A Chemical Transport Method for the Synthesis of Simple and Complex Inorganic Crystals—Survey of Applications and Modeling

Abstract

The chemical transport method is a process that occurs naturally; however, it is also very useful in the chemical laboratory environment for the synthesis of inorganic crystals. It was successfully used for the syntheses of simple and complex inorganic compounds, from binary (e.g., ZnS, CdSe) to quaternary (e.g., Cu₂ZnSnS₄) compounds. Many experimental parameters influence the quality of products of chemical transport reactions, and among them, one may distinguish the used precursors and applied temperature gradient. The careful selection of experimental conditions is crucial for the production of high-quality crystals. Mathematical descriptions of the chemical transport phenomena, however, may potentially help in the design of proper conditions.

1. Introduction

Chemical vapor transport (CVT) is a technique used to grow single crystals. It is also known as a chemical transport method or a chemical transport reaction. In a typical synthesis performed by this method, precursors and transport agents are placed in one location in an ampoule (the so-called source zone), where they are heated while the opposite side remains slightly cooler, creating a temperature gradient [1]. Heating facilitates the reversible reaction of the transport agent with the precursor, creating volatile gaseous compounds that decompose in a cold zone called the crystallization zone (or the so-called sink zone), releasing the transport agent. However, the temperature at the sink zone may be higher than that at the source zone (Figure 1). Numerous parameters influence this process, with the most important of them being the selection of a proper transport agent and precursor, as well as temperature and pressure. CVT is often utilized to create highly crystalline materials; however, its application is limited as it is relatively slow process due to the constraints of mass transport by gas diffusion.

The aim of this review is to present a survey of the influence of different experimental parameters of the chemical transport method on the product, and to present a collection of applications of this method for the synthesis of inorganic compounds. The review also collects information on mathematical descriptions of the chemical transport method, allowing for improved control and design of this process.

2. Examples of Simple and Complex Inorganic Compounds Obtained with the Assistance of the Chemical Transport Method with Emphasis on the Experimental Conditions

The CVT process is determined by the rate of formation of volatile gaseous compounds, which are also called transport species, and their further decomposition. The partial pressure of transport species can be estimated from the Gibbs free energy [3] or by using dedicated software, for example, TRAGMIN [4]. Therefore, the transport agent, precursor, and temperature should be chosen for each system to provide the desired partial pressure of the transport species. The temperature gradient should be chosen in such a way as to ensure a lower equilibrium constant and, therefore, lower partial pressure of the transport species in the growth area, which causes decomposition. A relatively fast process typically leads to higher yields; however, more defects will be present in its electronic structure. In addition, a high nucleation rate causes the formation of crystals that are of a smaller size [5]. Parameters should be optimized for each system to achieve the desired quality and quantity of the crystals.

2.1. Germanium

Thermodynamical modeling can be useful when analyzing the composition of phases, but it does not provide information about non-equilibrium processes. High-temperature gas balance measurements give information about mass transfer between condensed and gas phases, which presents data about evaporation and solid–gas reactions depending on temperature. However, it does not provide details about the composition of phases. Therefore, Heinemann and Schmidt [4] used a combination of both thermodynamic modeling and high-temperature gas balance measurements to investigate the equilibrium between gas and condensed phases during the synthesis of single germanium crystals from germanium, with iodine as the transport agent. The first step is iodine sublimation; then, depending on the temperature, a few different equilibriums are present in the system, but the main equilibrium responsible for the whole transport process is the reaction of gaseous GeI₄ with germanium, producing GeI₂ as a transport-effective species. To obtain chemically pure germanium crystals, a deposition temperature of above 460 °C should be used. Moreover, at higher deposition temperatures (about 600 °C), single-crystal growth is favored, as was concluded from morphological analysis.

2.2. Phosphorus

CVT was applied to obtain crystals of different allotropic forms of phosphorus, e.g., black phosphorus (BP) or violet phosphorus (VP). This method may also be used to obtain doped phosphorus crystals.

