Equilibrium distribution of defects in cadmium telluride before exposure to external factors

Modern technologies make it possible to produce cadmium telluride of sufficiently high quality, nevertheless, due to thermal and mechanical stress; structural defects in small quantities are formed in the semiconductor during and after cultivation. Possible single intrinsic point defects existing in a CdTe binary compound include neutral Cd or ionized interstitial Cdi, cadmium vacancy VCd, interstitial tellurium Tei, tellurium vacancy VTe, Te in the Cd node. Among them, donors are Cdi, VCd, and acceptors are Tei and VTe [1]. In the presence of sufficient energy, structural defects form clusters, the cause is thermomechanical stresses.

The features of the formation of clusters of vacancy and interstitial types in cadmium telluride under electron irradiation were discussed in [2], and a model of the distribution of point defects (vacancies and interstitial atoms) and their clusters (pores and precipitates) under nonequilibrium conditions caused by electron irradiation was constructed [3; 4].

The purpose of this work is to create a model of the distribution of point defects before exposure to any ionizing radiation. In fact, the influence of terrestrial conditions on the material is determined, before its use in space environment where materials are exposed to radiation, the evolution of point defects and their clusters in CdTe is described. The obtained results will then allow us to proceed to the study of heterogeneous conditions – the conditions in which aerospace vehicles operate in space.

Modeling of defect formation in CdTe

To describe the dynamics of defects in a CdTe crystal, the following values are introduced: с 0 and с n – concentrations of crystal nodes before irradiation and during irradiation; concentrations of point defects с I – interstitial and с V – vacancies; cluster concentrations с L – dislocation loops and с B – pores; averaged cluster sizes r L – radii loops and r B – are the radii of the pores. The rate of generation of point defects (Frenkel pairs) under electron irradiation in unit volume c 0 G , where G = ст- j - the probability of the appearing of one Frenkel pair per second per node, ст - the electron scattering cross section at the crystal node, j = 10 19 см ^{- 2} с ^{- 1} - the electron flux density. The probabilities of reactions per second to one pair of defects are proportional to the following values: K ₀ – recombination of the Frenkel pair (in this case, the node is restored), K_I , K_V – the appearing of minimal clusters (double point defects) or di-interstitial and divacancy (in this case, two point defects disappear). Reaction rates per second per unit volume and diffusion coefficients are calculated using the formulas

( K о , К , K v ) = b ³ v- exp

( E 0 , E I , E V ) kT

( D I , D_V ) = b ² v- exp

^{( E} mI ^{, E} mV ⁾ kT

where ( E ₀, E_I , E_V ) are the activation energies of the corresponding reactions; E_mI , E_mV – the migration energies of point defects; v - the frequency of vibrations of atoms in the lattice nodes; b - the lattice constant CdTe or the value of the Burgers vector; k – Boltzmann constant; T – absolute temperature. Numerical values of CdTe crystal values are used for calculations (see the table).

Numerical values of a number of parameters of a cadmium telluride crystal

c 0	CT	b	v	E 0	E 1	E V	E mI	E mV
cm–3	2 cm 2	nm	c–1	eV	eV	eV	eV	eV

1,5×1022     3×10-22      0.648        1013         0.25        0.45         0.5         0.320.6

The evolution of point defects and their clusters under electron irradiation is satisfactorily described by the following system of equations [2; 3; 5; 6]:

= - Gcn + K0 cicv,(3)

dt dcr     52cr                 „^2

= DI “T + Gcn — K0cIcV — 2KIcI    T~ KIcIcL , dt       dz2b dcV     d CV              _2K rn B 4П rB

~ = DV _ 2 + Gcn  K0cIcV  2KVcV       KVcVcB, dt       dz2b

^CL = KiJ.(6)

d t

d;B = KicV,(7)

dt dL = KiCib,(8)

d t

-B = KvCvb.(9)

d t

Spatially homogeneous distributions of point defect concentrations and zero conditions for cluster concentrations and sizes are taken as initial conditions:

cI = cV = 1012 см-3, cL = cB = 0, rL = rB = 0.(10)

Such a choice of initial conditions is rough, but convenient and can be justified if the result is compared with experimental data corresponding to long times. Indeed, due to the high rate of generation of Frenkel pairs, the initial conditions have little effect on the dynamics at long irradiation times. At short irradiation times, the initial conditions noticeably affect the dynamics, so they must be stationary solutions of the initial system of equations. However, in order for physically acceptable stationary solutions to exist, the system of equations should be refined, for example, to take into account the change in the number of nodes. Calculations based on first principles are very complex, require large computer resources or use model simplifications [7]. Experimental data from different authors give a large variation depending on the method [8]. The simplest and most intuitive method is the principle of detailed equilibrium. If some data are reliably established, then the remaining unknown quantities can be calculated without much difficulty, which we will use in this work.

