Физ.-Хим. основы курсач / Thermodyn_MBE / GaInPAs_

1. INTRODUCTION

Molecular-beam epitaxy ~MBE! is presently the most ef- ®cient epitaxial technology, which permits the fabrication of semiconductor heterostructures of ternary III-V compounds with one group-V element and is distinguished by the high accuracy of the composition pro®les. At the same time, in the MBE of ternary compounds with two group-V elements and especially of quaternary compounds, controlling the composition of the growing layer becomes a very complex task and requires a large number of growth experiments together with postgrowth testing. For this reason, the creation of an adequate theoretical model of the MBE growth of multicomponent compounds is still of great interest. In 1985, Heckingbottom showed that MBE can be regarded as a quasinonequilibrium process, and that it can be described within a thermodynamic model.1 This approach was developed further in Ref. 2, which describes a method for calculating the conditions that control the growth process and presents the results of a thermodynamic analysis for the growth of binary and ternary III-V compounds.

In this paper we describe a calculation method for the growth of ternary compounds with two group-V components and present the results of a thermodynamic analysis of the growth of ternary and quaternary compounds in the In±Ga± P±As system of materials.

2. THERMODYNAMIC MODEL OF THE GROWTH OF THE QUATERNARY COMPOUNDS GaxIn12xPyAs12y

The thermodynamic analysis of the MBE growth of quaternary solutions is based on the following basic approximations. Molecular-beam epitaxy is regarded as a quasiequilibrium process, in which equilibrium between the gas and solid phases is established on the surface of the growing layer.1 The substrate temperature is taken as the temperature of the system. The pressures corresponding to the ¯uxes of atoms and molecules from the substrate surface are taken as the equilibrium partial pressures. The thermodynamic properties of the quaternary systems are approximated by a regularsolution model.3

Assuming for de®niteness that the group-III elements reach the growth surface in atomic form and that the group-V elements reach it in the form of dimers, we consider the

where g refers to substances in the gas phase, and s refers to substances in the solid phase.

The solid solution GaxIn12xPy As12y is regarded as a homogeneous mixture of four binary compounds with corresponding activity coef®cients. The activity coef®cients characterize the degree of deviation of the properties of each binary component in a real solution from its properties in an ideal solution. Then, according to the law of mass action, the reaction indicated will conform to the system of equations

where the Ki is the equilibrium constant of the formation reaction of the respective binary compound, P is the equilibrium partial pressure of the respective element at the substrate surface, and ai is the activity of the respective binary compound in the quaternary solid solution.

The activities of the binary compounds in the quaternary solid solution can be expressed in terms of the interaction parameters of the binary compounds in ternary solid solutions3 and are given by the expressions

kT ln aInAs5kT ln@~12x !~12y !#1@y a11~1

2y !a2 #x21@xa31~12x !a4 #y 22@ac2~1 2x !~a22a1 !2~12y !~a42a3 !#x y , ~3.1!

kT ln aGaP5kT ln x y @y a11~12y !a2 #~12x !21@xa3

1~12x !a4 #~12y !22@ac1x~a22a1 !

kT ln aGaAs5kT ln@x~12y !#1@y a11~12y !a2 #~1 2x !21@xa31~12x !a4 #y 21@ac1x~a2

2a1 !2~12y !~a42a3 !#~12x !y , ~3.4!

where a1 is the interaction parameter between GaP and InP in a ternary GaAsP solution; and a2, a3, and a4 are the interaction parameters in GaAs±InAs, GaAs±GaP, and InAs±InP, respectively. Of the four equations ~3.1!±~3.4!, only three are linearly independent, since a restriction is imposed on the system of equations in the form of the relationship between the chemical potentials m of the binary components in the solid phase

which speci®es the parameter ac . Equation ~4! is actually the condition for equilibrium in the solid-phase exchange reaction

Then, using the system of equations ~2! and ~3!, we can express ac in terms of the equilibrium constants of the formation reactions of the binary compounds

In the case of the growth of the stoichiometric compound, the ¯uxes of the group-III and group-V atoms utilized for growth are equal. According to the mass conservation law, we can write the expression

F0Ga2FGa1F0In2FIn52~FAs0 22FAs21FP022FP2 !, ~7!

where the F0i are the incident ¯uxes of atoms or molecules onto the growth surface ~the external ¯uxes!, and the Fi are the re-evaporated ¯uxes that create the equilibrium partial pressures of the elements over the growth surface.