2.2.1. Violet Phosphorus

To investigate the properties of violet phosphorus (VP) and phosphorene, a single layer exfoliated from VP (VP structure is shown in Figure 2) for the efficient synthesis of pure and highly crystalline VP is necessary. Zhang et al. obtained VP with 80% yield using amorphous red phosphorus (aRP), Sn, and SnI₄ as the transport agents. SnI₂ is formed as a byproduct. Only minute amounts of black phosphorus (BP) and P-Sn-I composites were found [6].

During the CVT process, Sn and SnI₄ decompose in a hot area and react with aRP vapor, forming composites from which VP is formed. When the source zone temperature, reaction zone temperature, cool zone temperature, and temperature gradient are changed, it is possible to obtain BP, e.g., when the reaction zone temperature is 530 °C and temperature difference is 20 °C, VP nucleation is favored, and when reaction zone temperature is 490 °C and temperature difference is 60 °C, BP nucleation is favored.

2.2.2. Orthorhombic Black Phosphorus

Zhang et al. [9] developed a CVT synthesis of orthorhombic black phosphorus from red phosphorus as a precursor, with a mixture of tin and iodine as the transport agent. A two-step heating process with a temperature difference of 50 °C during the whole process was used. Firstly, the hottest side of the silica ampoule with the reagents inside was heated from room temperature to 460 °C for 7 h and then maintained for 5–10 h. The second step was heating to 630 °C in the high-temperature zone at a heating rate of 25 °C per hour, and then the parameters were maintained for 5–10 h. Finally, the ampoule was cooled to room temperature with a 50 °C–per-hour cooling rate. The obtained post-reaction mixture contained only black phosphorus with tin phosphide as the only by-product, which could be separated by reaction with mineral acid. The exact mechanism of the process is not known; however, during the first-step heating reaction, only tin phosphide (Sn₄P₃) and tin phosphide–iodide (Sn₂₄P_19.3I₈) compounds were collected, so both substances were responsible for phosphorus transport.

2.2.3. Doped Black Phosphorus

The properties of BP can be improved by doping. Fu et al. investigated the effect of doping with Sb and Bi. Therefore, they synthesized BP, Bi-BP, and Sb-BP from RP, Sb, and Bi as substrates, and used SnI₄ as the transport agent using the CVT technique [10].

Both pure and doped BP were obtained with a yield exceeding 90%. Doping improved the light-absorption intensity. Doping with Sb resulted in an altered work function (WF) from 4.54 eV for BP to 4.71 eV for Sb-BP. Furthermore, Sb-doping elongated storage time, and the degradation over time was much smaller.

Later, the same group [11] achieved tellurium-doped black phosphorus, synthesized from red phosphorus, tin, and tellurium (IV) iodide. The hot zone of the ampoule was heated to 870 K for 120 min and maintained for 90 min with a temperature difference of 200 °C; after that, ampoule was naturally cooled to ambient temperature. XRD and Raman’s investigations proved the high crystallinity of the obtained crystals. The atomic ratio of the Te doping was up to 0.5%. In addition, electrocatalytic investigations suggest a higher activity of Te-doped black phosphorus in comparison to undoped black phosphorus.

2.3. PbGa₂Se₄ Crystals for Photodetection Applications

The photosensitivity, high resistivity, and nonlinear properties of ternary chalcogenide PbGa₂Se₄ seem to make it a useful material in optoelectronics. When its crystals are used in photodetectors, the detectors are highly photosensitive [12].

Ji et al. investigated the use of the CVT method to perform single-crystal growth. They failed when using a stoichiometric ratio of elements. However, they succeeded when the CVT method was used to convert pre-synthesized (from elements) polycrystalline PbGa₂Se₄ (both pure and containing PbSe) to monocrystals. Transport efficiency was not optimal when the starting reagent contained PbSe. Iodine was used as the transport agent:

$$PbG{a}_{2}S{e}_4\left(s\right)+4I_2\left(g\right)\rightleftharpoons Pb{I}_2\left(g\right)+3Ga{I}_3\left(g\right)+2Se\left(g\right)$$

The obtained crystals were investigated. Their crystal structure exhibited Fddd symmetry. The atomic ratio of the elements was in good agreement with the stoichiometric compound. A bandgap of E_g = 2.56 eV, resistivity of 6.59 TΩ × cm, and peak detectivity of 3.3 × 10⁸ Jones were found.

