Heteroepitaxy of diamond semiconductor on iridium: a review

Abstract As one of the representatives of carbon-based semiconductors, diamond is called the “Mount Everest” of electronic materials. To maximize its properties and realize its industrial applications, the fabrication of wafer-scale high-quality diamonds is critical. To date, heteroepitaxy is considered as a promising method for the growth of diamond wafers with considerable development. In this review, fundamentals of diamond heteroepitaxy is firstly introduced from several perspectives including nucleation thermodynamics and kinetic, nucleation process at the atomic level, as well as the interplay between the epitaxial film and substrate. Second, the bias enhanced nucleation (BEN) method is reviewed, including BEN setup, BEN process window, nucleation phenomenology (mainly on Iridium), nucleation mechanism by ion bombardment, and large-scale nucleation realization. Third, the following textured growth process is presented, as well as grain boundary annihilation, and dislocation and stress reduction technologies. Fourth, the applications of diamonds in electronic devices are studied, showing its excellent performances in the future power and electronic devices. Finally, prospects in this field are proposed from several aspects.


Research background
Silicon-based semiconductors, for example Si and Ge, are currently the most mature and widely used materials, which have triggered great changes in the information age [1][2][3]. As the development of silicon-based devices approaches physical limits (device size, device performance, device power, processing cost, etc.), exploring new wide-bandgap semiconductors becomes an urgent issue [4,5].
In the post-Moore era, carbon-based semiconductors have gradually attracted attention, including graphene, carbon nanotubes, diamond, etc. [6][7][8][9][10]. Among them, diamond is treated as an insulator for a long time. However, it can present conductive properties by doping. To date, diamond has been regarded as the ultimate semiconductor owing to its ultra-wide bandgap of 5.47 eV [11].
Except for the wide bandgap, diamond also presents many intrinsic advantages due to its unique atomic configuration, such as the high hardness, high elastic modulus, high thermal conductivity of more than 2000 W/(m K) [12][13][14], high carrier mobility (4500 cm 2 V/s for electrons; 3800 cm 2 V/s for holes) [15,16], high breakdown electric field [17], high carrier-saturated drift velocity and low dielectric constant of 5.7 [18]. Therefore, diamond is a quite promising candidate of the next-generation electronic devices in the semiconductor industry [18][19][20].

Technology bottlenecks
Diamond was fabricated by the high-pressure high-temperature method for the first time in 1955. Despite it has been discovered and synthesized for almost seventy years, the application of diamond is still in the ascendant stage. Now, the thermal and electrical applications of diamond are far from reaching the expectation of broad market, especially some extreme conditions generally require the devices to be voltage-resistance, high temperature-resistance and radiation-resistance materials [21]. Due to the absence in grain boundaries, single crystalline diamond is always superior than polycrystalline one not only in electrical properties but also in thermal stability [16,[22][23][24]. Therefore, the synthesis of single crystalline diamond is essential.
Compared with Si-based and other semiconductors, the size and doping technique are two bottlenecks restricting diamond applications. On one hand, Si wafer is already scaled up to 12 in. with very mature processes, while,), 4-6 in. is also realizable for other compound semiconductors (Ga 2 O 3 , SiC, GaN, etc.) [25][26][27][28]. In the contrast, the maximum size of diamond was reported to be only 1-3.5 in. [29][30][31]. On the other hand, Si wafer conductivity can be easily improved by n-type or p-type doping, which provides fundamentals for the electronic applications (detectors, field-effect transistor, diodes, nuclear battery, etc.) [32]. As a comparison, it is still difficult for diamond to realize the good conductivity by n-type doping, because p-type bulk doping by boron is very well controlled over a wide range of concentrations [33] and the formation of two-dimension hole gas (2DHG) on hydrogen-terminated (C-H) diamond can provide a second method to improve the surface conductivity of p-type [34][35][36][37][38]. The hydrogen termination is able to induce the conductivity channel with the interface charge in the surface. As a result, C-H diamond with p-channel conduction is effectively obtained. Positively charged hydrogen atoms of surface C-H dipoles have the negative charged adsorbates from the atmosphere attracted and adsorbed at the diamond surface. The surface negative charge sheet induces the 2DHG layer with a high hole carrier density around 10 13 cm −2 . Once the diamond surface is terminated by oxygen, the surface conduction originated from 2DHG disappears [36,39,40]. Currently, many electronic devices have been fabricated, including detectors, field-effect transistor, diodes, nuclear battery, etc. [41][42][43][44][45][46].

