Kinetic analysis of 2D Mo2C crystal growth via CVD

We investigated the growth mechanism of 2D Mo2C crystals by chemical vapor deposition (CVD) under various time and temperature conditions. The growth kinetics and mechanism of Mo2C on Cu via chemical vapor deposition (CVD) were investigated using a modified Johnson–Mehl–Avrami–Kolmogorov (JMAK) model. To analyze the surface coverage, we employed scanning electron microscopy (SEM) and applied the modified JMAK model to determine the growth rate and activation energy. The growth rate of Mo2C exhibited temperature-dependent behavior, described by the Arrhenius relationship, with an apparent activation energy of 4 eV. The Avrami plot exhibited an exponent of 3 indicating a complex process with nucleation and growth.


Introduction
Two-dimensional (2D) materials have emerged as a captivating area of research due to their unique electronic, optical, and mechanical properties, offering tremendous potential for a wide range of applications in nanoelectronics, catalysis, and energy storage [1].Among the various 2D materials, transition metal carbides have garnered significant attention due to their intriguing properties and versatile nature.In particular, molybdenum carbide (Mo 2 C) has demonstrated promising characteristics, including superconductivity, exceptional thermal stability, and high catalytic performance [2][3][4][5][6].
Realizing the full potential of Mo 2 C in various applications requires the controlled synthesis of Mo 2 C with tailored properties.Chemical vapor deposition (CVD) has proven to be a versatile technique for precise growth of 2D materials [7].For this CVD method, a Cu foil on top of the Mo substrate (Mo-Cu stack) is heated and used as substrate for Mo 2 C. At temperatures exceeding 1356 K, Cu melts, and Mo diffuses into Cu, forming a diluted Mo-Cu alloy.Concurrent to Mo-Cu alloy formation, CH 4 dissociates, depositing carbon atoms on the Mo-Cu alloy surface, where they combine with Mo atoms to form Mo 2 C crystals [8].While several research groups have investigated the effects of processing parameters such as the type [9][10][11][12][13] and thickness of metals (mostly Cu) [11,12,[14][15][16], H 2 :CH 4 ratio [9,14,15,[17][18][19][20][21], and duration on crystal size and morphology [4,11,12,22], the growth dynamics of Mo 2 C using the CVD method have not been sufficiently studied, resulting in inadequate understanding in growth mechanism.In recent developments, a physico-mathematical approach has been employed to construct an analytical forward model, encompassing factors such as bulk diffusivities, surface diffusivities, and solubility gradients, in the context of Mo 2 C crystal growth.A set of coupled nonlinear flow equations has been formulated to describe the intricate dynamics within the Mo-, Cu-, and Mo 2 C layers.This innovative framework successfully anticipates the growth rate of Mo 2 C crystals, providing accurate predictions for both vertical and lateral growth directions [8].
Building upon the aforementioned research, this manuscript offers an in-depth exploration of a kinetic analysis concerning the growth of 2D Mo 2 C crystals through chemical vapor deposition (CVD).We explore the influence of growth temperatures and times on the growth kinetics, employing an analytical model for growth that has been validated.This combined analytical-experimental toolkit is expected to provide new fundamental insights into the CVD growth of Mo 2 C and enabling the controlled production of high-quality 2D Mo 2 C crystals.These advancements have the potential to disrupt accepted perspectives by expanding the boundaries of transition metal carbide engineering, opening new applications in materials, electronics, optics, and possibly even as superconducting materials for quantum information sciences.

CVD of Mo 2 C on Cu
Copper foils (Alfa Aesar, 13 380), measuring 0.127 mm in thickness, boasting a purity of 99.9%, and featuring a diameter of 10 mm, were positioned atop molybdenum foils (Nanografi, NG06BPM0190P1), which were 0.1 mm thick, possessed a purity of 99.95%, and had a diameter of 10 mm.The assembly was then subjected to the desired experimental temperatures (1363 K, 1373 K, 1383 K, 1403 K), maintaining various growth durations as part of the systematic investigation, while being exposed to gas flows of 50 sccm N 2 , 50 sccm H 2 , and 5 sccm CH 4 .Following the completion of the growth process, the supply of CH 4 was terminated, and upon the furnace temperature descending to 1358 K, the samples were rapidly cooled by moving them out of the heated zone, all while being exposed to N 2 and H 2 gases.Detailed information regarding the precleaning process and other specifics can be found elsewhere [8].