2.4. NbSe₂—A Two-Dimensional Compound

Two-dimensional NbSe₂ has three common polytypes: 2H-NbSe₂, 4H-NbSe₂, and 3R-NbSe₂. 2H-NbSe₂ is reported to have superconducting properties. Crystals of 2D NbSe₂ can be grown with the aid of the CVT technique. Dimitrov et al. obtained crystals containing 2H-NbSe₂ and 4H-NbSe₂ phases [13].

Crystals were grown from pre-synthesized (from raw elements) polycrystalline NbSe₂ using Br₂ as the transport agent. The method setup is shown in Figure 3. In the cold zone, hexagonal P6₃/mmc 2H-NbSe₂ (Figure 4) and P-6m2 4H-NbSe₂ were grown and identified with XRD analysis. The material’s photoluminescence was observed when the material was exposed to air.

2.5. RuS₂—A Two-Dimensional Compound

Ruthenium disulfide is an example of a semiconducting transition metal dichalcogenide that is used in catalysis and optoelectronics. Sai et al. applied the CVT technique to obtain different crystals depending on the temperature [14].

A mixture of RuS₂, RuO₂, and Cl₂ was used as the transport agent. The scheme of the method setup is shown in Figure 5. It was found that the best quality of crystals was achieved when the sample was obtained at a lower temperature of 900 °C, in comparison with the 1050 °C for the first sample. Moreover, the sulfur concentration was closer to stoichiometric when a temperature of 900 °C was applied. As the synthesis temperature decreased and the sulfur concentration increased, the band gap (E_g) of the product increased from 1.25 eV at 1050 °C to 1.68 eV at 900 °C.

2.6. ZnSe Crystals Doped with Fe²⁺ and/or Cr²⁺ for Mid-IR Applications

Mid-IR laser finds its application in various fields, e.g., spectroscopy, communication. Doped ZnSe crystals may be useful in these types of applications due to their optical properties. Huang et al. synthesized and investigated the properties of Fe²⁺- and Cr²⁺-doped ZnSe crystals obtained with the aid of the CVT technique [15].

The synthesis of doped ZnSe was conducted in two steps—the synthesis of a polycrystalline material and subsequent single-crystal growth. The polycrystalline material was synthesized from ZnSe and FeSe or FeSe and CrSe. Then, the obtained material was used to grow a single crystal, with a modified CVT method using iodine as the transport agent. The polycrystalline material was ground and put in an ampoule alongside the transport agent and was then sealed under a vacuum. The tip of the ampoule was put in the hottest area and then raised a few centimeters, which caused the tip to become the cold zone. The rise maintained to preserve the crystal’s growth [15].

In comparison with pure ZnSe, the doped samples were not transparent in the range 2.5–4.5 μm and were barely transparent in the range 4.5–5.0 μm. A decrease in bandgap from 2.52 eV to 2.27 eV for the Fe²⁺-doped material and to 2.35 eV for the Cr²⁺-doped material was observed.

2.7. NbCo_1.1Te—A Ferromagnetic Non-Stoichiometric Intercalated Compound

Even though some properties of NbCoTe₂ are known, the effects of its intercalations were unknown. Feng et al. decided to investigate the effects of the Co intercalation [16].

To obtain a single crystal, a two-step method was used. First, polycrystalline NbCoTe₂ was synthesized from elements in the form of powder via a solid-state reaction. Then, the obtained polycrystalline powder was transformed into single crystals by applying the CVT method and using TeCl₄ as the transport agent. The material was found to exhibit room-temperature ferromagnetism and magnetic anisotropy.

2.8. CuInS₂—A Semiconducting Material

Giri et al. investigated CuInS₂ crystals, which were obtained with the CVT method. Initially, CuInS₂ was synthesized from a stoichiometric mixture of elements and subsequently ground. Then, I₂ was used as the transporting agent [17].

The obtained crystals were investigated. It was found that the chemical composition differed only slightly from the stoichiometric one—the amount of S was slightly higher. Its electrical and thermal properties were investigated. The estimated E_f value was 0.22 eV for the four-probe method and 0.23 eV for the two-probe method. The E_f values confirms that CuInS₂ is an n-type semiconductor. The found values of bulk resistance (ρ_bulk), bulk conductivity (σ_bulk), Seebeck coefficient (S), and thermal conductivity (k) are shown in

Table 1. Properties of CuInS₂ single crystals obtained with the chemical transport method [17].

