vcd). When
-
this is not possible, then the mechatlism involving (VTe Vcd) will be studied, a s Shaw and Watson suggested. But it is evident that the contribution from Frenkel defect pairs in the case of C1 diffusion in CdTe is appreciable. The work in this direction is in progress.
K88
physica status solidi (b) 160 References
/I/ D. SHAW, J. C r y s t a l Growth S , 778 (1988). /2/ V. SURIYANARAYANAN, Y. MADHAVAN, and K. RAMACHANDRAN ; phys. stat. sol. (b) 125, 513 (1984). /3/ Y. MADHAVAN, K. RAMACHANDRAN, and T.M. HARIDASAN, phys. stat. sol. (b) 154, 55 (1989). ?, /4/ D. SHAW and E. WATSON, J. Phys. C l
4945 (1984).
/ 5 / A.A. MARADUDIN, E.W. MOTROLL, G.H. WEISS, and I.P. IPATOVA, T h e o r y of Lattice Dynamics in the Hamonic Approximation, Academic Press, New Y o r k 1971. I S / B.N.N.
ACHAR, Phys. R e v . B
lo, 3848
(1979). (Received April 9, 1990)