Structural characterization of Mo 2 C crystals
To investigate the structural properties of Mo 2 C crystals grown on Cu substrates, a comprehensive characterization approach was employed.Thin sections of Mo 2 C grown on Cu samples were meticulously prepared using Focused Ion Beam (FIB) processing for subsequent Transmission Electron Microscopy (TEM) analysis.The morphological features of the synthesized Mo 2 C crystals were thoroughly examined using both Scanning Electron Microscopy (SEM) (Quanta 200 FEG, FEI) and TEM (FEI Tecnai G2 F20).Graphene and Mo 2 C crystals were elucidated through Raman spectroscopy measurements, conducted using a Witec Alpha300S instrument with an excitation wavelength of 532 nm.Furthermore, the phase analysis of the Mo 2 C crystals was undertaken utilizing x-ray diffraction (XRD) on a D8 Advance Bruker instrument equipped with CuKα radiation.

Measurement of Mo 2 C area coverage
After 2D Mo 2 C crystal growth, the samples were examined using SEM (Phenom XL).The acquired SEM images were processed and the fractional surface (area) coverage of Mo 2 C was obtained by using 'ImageJ' software.(c)).Our analysis using XRD (ICDD: 04-003-0962) and Raman spectroscopy (figure 1(d)) confirmed the orthorhombic structure of the growing Mo 2 C crystals which agrees with published studies [14].XRD results exhibited negligible differences despite varying growth temperatures.The results of Raman spectroscopy not only corroborated with the XRD conclusions pertaining to α-Mo 2 C formation but also unveiled the concurrent formation of graphene during Mo 2 C crystal growth.This finding is reinforced by the 2D signature peak of graphene in figure 1(d).These findings agree with previous studies [14].