Property	T = 303 K	T = 423 K
ρbulk [Ω·cm]	16.78	5.26
σbulk [ʊ·cm]	5.96 × 10−2	0.19
S [μV∙K]	−148	−372
k [W∙m−1∙K−1]	0.37	2.83

. The material seems to have promising properties for thermoelectric applications.

2.9. Centimeter-Sized UTe₂ Single Crystals

UTe₂ potentially exhibits spin-triplet superconductivity. Therefore, it grabs the attention of researchers. Its properties are closely related to its degree of crystallization. Yao et al. investigated crystal growth using the CVT method [18].

Yao et al. used different ratios of U/Te—1:1.5, 1:2, and 1:2.5. I₂, AgBr, or TeBr₄ were employed as transport agents. Different temperatures were applied. An increased U/Te ratio resulted in an increase in the crystals’ sizes. When I₂ was used, the crystals were the largest. When TeBr₄ was used, the crystals were slightly larger than when AgBr was used. The obtained crystals were not air-stable and were easily oxidized.

2.10. ZrSe₃—A Quasi-1D van der Waals Materials

Quasi-1D ZrSe₃ has unique optical and electrical properties, which makes it potentially useful in electronics. Xu et al. synthesized it via the CVT method and investigated its properties [19].

ZrSe₃ was synthesized from raw Zr and Se powders mixed in a stoichiometric ratio (1:3), and I₂ was used as the transport agent. The obtained silver flakes were characterized with XRD. The XRD patterns showed that tiny amounts of ZrO₂ were formed as a product of a side reaction with atmospheric oxygen present in the ampoule during synthesis. With the aid of quantitative energy-dispersive spectroscopy, a 1:2.9 ratio of Zr to Se was found. The crystal structure and XRD patterns are shown in Figure 6.

For the purpose of preparing samples for an investigation of the material’s thermal properties, crystals were ground, cold-pressed, and annealed. The obtained heat capacity (300 K) value was 0.311 J∙g⁻¹∙K⁻¹, the thermal conductivity constant (300 K) was 5.4 ± 1.3 W∙m⁻¹∙K⁻¹, and the average speed of sound was 2122 m∙s⁻¹. The obtained data are in good agreement with those previously reported.

2.11. Cubic BAs

Cubic boron arsenide (c-BAs) single crystals were synthesized by Xing et al. [20] with a chemical vapor transport method, using I₂ and extra As as the transport agents. The maximum size of crystals was achieved with an As/I ratio of 1:3. This ratio suggests that AsI₃ may contribute to the transport process. The gas pressure significantly influences the crystal size. Crystals synthesized under a pressure of 35 atmospheres reached a size of 1000 µm, while those grown at 25 atm were 800 µm and 500 µm under 10 atm. At such a high pressure, the main growth mechanism is convection rather than diffusion. Despite good crystallinity, the thermal conductivity value at room temperature was 132 W/m-K for both crystals synthesized under 25 atm and 35 atm, while the theoretical is 1000 W/m-K. The experimental results are significantly lower than those theoretically predicted due to the presence of electronic defects.

2.12. TiSe₂—A Two-Dimensional Compound

Wang et al. [21] synthesized 2D TiSe₂ from Ti and Se powders. Instead of the popular I₂, AgCl was chosen as the transport agent because it allows for slow mass transport decomposition to Ag and Cl₂ at suitable temperatures, while the use of iodine causes faster mass transport and leads to thicker flakes. During the synthesis, the temperatures were 800 °C and 780 °C in source and growth (i.e., sink) regions, respectively. Saphire substrate was placed in the growth region. Such conditions allowed for the growth of ultrathin TiSe₂ flakes with a thickness of 3–8 nm and a length of 4–10 µm. The results of temperature-dependent electrical resistance measurements suggest that these synthesized nanoflakes are better quality than those mechanically exfoliated.

