REMARKS ON THE METHODOLOGY OF GROWING ALKALI-HALIDE CRYSTALS FROM THE MELT

V. D. Kuchin

(Presented by Professor, Doctor A. A. Vorob’ev)

In the mass preparation of specimens from crystals grown from the melt by the Czochralski method [1] in the High-Voltage Engineering Laboratory of the Tomsk Polytechnic Institute, it was observed that, when cleaving NaCl, KCl, KBr, and KJ crystals in the \([100]\) direction, the cleavage surfaces very often are not plane-parallel, but wave-like. It may be supposed that in grown alkali-halide crystals, just as in metals and semiconductors [2], deformation of the crystals is caused by thermal stresses owing to their nonuniform or excessively rapid cooling. Often this fact is not taken into account. Therefore ionic crystals grown from the melt always have a considerably larger number of defects than the same crystals obtained from the vapor phase [3].

Let us consider the thermal state of a rock-salt crystal grown from the melt. The heat received by the crystal from the melt is partly removed by cooling water and partly dissipated through radiation into the surrounding medium. We shall find the radial and axial temperature variations in the crystal. The temperature distribution over the crystal was assumed to be calculated by the formula [4]:

\[ T = T_n e^{-(at)^3}, \tag{1} \]

where \(T_n\) is the melting temperature;
\(\quad t\) is time;
\(\quad a\) is a coefficient determined from experiment.

For alkali-halide crystals the coefficient \(a\) is unknown, and therefore it does not appear possible to carry out a calculation using formula (1).

The heat-conduction equation for a crystal having the form of a circular cylinder of radius \(r\), in cylindrical coordinates, is written in the form of a second-order differential equation:

\[ \frac{\partial^2 T}{\partial \rho^2} + \frac{1}{\rho}\frac{\partial T}{\partial \rho} + \frac{\partial^2 T}{\partial z^2} =0, \tag{2} \]

where \(\rho\) is the current radius vector, \(z\) is the ordinate. The boundary conditions in solving this differential equation are as follows:

1) at \(z=0\) the temperature of the crystal \(T\) is equal to the temperature of the melt \(T_n\):

\[ T = T_n \quad \text{for } z=0 \text{ and } 0 \le \rho \le r; \tag{3} \]

2) the quantity of heat supplied to the crystal through thermal conduction from the melt is equal to the quantity of heat dissipated by the surface of the crystal:

\[ \lambda \frac{\partial T}{\partial \rho} = \alpha T, \tag{4} \]

where \(\lambda\) is the coefficient of thermal conductivity, and \(\alpha\) is the heat-transfer coefficient. The solution of equation (2) will have the form:

\[ T=\frac{2}{\left(\frac{\alpha r}{\lambda}\right)^2+(a_n r)^2}\cdot \frac{\alpha r}{\lambda} \sum_{n=1}^{n=\infty} \frac{I_1(a_n r)}{I_0(a_n r)} \cdot \frac{T_n}{e^{a_n z}}, \tag{5} \]

which contains \(I_0\) and \(I_1\)—Bessel functions of the zeroth and first order, respectively, while the argument \((a_n r)\) corresponds to the boundary condition (4). Calculations carried out by this formula are extremely cumbersome. With sufficiently large approximations, in particular,

Fig. 1. Axial temperature distribution during growth of an NaCl crystal.

Fig. 2. Radial temperature distribution during growth of an NaCl crystal.

excluding the infinite series of Bessel functions, from equation (5) one can obtain an expression of the type:

\[ T \cong \left(1-\frac{\rho}{2r}\cdot\frac{\alpha}{\lambda}\right) \frac{T_n}{e^{z\sqrt{\frac{2}{r}\cdot\frac{\alpha}{\lambda}}}}, \tag{6} \]

which makes it possible to determine the temperature at any point of the crystal. Analysis of equation (6) shows that the temperature distribution in the direction of the \(z\) axis obeys an exponential law, and in the radial direction—a parabolic one.