Results and discussion
The growth of Mo 2 C crystals was investigated to determine the kinetics at the temperature range of 1363 to 1403 K, employing various durations.The lower limit of the processing temperature was determined by the melting point of Cu (T m, Cu = 1358 K), while the upper limit was imposed by the excessive evaporation of Cu at temperatures above 1403 K due to its high vapor pressure.Figure 2 displays SEM images of Mo 2 C crystals grown on Cu substrates at different temperatures (1363 K, 1373 K, 1383 K, 1403 K) and durations (8-80 min).As shown in figure 2, the Mo 2 C crystals begin to impinge on neighboring crystals at shorter processing times as the temperature increases.
Modified Johnson-Mehl-Avrami-Kolmogorov (JMAK) model used to analyze the growth mechanism and kinetics of Mo 2 C growth.To account for deviations from the original assumptions of linear time-dependent isotropic one-dimensional growth rate (Ġ) and constant nucleation rate (N ⋅ ), a modified JMAK equation was employed.The fractional transformed area (ξ) incorporates the quadratic time-dependent isotropic growth rate and the inverse time-dependent nucleation rates, as presented the following formula [23].
is the total area of Mo 2 C crystals; N  is the nucleation rate per unit area per unit time [#/cm 2 s]; G  is the growth rate in terms of distance per unit time [cm/s], t represents time and n is the Avrami exponent.Traditionally, the JMAK equation is utilized to calculate the untransformed area under the assumption that nucleation and growth rates remain constant, regardless of time, at a given temperature.In this study, an assumption is made that the JMAK model can be applied to account for the time-dependent nucleation and growth rates by incorporating rate equations that describe diffusion-assisted radial growth.This modification for time-dependent nucleation is more representative of the nucleation and growth dynamics of the Mo 2 C crystal growth process.
Diffusion controlled growth rate for a disc shaped crystal with a radius R follows inverse quadratic time dependence from known relationship, and denoted by [24,25].
where D is the diffusion coefficient, α is the constant that governs the growth rate of the crystal.The growth rate can be written as follow: where R is the gas constant (8,63 = is called as the apparent activation energy for Mo 2 C growth.The constants { N D . . . T T m / is the inverse homologous temperature of Cu substrate with T m Cu = 1358 K.By utilizing equation (4), a graph that illustrates the relationship between lnln[1/(1−ξ)] and ln(t) is plotted (figure 3(a)).By employing this approach, we determined the Avrami exponent (n) governing the observed phase transformation (Mo 2 C formation) dynamics.The graphical representation of this correlation is visually depicted in figure 3(a), with the slope of the graph -directly equivalent to n-2 -being determined as 1.This determination allows us to establish the Avrami exponent, commonly denoted as 'n,' at a value of 3.This Avrami exponent value points to the involvement of a complex mechanism characterized by nucleation and growth phenomena.This suggests that the progression of the phase transformation is not merely governed by linear growth or simple diffusion-controlled kinetics, but rather a more intricate interplay between nucleation events and three-dimensional growth of the new phase.
Remarkably, the integration of the determined Avrami exponent (n) value of 3 into equation (1) yields a relationship: Furthermore, the apparent activation energy is calculated from the slope of a ln ln[1/(1−ξ)] graph plotted against the reciprocal of the homologous temperature (as per equation ( 4)).This analysis is illustrated in figure 3(b), revealing a determined activation energy of E A = 4 eV.The observation of such a high activation energy necessitates further discussion to elucidate the underlying mechanisms involved.As previously discussed in the context of Raman spectroscopy analyses (figure 1), the physical processes governing Mo 2 C crystal growth accompanied by graphene formation provide an initial clue for the observed high activation energy.This aspect will be further discussed using the schematic diagram depicted in figure 4.
When graphene formation is not controlled, different scenarios depicted in figure 4 can occur simultaneously, a phenomenon that we have extensively discussed in our previous works [8,15,26,27].If graphene forms on Cu surface before or simultaneously with Mo 2 C crystal formation (figures 4(b), (c), (e)), the lateral growth of Mo 2 C crystals involves a sequence of diffusion processes, including:  As these processing steps occur sequentially, the activation energy is expected to primarily correspond to the rate-limiting step with the highest energy barrier.Based on this reasoning, Mo diffusion process through the liquid copper (D Cu Mo ) (as depicted in figure 4(a)) can be excluded as a rate-controlling process since Mo in Cu exhibits high mobility at temperatures exceeding T m Cu = 1358 K.In contrast, the diffusion of Mo through graphene (D G Mo ) and the diffusion of C through graphene (D G C ) are likely to exert a significant rate-limiting effect on Mo 2 C growth.These processes contribute to an increase in the activation energy and play a substantial role in shaping the overall growth kinetics of the Mo 2 C crystal growth process.
In our previous study [26], it was aimed at calculating the apparent activation energies for the growth of both Mo 2 C crystals and graphene, analogous growth experiments were conducted in a comparable setup between 1360-1373 K.The determination of activation energies for lateral growth involved the utilization of the Arrhenius equation, R = Aexp(−E A /kT), where R represents the rate of change in lateral size of the crystal at a given temperature T and duration t.In that investigation, an activation energy of 1 eV was obtained for graphene growth, exhibiting close proximity to values reported in existing literature [28].The rate-limiting step for graphene growth was attributed to the surface diffusion of carbon on Cu, as corroborated by this quantitative alignment.In the exploration of Mo 2 C activation energy (E A ), the growth rate was assessed by focusing on individual Mo 2 C crystals, as opposed to the area coverage approach adopted in this study.Calculating the activation energy for Mo 2 C growth yielded a value of 3.76 eV [26], a result closely resembling the findings of this study.To deduce the rate-limiting step, the successive stages in the lateral growth of the Mo 2 C crystal were considered.Based on the literature data [9,17,29,30], which provide insights into the theoretically calculated energy barrier of 0.07 eV concerning the surface diffusion of carbon on solid Cu, and considering the work by Ejima et al [31], which establishes activation energies ranging from 0.33 eV to 0.54 eV for the diffusion of diverse transition metal solutes (such as Fe, Co, Ni, Cu, Ru, Ag, etc) in liquid Cu, it becomes apparent that certain possibilities can be ruled out.This leads to speculation that the diffusion of Mo to the Cu surface through graphene defects might be the step that limits the rate of the process.
To validate this model, an additional process step is introduced to facilitate the formation of graphene prior to Mo 2 C crystal growth.Figure 5 presents a comparison of the crystals formed with and without the preceding growth of graphene on the Cu substrate.It is observed that when the surface is covered with graphene before Mo 2 C growth, there is an increase in nucleation density and a decrease in the growth rate.These findings are in agreement with previous literature [9].
The findings presented in this study suggest that the presence and morphology of graphene exert a profound influence on the growth dynamics of Mo 2 C crystals.This observation underscores the importance of delving deeper into the intricate relationship between the properties of graphene and the resulting characteristics of Mo 2 C crystals.Notably, effective control over Mo 2 C growth can be achieved through strategic manipulation of graphene's presence and morphology as a pivotal parameter.In the existing literature, several studies have offered strategies for achieving diverse patterns of graphene through growth and etching using CVD [32][33][34].By thoroughly exploring the intricate interplay between graphene and Mo 2 C crystal growth, a wealth of invaluable insights can be gained, serving to refine the optimization of growth techniques and elevate the overall quality and properties of the resulting crystals.This not only holds promise for enhancing the performance of Mo 2 C-based applications but also paves the way for similar advancements in other 2D materials.