2.13. ReSe₂—A Two-Dimensional Compound

Xing et al. [22] obtained ReSe₂ ultrathin flakes by starting with Re and Se powders and with different transport agents (KCl, AgCl, TeCl₄). Saphire or mica were used as the growth substrate. The quartz ampoule was heated in a single-heating-zone furnace at a controlled rate of 10 °C/min, reaching a maximum temperature of 1000 °C. The optimal temperature of the crystal growth zone is 600 °C. After maintaining the ampoule at 1000 °C for 30 min, it was allowed to naturally cool to room temperature. Figure 7a–f presents the variety of shapes of the products depending on the transport reagent. The use of KCl led to a triangular crystal shape, while AgCl and TeCl₄ promoted growth on one axis alone, causing the formation of star-like and stripped crystals, respectively. ReSe₂ grown on sapphire was slightly thinner than that grown on mica substrate.

2.14. Ta₂Pd₃Se₈—A One-Dimensional Compound in the Form of Nanowires

Zhang et al. [23] fabricated one-dimensional Ta₂Pd₃Se₈ nanowires. They were produced from Ta, Pd, and Se. Four transport agents were compared: I₂, PdCl₂, PdBr₂, and Se. A comparison of the different transport agents revealed that the yields achieved with PdCl₂ and PdBr₂ were 4.27 and 3.75 times higher, respectively, than those obtained with I₂, while Se resulted in the lowest yield. Additionally, a 700 °C synthesis temperature led to the highest yield, while the maximum crystallite size was noticed at 900 °C. Thermodynamical modeling revealed that the decrease in yield at temperatures of more than 700 °C can be explained by the decrease in TaCl₄ partial pressure as the temperature grew above 680 °C; thus, TaCl₄ is the transporting species that determines the rate of the whole process. Investigation of the effect of growth time showed that longer times lead to higher yields; however, the transport rate decreases with time. The electrical property analysis showed that Ta₂Pd₃Se₈ demonstrated a high current on/off ratio of about 10⁵ and mobility of 13.8 cm² V⁻¹ s⁻¹, suggesting potential use in electronics.

2.15. WS₂-MoS₂ Heterostructure

The synthesis of a few layers of different compounds on top of each other may be utilized as a method for heterostructure fabrication. Employing this approach, Zhao and Jin [24] achieved a WS₂-MoS₂ heterostructure, using water vapor as the transport agent. CaSO₄· 2 H₂O was used as a precursor of water vapor, enabling the precise control of the amount of water vapor released. A three-zone furnace was used. MoS₂ powder was positioned upstream of WS₂. The heterostructure growth process consisted of two main steps. During the first step, for WS₂ growth, CaSO₄ · 2 H₂O was heated to 110 °C, while WS₂ was kept at 1200 °C in Zone 2, and MoS₂ remained cool in Zone 1. Once the WS₂ growth phase was complete, the temperature of the CaSO₄·2H₂O (T_w) was reduced to approximately 85 °C. At this point, the MoS₂ precursor was introduced into Zone 2, while the WS₂ precursor was removed. The WS₂ growth phase lasted about 15 min, followed by an 8-min MoS₂ growth phase. Afterward, the furnace was opened to enable rapid cooling. Photoluminescence and Raman mapping confirmed three different types of heterostructure, depending on temperature and layer growth time: monolayer lateral; few layers, lateral; and lateral + vertical.

2.16. Magnetite (Fe₃O₄) Coated with Fe_3−xSn_xO₄

Su et al. [25] achieved Fe_3−xSn_xO₄-coated magnetite from SnO₂ and Fe₃O₄ under a CO-CO₂ atmosphere with 15 vol. % CO at 950 °C. The mechanism of the process is shown in Figure 8. Carbon monoxide causes the reduction of tin (IV) oxide to gaseous tin (II) oxide, and then SnO reacts with Fe₃O₄, producing Fe_3−xSn_xO₄. The main factor responsible for the whole process is the partial pressure of SnO. SEM-EDS analyses showed that a 5 µm layer of Fe-Sn spinel outer layer is formed on the magnetite.

3. Mathematical and Theoretical Description of the Chemical Vapor Transport Process

3.1. Exact Analytical Description of Mass Transport Coupled with Chemical Reactions

During the CVT process, three distinct phenomena should be taken into account: heat transport, mass transport, and chemical reactions. All of them are connected because region-selective heating resulting in a temperature gradient may cause the flow of liquid as well as influence (that is, promote or inhibit) chemical reactions. Overall, the mathematical description of chemical vapor transport is a tough task, especially when many reacting elements and compounds are involved.