Figures 1 and 2 present the results of calculations, respectively, of the axial and radial temperature distributions in an NaCl crystal, under the assumption that \(r=2\) cm, \(\alpha=1\) kcal/cm\(^2\)·sec·deg [5] and \(\lambda=240\cdot10^{-5}\) kJ/m·sec·deg [5]. As can be seen from Fig. 1, the maximum temperature difference in the crystal in the radial direction occurs at \(z=r/2=1\) cm and reaches \(\Delta T=124^\circ\)C. In this case the calculated value of the temperature at the center of the crystal is \(716^\circ\)K, and at the surface of the same layer it is \(592^\circ\)K.

The temperature gradient in the crystal is determined by formula (2):

\[ \frac{dT}{d\rho}=\frac{\sigma}{\lambda}T^4, \tag{7} \]

where \(\sigma\) is the Stefan–Boltzmann constant.

Calculation shows that at \(z=r/2=1\) cm the temperature gradient is 314 deg/cm at the surface of the melt and falls to 42.5 deg/cm at the apex of the crystal. The larger the temperature gradient, the greater the magnitude of the mechanical stresses arising in the crystal upon cooling. Since the temperature in different parts of the crystal is different, the mechanical stresses in different regions of the crystal will also be unequal. The most heavily loaded region will be the central part of the crystal. The difference in the linear dimensions of two neighboring regions of the same-

of the same layer of the crystal in the direction of the radius \(r\) is proportional to the temperature difference between these portions, i.e. \(\Delta l=\beta \Delta T\). These contractions are small, but nevertheless sufficient to cause plastic deformation of the crystal. The magnitude of the mechanical stresses in the crystal can be determined from the formula

\[ F=E\Delta l=E\beta \Delta T, \tag{8} \]

where \(E\) is the modulus of elasticity (Young’s modulus).

For NaCl \(E=4.2\cdot 10^5\ \mathrm{kg/cm^2}\) [6] and \(\beta=4\cdot 10^{-5}\ 1/\mathrm{degree}\) [7]. At \(\Delta T=124^\circ\mathrm{C}\), mechanical stresses are obtained in a NaCl crystal that considerably exceed its mechanical strength (according to the data of V. D. Kuznetsov [8], the tensile strength of NaCl crystals is \(0.5\)--\(0.6\ \mathrm{kg/mm^2}\)). If \(\Delta T=10^\circ\) is adopted, then the magnitude of the mechanical forces in NaCl crystals, calculated from formula (8), is \(1.68\ \mathrm{kg/mm^2}\), i.e. of the same order as in some metals [2]. Optical investigations [3] confirm the existence of mechanical stresses due to nonuniform cooling of the crystal during growth. Under the action of such considerable mechanical stresses, displacement (“slip”) of some layers of the crystal relative to others occurs in a plane parallel to the surface of the melt. Defects appear in the structure of the crystal, the density of which in the direction of the radius \(r\) is determined from the relation [2]

\[ N=\frac{\beta}{a_0}\cdot \frac{\Delta T}{\Delta \rho}, \tag{9} \]

where \(a_0\) is the lattice constant.

Recent optical and electrical measurements [3] have confirmed the assumption that microdisturbances of the structure exist in deformed crystals. Lattice defects change the mobility of electrons, contribute to the accumulation of space charge, and affect the crystallizability of the substance. Calculation for our case gives \(N=1.75\cdot 10^5\ 1/\mathrm{cm^2}\). With an increase in the radial temperature difference, the number of lattice disturbances increases. Therefore, when growing crystals from the melt, the cooling rate of the crystal during the growth process plays an essential role; it can be determined from the formula [2]:

\[ \frac{dT}{dt}\simeq \frac{2\lambda}{3cr}\cdot \frac{dT}{d\rho}, \tag{10} \]

where \(c\) is the heat capacity of the crystal; for NaCl \(c=0.24\ \mathrm{cal/g\cdot degree}\) [9].

The cooling of the crystal must be carried out so slowly that the temperature gradient is, as far as possible, minimal. In other words, the isothermal surfaces in the crystal must be parallel to the surface of the melt. Let us take \(dT/d\rho=10\ \mathrm{degree/cm}\) [10]. At \(dT/d\rho<10\ \mathrm{degree/cm}\) the time required to grow the crystal increases sharply and may reach several tens of hours. Moreover, a value of \(dT/d\rho<10\ \mathrm{degree/cm}\) is practically difficult to obtain. The cooling rate of a NaCl crystal, determined by formula (10) at \(dT/d\rho=10\ \mathrm{degree/cm}\), is \(dT/dt=2.4\ \mathrm{degree/min}\).