Conclusions
The growth of 2D materials via CVD requires a comprehensive understanding of the underlying kinetics and mechanisms involved.This investigation was undertaken with the objective of exploring the growth mechanism underpinning Mo 2 C crystal formation on a Cu substrate, leveraging a modified Johnson-Mehl-Avrami-Kolmogorov (JMAK) model.
Through an array of CVD experiments encompassing diverse time and temperature parameters, we embarked on an exploration of the growth dynamics inherent to 2D orthorhombic Mo 2 C crystals.The quantification of surface coverage for these Mo 2 C crystals was accomplished through SEM analysis.Employing the JMAK model, we investigated the growth rate and determined the activation energy of the process.The unveiling of an Avrami exponent of 3 indicated a complex process with nucleation and growth.The calculated apparent activation energy for Mo2C growth was found to be 4 eV, a notably elevated value that can be attributed to the concurrent formation of graphene alongside Mo 2 C.
Building upon these insights, we propose that the precise control of Mo 2 C growth can be notably achieved by manipulating the presence and quality of graphene as a pivotal parameter.This deduction stems from the intertwined nature of Mo 2 C growth and the interplay with graphene.Further investigations on the influence of graphene structure and quality on Mo 2 C crystal growth are warranted, as they could provide valuable insights for optimizing growth conditions and enhancing the overall quality of Mo 2 C crystals.This study contributes to the fundamental understanding of Mo 2 C growth kinetics via CVD, opening new possibilities for tailoring the growth process and ultimately advancing the applications of 2D materials in various fields.

Figure 1 (
Figure 1(a) illustrates the schematic arrangement for growth of Mo 2 C. TEM images in figure 1(c) depict crosssectional views of different parts of the Mo 2 C crystal on a Cu foil substrate (figures 1(b), (c)).Our analysis using XRD (ICDD: 04-003-0962) and Raman spectroscopy (figure 1(d)) confirmed the orthorhombic structure of the growing Mo 2 C crystals which agrees with published studies[14].XRD results exhibited negligible differences despite varying growth temperatures.The results of Raman spectroscopy not only corroborated with the XRD conclusions pertaining to α-Mo 2 C formation but also unveiled the concurrent formation of graphene during Mo 2 C crystal growth.This finding is reinforced by the 2D signature peak of graphene in figure1(d).These findings agree with previous studies[14].The growth of Mo 2 C crystals was investigated to determine the kinetics at the temperature range of 1363 to 1403 K, employing various durations.The lower limit of the processing temperature was determined by the melting point of Cu (T m, Cu = 1358 K), while the upper limit was imposed by the excessive evaporation of Cu at temperatures above 1403 K due to its high vapor pressure.Figure 2 displays SEM images of Mo 2 C crystals grown on Cu substrates at different temperatures (1363 K, 1373 K, 1383 K, 1403 K) and durations (8-80 min).As shown in figure 2, the Mo 2 C crystals begin to impinge on neighboring crystals at shorter processing times as the temperature increases.Modified Johnson-Mehl-Avrami-Kolmogorov (JMAK) model used to analyze the growth mechanism and kinetics of Mo 2 C growth.To account for deviations from the original assumptions of linear time-dependent isotropic one-dimensional growth rate (Ġ) and constant nucleation rate (N

Figure 1 .
Figure 1.(a) Schematic representation of the CVD setup employed for Mo 2 C growth, along with a side view and a top view of the sample.(b) SEM images showing the process of TEM sample preparation using FIB (c) TEM images showing the cross-sectional structure of various regions of Mo 2 C crystal on Cu foil, (d) XRD and Raman spectroscopy analyses of Mo 2 C crystals grown on a Cu substrate at a temperature of T = 1363 K and a growth duration of t = 64 min.

Figure 2 .
Figure 2. The scale bar is 100 μm in each image.Mo 2 C crystals start impinging on each other at earlier processing times with increasing temperature.

Figure 4 .
Figure 4. Schematic illustration of potential scenarios for graphene and Mo 2 C crystal locations: (a) absence of graphene formation, (b) competitive growth of graphene and Mo 2 C on the same layer, (c) crystal formation on graphene (d) Crystal formation under graphene, (e) crystal formation between graphene layers.