General considerations regarding the coupling of mass transfer with reversible chemical reactions are relatively old. Olander theoretically investigated the effect of several types of equilibrium chemical reactions on liquid-phase mass transfer [26]. The following reversible reactions were considered:

$$A\rightleftarrows E$$

(1)

$$A\rightleftarrows 2E$$

(2)

$$A+B\rightleftarrows E$$

(3)

$$A+B\rightleftarrows E+F$$

(4)

Olander assumed that reacting chemical individuals were in equilibrium concentrations through the diffusion path [26]. Based on this, the following reaction-diffusion equations may be written (example for the equilibrium presented in Equation (3)):

$$D_A{\displaystyle \frac{{\partial }^{2}A}{\partial x^2}}=kAB-k^{\prime }E+{\displaystyle \frac{\partial A}{\partial t}}$$

(5)

$$D_E{\displaystyle \frac{{\partial }^{2}E}{\partial x^2}}=-kAB+k^{\prime }E+{\displaystyle \frac{\partial E}{\partial t}}$$

(6)

$$K={\displaystyle \frac{E}{A\cdot B}}={\displaystyle \frac{k}{k^{\prime }}}$$

(7)

where $D_A,D_E$ are diffusion coefficients of reagents A and E, respectively; $A,B,E$ are concentrations of the reagents; $K$ is the equilibrium constant; $k, k^{\prime }$ are the kinetic parameters (forward and backward, respectively).

The set of two partial differential equations and one algebraic equation presented above were solved analytically by applying two additional assumptions: film theory mass transfer and surface renewal mass transfer [26]. The procedure of analytical solutions is tedious and is thoroughly described in ref. [26]. The equations were solved for each considered equilibrium (Equations (1)–(4)). The outcome was equations of mass transfer coefficients as a function of the concentrations, equilibrium constant, and diffusion coefficients of reacting species [26]. Later, Huang and Kuo presented similar theoretical investigations but for irreversible first-order reactions between two phases and for reversible reactions of general form [27,28]:

$$A+nB\rightleftarrows E$$

(8)

The authors showed, for first-order irreversible chemical reactions, that if an accurate physical mass transfer coefficient is available, three distinct approximations (namely: the film-penetration theory, the film theory, and the surface renewal theory) give practically the same predictions of the effect of a chemical reaction on the mass transfer rate [28]. On the other hand, if the physical mass transfer coefficient is not known, the choice of theory or the underlying mechanism may be important [28]. In the case of reversible chemical reactions, the choice of mechanism or theory is important, except in cases when the diffusivities of the reactant and product are nearly the same [27].

These were the cases that could be treated analytically; however, they did not include the temperature effect and considered only relatively simple chemical reactions. Including more chemical species (i.e., more chemical reactions) incorporates more mathematical equations into the model; thus, numerical methods need to be applied in more complicated cases [29]. Models including relatively large amounts of chemical reactions may still be computationally demanding; however, a reasonable assumption can be made [29]. They may concern the reduction of dimensionality of the model (if relevant symmetry is present), as well as the time scales of physical and chemical phenomena [29]. For example, the time scale of many chemical reactions in reacting flows is significantly different from the time scales of physical phenomena like molecular transport (diffusion) [29]. Taking this into account, further simplifications of model equations may be reached, as demonstrated, for example, by Maas [29].

In an extreme case, when the chemical effects may be neglected, the problem can be reduced to the physical vapor transport (PVT), i.e., the process is limited by diffusion (mass transfer) [29,30]. Numerical modeling of diffusive PVT in cylindrical ampoules was presented by Greenwell and cooperators, who considered the formation of crystals of component A through transportation with an inert component B [30]. Researchers considered closed cylindrical ampoules with an aspect ratio (length/radius) in the range of 0.5 to 10 [30]. Additionally, two cases were taken into account: first, when the partial pressure of component B was comparable to that of component A; and second, when the same partial pressure was relatively very low (authors called them “impurity cases”) [30]. The main outcomes of this numerical study were the recirculation of component B even when the gravity interaction was not present; the occurrence of the concentration gradients in a direction normal to the main transport direction in the vapor space; and non-uniform concentration gradients at the interface that cause the non-uniform crystal growth rates [30]. Similar investigations were presented by the same team of researchers but regarding cylindrical vertical ampoules [31].