Spatially homogeneous model

First, consider a model of the "infinite crystal" type, the goal is to exclude boundary conditions, i.e. the initial conditions are spatially homogeneous. Instead of the value G describing the generation of Frenkel pairs under irradiation, we introduce an unknown effective value P describing the thermal generation of Frenkel pairs. If, in the system (3)–(9), the partial derivatives in time are turned to zero and diffusion is discarded, then a trivial solution will be obtained – all the desired variables are equal to zero as at absolute zero temperature. At the temperature T = 300 K and the expected thermal equilibrium, one can not consider large clusters and consider their radii to be zero, although the radius cannot be less than the lattice constant. In this case, the first three equations of the system (3)–(5) give the same equation for three unknowns: Pcn = K0CICV . Only recombination reactions are taken into account here, and double defects are discarded. The number of point defects is much less than the number of nodes and it can be assumed that cn = с0 . The magnitude is K0 known, but the magnitudes are cI, cV not experimentally ob- servable, and the physical processes described by the magnitude P are very diverse and have not yet been studied much [9; 10]. Thus, it is not enough to discard some terms in the system (3)–(9), it is necessary to refine, i.e. to take into account additional reactions that can provide thermal equilibrium and do not contradict physical conditions.

Consider the following model: we will take into account the balance between the processes of generating point and double defects and the processes of their recombination. It is convenient to enter dimensionless quantities for calculations. We divide all concentrations by the value с , then using substitutions: c_n = n • c₀ , c_} = u • c ₀, c_v = v • c ₀ and assuming n = Ithat it is true under normal conditions

( T = 300 K), we rewrite the first three equations of the system as

P = c0K0uv + c0KIu 2 • c0K0v + c0KVv2 • c0K0u,(11)

P = c0K0uv + 2c0KIu 2 - c0KIu2 • c0K0v + c0KVv2 • c0K0u,(12)

P = c0K0uv + 2c0KVv2 + c0KIu2 • c0K0v - c0KVv2 • c0K0u.(13)

Here, in the first equation (11), the processes of restoring nodes are taken into account: recombination of point defects, recombination of a vacancy with a dime node and recombination of an interstitial with a divacancy. In (12), the processes that give birth to the interstitial (with a minus sign) and destroy the interstitial (with a plus sign) are taken into account. Similarly, (13) takes into account the processes that create or destroy a vacancy. Subtracting (11) from (12) and (13), we obtain simple analytical expressions for the quantities u and v , and substituting them back into (11) gives a simple expression for the quantity R. Let's put these results together and write in the following form:

u = v = — * 1,0562165 X10-10 ^ с = cv *1,6 X1012 см-3,(14)

• c0K0

P = K0 + KI + KV * 1,05674436 X10-10 с-1.(15)

c 0 K 0 ²

According to the formula, P = v- exp ( - E_p / kT ) we can calculate the effective thermal activation energy of the Frenkel pair E_p = kT • ln ( v / P ) * 1,36767 эВ.

Conclusion

As a result of the study, the effective thermal activation energy of the Frenkel pair was calculated – 1.37 eV, which is the first stage of a comprehensive study of the conditions for the formation of structural defects and their clusters in cadmium telluride crystals under normal conditions with a temperature of 300 K. The described results coincide with the theoretical data in order of magnitude, but have small differences. This is inevitable, since our model is based on the representation of an initially "ideal" crystal. Nevertheless, the presented model confirms the data on the appearance and evolution of a defective network in a crystal without external influence and makes it possible to further use the data obtained for an inhomogeneous binary crystal model, with the receipt of fundamentally important numerical data on the thermal activation energy.