The ratio between the mole fractions x and y of the components of the quaternary solution is also determined by the ratio between the ¯uxes utilized for growth:

Using Eqs. ~8! and ~9!, we can write ~7! in terms of the In and P ¯uxes alone:

Expressing the ¯ux in terms of the pressure according to the formula

where Pi is the pressure corresponding to the ¯ux, and mi is the molecular weight, and using ~2!, we obtain

Now, assigning the incident Ga and P ¯uxes, i.e., F0Ga and F0P2 , as well as the composition of the quaternary compound, i.e., the values of x and y , we can ®nd the solution of this equation relative to PP2. Substituting the values found for

PP2 into system of equations ~2!, we ®nd all the pressures corresponding to the re-evaporated ¯uxes, i.e, PGa , PIn , and PAs2, and then utilizing ~11!, we ®nd the re-evaporated ¯uxes themselves. Finally, using Eqs. ~8! and ~9!, we ®nd the required incident In and As ¯uxes: F0In and F0As2 .

Thus, all the external ¯uxes of the group-III and group-V elements needed to obtain a quaternary solid solution GaxIn12xPy As12y of de®nite composition, as well as the reevaporated ¯uxes of the elements, have been established.

In the case of the growth of a ternary compound with two group-V elements, for example, GaPy As12y ~i.e., when x51), Eq. ~12! transforms into

3. CALCULATION RESULTS AND DISCUSSION

The equilibrium constants of the formation reactions of the binary compounds used in the calculations were taken from Ref. 4, and the interaction parameters of the binary compounds in the ternary solid solutions were borrowed from Ref. 5.

The results of the calculations in the case of the growth of GaPy As12y on a GaAs substrate are presented in Figs. 1 and 2. The calculations were performed under the assumption that an elastically stressed layer of GaPy As12y forms on the GaAs surface and that its crystal lattice constant in the

FIG. 1. Relationship between the fraction of the external phosphorus ¯ux

400, 2 Ð 500, 3 Ð 600. The dashed line is a plot of the calculation results from Ref. 2. The points are experimental data from Ref. 6.

growth plane is determined by the lattice constant of GaAs. The in¯uence of the variation of the Gibbs free energy of formation of the binary compounds in the case of elastically stressed growth on the equilibrium constants of the reactions was taken into account in these calculations according to Ref. 7.

FIG. 2. Dependence of the incorporation coef®cient of phosphorus on the composition of the compound GaPy As12y grown @FGa0 51014 (cm2•s)21, T5520 °C # for various values of the external arsenic ¯ux FAs0 2,(cm2•s)21: 1 Ð 1014, 2 Ð 331014, 3 Ð 1015.

Figure 1 shows the relationship between the fraction of the external P ¯ux in the total ¯ux of group-V elements and the mole fraction of P in the ternary solution grown for ®xed external ¯uxes of As2 and Ga and various growth temperatures. For comparison, the ®gure also presents experimental data taken from Ref. 6 and calculation results from Ref. 2, in which the in¯uence of the elastic stresses on the Gibbs free energy of formation of the binary compounds was disregarded. It is seen from Fig. 1 that allowance for the in¯uence of the elastic stresses leads to better agreement between the theoretical and experimental data.

Figure 2 presents the dependence of the incorporation coef®cient of phosphorus KP2, which is de®ned as the ratio of the part of the incident P ¯ux that is incorporated into the

growing layer to the total incident P ¯ux, on the composition of the layer grown at a ®xed ¯ux F0Ga51014(cm2•s)21 for various values of F0As2 under the condition of enrichment

with respect to the group-V elements

2~F0As21F0P2 !.F0Ga.

The incorporation coef®cient of phosphorous is given by the following expression at T,620 °C:

As FP02 approaches the value of y F0Ga/2 in the region of compositions with a small mole fraction of P, the incorporation coef®cient becomes equal to unity, i.e., nearly all of the incident P is incorporated. The decrease in the incorporation coef®cient as the mole fraction of P increases is associated, as is seen from Fig. 1, with an increase in the external P2 ¯ux, which is needed to obtain a compound of de®nite composition at a ®xed incident As2 ¯ux, while the Ga ¯ux remains unchanged. Similar arguments also account for the decrease in KP2 as FAs0 2 increases.

Plots of the ratios between ¯uxes of group-III and group-V elements as functions of the mole fractions of the components (x and y ) in the quaternary solid solution in the case of the growth of GaxIn12xPy As12y that is latticematched to GaAs at various growth temperatures in the range 500±600 °C under the conditions of enrichment with respect to the group-V elements are shown in Figs. 3 and 4.