As may be seen from the examples described above, analytical and numerical approaches can give interesting insights into the CVT process; however, they demand tedious mathematical calculations and/or computationally demanding numerical treatment [32,33].

3.2. Qualitative, Simplified Description of Chemical Vapor Transport—The Schäfer Equation

Let us recall that the overall chemical transport process can be seen as following three steps:

-

at the source, phase change of reagents from solid to gaseous or liquid phase (sometimes connected with additional chemical reaction);

-

liquid or gas movement;

-

at the sink, deposition of the solid from the gaseous or liquid phase (sometimes connected with additional chemical reaction).

It turns out that in the vast majority of cases, the movement of liquid or gas is the rate-limiting step; thus, it determines the rate of the whole CVT reaction. Schäfer analyzed the gas diffusion process and proposed the following equation to describe the rate of transportation of substance A in a closed system [34,35]:

$$\dot{n}\left(A\right)={\displaystyle \frac{n\left(A\right)}{t^{,}}}={\displaystyle \frac{\frac{\frac{i}{j}\cdot \Delta p}{\sum p}\cdot \overline{T^{0.75}}\cdot q}{s}}\cdot 0.6\cdot {10}^{-4}$$

(9)

In Equation (9), $\dot{n}\left(A\right)$ is the transport rate (in unit mol × h⁻¹); $t^{,}$ is the transport duration (h); i and j are the stoichiometric coefficients; $\Delta p$ is the difference in partial pressure at the source and sink of chemical species participating in the transport process; $\sum p$ is the total pressure (in bars); $\overline{T}$ is the mean temperature along the diffusion path (K); $q$ is the cross-section of the diffusion (cm²); and $s$ is the diffusion path (cm).

The equation described above may be successfully used for the simplified, qualitative design of chemical vapor transport processes. However, in complex CVT reactions, such an approach (based on Equation (9)) may fail, and one may need to consider a numerical approach.

3.3. Thermodynamic Modeling of the Chemical Vapor Transport

The modeling described in Section 3.1 and Section 3.2 deals with the description of the CVT process in the domain of time and, eventually, space. Another approach which was proven to be useful uses chemical thermodynamics (see, for example, the cases of germanium and nanowires of Ta₂Pd₃Se₈ described in Section 2). Many reagents being involved in the process (starting materials, transporting agents, and catalysts) cause the occurrence of an even greater number of chemical equilibria. To perform such modeling, one has to assume the chemical composition of intermediate compounds, from which some may act as transporting agents, and postulate the chemical reversible reactions in which these intermediates, potentially together with the reagents and additional transporting agents, are involved. Moreover, it is necessary to obtain thermodynamic data (enthalpies, entropies) for each reaction. If experimental data are not available, one needs to calculate these values using theoretical methods that may be inaccurate, thus leading to false conclusions.

Nevertheless, such modeling was successfully applied in the chemical vapor transport synthesis of compounds such as ZnS, ZnSe, NbSe₂, MoCl₃, CrX₃ (X = Cl, Br, I), and MnBi₂Se₄ [36,37,38,39,40,41]. In each case, it was necessary to solve a set of algebraic equations resulting from taken assumptions. These equations may be solved analytically or numerically. For that purpose, one may also use a dedicated, open-source software called TRAGMIN [42].

4. Conclusions

Chemical vapor transport is a complex process. The course and result of the CVT process are influenced by the transport agent, precursor, and temperature, which determine the partial pressure of transport species, and by the temperature gradient, which affects the decomposition rate. Higher partial pressures of intermediates and a steeper gradient of transport species’ partial pressures accelerate the process, increasing yield, but may introduce more defects and reduce the crystal size due to higher nucleation rates. Thus, the experimental conditions of the CVT process must be carefully adjusted in each individual case. Nevertheless, by applying the CVT method, it is possible to achieve both bulk monocrystals and thin layers, as well as nanowires and heterostructures. Theoretical modeling and experimental investigations are widely used to investigate the mechanism of transport.