The crystal will grow only if the growth rate is higher than the cooling rate of the crystal. It is known that the rate \(v_k\) at which pure crystals grow from the melt is approximately determined by the formula

\[ v_k\simeq \frac{DS}{R_0T_n}\cdot \frac{\Delta T_z}{\Delta z}, \tag{11} \]

where \(D\) is the diffusion coefficient, equal for NaCl to \(0.14\ \mathrm{cm^2/sec}\);

\(S=38\ \mathrm{cal/mol\cdot degree}\) is the molar entropy of the melt;

\(\Delta z\) is the distance in the direction of pulling of the crystal over which crystallization occurs (taken as \(5\cdot 10^{-2}\ \mathrm{cm}\) [11]);

\(R_0\) is the universal gas constant;

\(\Delta T_z\) is the temperature drop at the crystallization boundary (over a distance \(\Delta z\)).

From Fig. 1, for \(\Delta z = 5 \cdot 10^{-2}\ \text{cm}\), at \(\Delta T/\Delta \rho = 10\ \text{deg}/\text{cm}\), we find \(\Delta T_z = 5\ \text{deg}\), and the crystallization rate \(v_k = 2.45\ \text{cm}/\text{sec}\), i.e., considerably greater than the cooling rate.

Knowing the cooling rate of the crystal, one can determine the increase in the crystal weight per unit time as follows:

\[ dM/dt = \pi r^2 v d, \tag{12} \]

where \(d\) is the density of the substance. For NaCl with \(dT/dt = 2.4\ \text{deg}/\text{min}\), we obtain \(dM/dt \simeq 28.8\ \text{g}/\text{h}\). Proceeding from the cooling rate of the crystal, we shall also find the time of crystal growth as:

\[ t = \frac{\Delta T_z}{dT/dt}, \tag{13} \]

where \(\Delta T_z = 690^\circ\text{C}\) for \(z = 5\ \text{cm}\) (Fig. 2). Calculation by formula (13) gives \(t = 4.8\ \text{h}\).

For the other alkali-halide crystals studied (KCl, KBr, and KJ), an analogous calculation can be carried out with some additional approximations. We were unable to determine the thermal-conductivity coefficients \(\lambda\) of the indicated crystals. For a NaCl crystal, the ratio of the coefficients \(a/\lambda\) is \(0.175\ \text{cm}^{-1}\). This ratio, roughly speaking, will be constant, since the coefficients \(a\) and \(\lambda\) themselves depend only slightly on temperature and are quantities of the same order. We shall take the ratio of these coefficients to be \(0.175\ \text{cm}^{-1}\) for the remaining alkali-halide crystals as well. We believe that the error introduced by this assumption is small and lies within the accuracy of the calculations being made.

If the indicated assumption is adopted, then the above calculation can be performed for KCl, KBr, and KJ crystals. The obtained quantities are summarized in the table.

The calculation performed shows that excessively rapid cooling of alkali-halide crystals grown from the melt can be the cause of the appearance in the crystals of significant mechanical stresses and the occurrence of structural deformations. The values found for the rate of crystal growth, as the experiments performed show, can be used in the practice of growing alkali-halide crystals from the melt. The crystal cooling rates obtained in our calculations may be taken as the greatest

rates that may be permissible in cooling and heating specimens of alkali-halide crystals in order to determine the temperature dependence of their electrical, mechanical, and other characteristics. The calculated crystal-growth times agree with the experimental values.

References

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5. A. F. Walter and L. D. Inge, ZhTF, 8, 295, 1938.
6. Ed. by Ya. G. Dorfman and S. E. Frish, Collection of Physical Constants, ONTI, 1937.
7. V. P. Zhuze, DAN SSSR, 99, 711, 1954.
8. V. D. Kuznetsov, Physics of the Solid State, Krasnoe Znamya Publ., Tomsk, 1941.
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