The relationship between the fraction of the external phosphorus ¯ux F0P2 in the total external group-V ¯ux and the mole fraction of P in the compound GaxIn12xPy As12y grown is presented in Fig. 3. In the temperature range indicated this relationship is virtually independent of the growth temperature. The difference between curves 1 and 2 in the range of compositions with a low P content is attributed to variation of the incorporation coef®cient of P, as in the case of the ternary solutions.

Figure 4 presents the relationship between the fraction of the external indium ¯ux F0In in the total external group-III ¯ux F0In1F0Ga and the mole fraction of Ga in the compound grown. It is seen from the ®gure that at a growth temperature

FIG. 3. Relationship between the fraction of the external phosphorus ¯ux in the total external ¯ux of group-V elements and the mole fraction of phosphorus in the compound GaxIn12xPy As12y grown under the conditions of enrichment with respect to the group-V elements @F0Ga51014 (cm2•s)21# for substrate temperatures in the range 500±600 °C and various values of the external arsenic ¯ux FAs0 2,(cm2•s)21: 1 Ð 231014, 2 Ð 231015.

of 500 °C the ratio of the external indium ¯ux to the total external group-III ¯ux F0In/(F0In1F0Ga) is a linear function of the mole fraction of Ga. This result is a consequence of the fact that at the growth temperature cited the re-evaporated In and Ga ¯uxes are negligibly small compared with the external ¯uxes of the respective elements; i.e., the sticking coef- ®cients of Ga and In are close to unity. However, when the growth temperature is increased, the re-evaporated In ¯ux

FIG. 4. Relationship between the fraction of the external In ¯ux in the total external ¯ux of group-III elements and the mole fraction of Ga

@FGa0 51014 (cm2•s)21# in the compound GaxIn12xPy As12y grown for growth under the conditions of enrichment with respect to the group-V

elements and various values of the substrate temperature T, °C, and the external arsenic ¯ux F0As2,(cm2•s)21: 1 Ð 500 and 231014, 2 Ð 550 and 231014, 3 Ð 600 and 231014, 38 Ð 600 and 231015.

FIG. 5. Re-evaporated In ¯ux from the growth surface of GaxIn12xPy As12y as a function of the mole fraction of Ga @FGa0 51014 (cm2•s)21# in the compound for various values of the substrate temperature T,°C, and the external arsenic ¯ux F0As2,(cm2•s)21: 1 Ð 500 and 231014, 2 Ð 550 and 231014, 3 Ð 600 and 231014, 38 Ð 600 and 231015.

increases signi®cantly, as is re¯ected in Fig. 5, which shows the dependence of the re-evaporated In ¯ux on the composition of the growing layer for three different growth temperatures. At the same time, the re-evaporated Ga ¯ux remains negligibly small and does not exceed 231011(cm2•s)21 at

600 °C. As a result, the dependence of F0In/(F0In1F0Ga) becomes nonlinear at high growth temperatures in the range

500±600 °C. An increase in the total group-V ¯ux FP021F0As2 leads to a decrease in the re-evaporated In ¯ux ~Fig. 5, curve 38). As a result, a smaller external In ¯ux is required to grow a compound of identical composition at high substrate temperatures. The dependence of F0In/(F0In1F0Ga) at an increased value of FP021F0As2 is shown in Fig. 4. The calculations were performed for an external Ga ¯ux F0Ga51014 (cm2•s)21, which corresponds to a gallium arsenide growth rate

VGaAs51 Å/s. The dependence of the growth rate of a compound on its composition in the case of the growth of a

quaternary solid solution for the dependences in this ®gure is speci®ed by the expression VGaAs /(12x).

4. CONCLUSIONS

Thus, a method for obtaining a thermodynamic description of the MBE growth of quaternary solid solutions with two volatile components has been proposed in this paper; a thermodynamic analysis of the growth of the compounds GaPy As12y and GaxIn12xPy As12y has been performed. It has been shown that allowance for the in¯uence of the elastic stresses appearing in the case of the pseudomorphic growth of GaPy As12y on GaAs on the equilibrium constants of the formation reactions of the binary compounds leads to better agreement between the experimental and theoretical data.

We thank S. V. Ivanov and N. N. Ledtsov for some helpful discussions.

This work was performed with the support of a NATO ISEP Linkage Grant.

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Translated by P. Shelnitz