Synthesis of high-quality diamond wafers
Epitaxy on a single-crystal substrate provides a feasible method for precisely controlling the grain orientation of epitaxial materials to achieve the large-area single-crystal growth. There are two basic concepts: homoepitaxy and heteroepitaxy [3,47]. Homoepitaxy [48][49][50] is a method to deposit diamond film on high-quality single-crystal diamond seeds in chemical vapor deposition (CVD) reactors, which include hot-filament CVD [51], microwave-plasma CVD [52][53][54], DC arc plasma jet CVD [55], etc. Though the gas excitation and activation methods are slightly different, the growth process is similar [56]. To enlarge the diamond size, three-dimension growth [53,57] and mosaic growth [51,[58][59][60][61][62][63][64][65] are generally adopted. Nad et al. [57] conducted the MPCVD growth of single crystalline diamond substrates with PCD rimless and expanding surfaces. The lateral SCD surface area increased up to two times greater than the initial seed surface area in one run. The lift-off method can be used to separate the freestanding single-crystal CVD diamond slice from the seed [66][67][68][69][70][71]. Yamada et al. [63,64] reported a 2-in. wafer (40 × 60 mm 2 ) growth by the mosaic growth, but the high cost and inevitable interface greatly hinder the development of this method.
Heteroepitaxy is a concept that an epitaxial layer is grown on a heterogeneous single crystal substrate, showing a great potential in preparing high-quality diamond wafers, and to date, the diamond wafer up to 3.5 in. has been fabricated by heteroepitaxy [29]. More details of heteroepitaxy growth of diamond by CVD methods as well as current status and strategies are discussed in this review.

Thermodynamic and kinetic
Classical nucleation theory is the most mature theoretical model to understand the nucleation of a new phase. Except for the classical nucleation theory, other important non-classical theories such as density-functional theory (DFT) and diffuse interface theory are also proposed in these years [72,73].
According to the classical nucleation theory, nucleation on a hetero-substrate is called heterogeneous nucleation. As illustrated in Figure 1 (right), the rate of heterogeneous nucleation is then studied in Equation (1) [74,75], where θ is the contact angle between the nucleus and substrate, N is the atom number per volume in gas, k B is the Boltzmann constant, h is the Planck constant, R is the molar gas constant, P is the actual vapor pressure, P 0 is the standard vapor pressure, σ f-s is the interface energy between the film and substrate, and T denotes the substrate temperature: where I 0 and ∆G n can be obtained from Equations (2)-(4), From this equation, the nucleation rate is related with the substrate temperature T, reactive atom number per volume (gas concentration) N, the gas pressure P, the contact angle θ, volume per film atom ν f , and the interface energy between reactive gas and film σ f-g , G f and G g are the free energy per film atom or per gas atom. Thus, by decreasing contact angle θ, increasing the substrate temperature T, gas concentration N, and gas pressure P, the nucleation rate I n can be enhanced greatly. This implies there is a process window for parameters mentioned above. This is confirmed in Section 3.2.

Atomic-scale process
The classical nucleation (Section 2.1) well describes mechanisms occurring during PVD methods. It cannot take fully into account the interactions occurring between the substrate and reactive species like radicals from the CVD plasma. At the atomic scale, heteroepitaxy using CVD (or physical vapor deposition) [76][77][78][79] generally includes diffusion, adsorption and desorption of species, species coalescence, and cluster formation and growth, the creation of preferential nucleation sites on the substrate surface, disruption of small clusters, increase in the effective adatom mobility or migration rate, and increase in the substrate temperature ( Figure 1).
Chen et al. [76,77] obtained the nucleation rate from gas based on the atomic process illustrated above. The dynamic equilibrium between adsorption and desorption process of species, and the kinetic process of atomic diffusion on the substrate surface are considered in their model. When the cluster formation and growth from one single atom is studied, the nucleation rate from the gas can be calculated in Equation (5), where C is approximately constant over a reasonable range of P and T, J is the imping flux rate, m is the atom mass, v m is the film atom volume, E a is the adsorption energy, E d is the activation energy of surface diffusion: where K 1 and K 2 can be obtained from Equations (6) and (7), From the equations above, the nucleation rate can be enhanced by increasing the atom imping rate J, the surface adsorption energy E a , the interface energy between the gas phase and film σ f-s , and decreasing the activation energy of surface diffusion E d . Through the ion bombardment under the electric field, the imping rate and the surface adsorption energy can be further enhanced, the nucleation rate is thus improved.
Moreover, according to the Arrhenius equation, the nucleation rate is also related the temperature and the nucleation activation energy; thus, the energy barrier has to be overcome by either increasing the temperature or increasing the energy of species from other pathways (the kinetic energy provided by the ion bombardment during BEN). The analysis about how diamond nucleates epitaxially in Section 3.3 also confirms this point.
How these atomic-scale phenomena affect the nucleation and growth was studied by Matthews [80]. Based on the proposed kinetics theory for the thin film deposition, supposing the deposition rate of the species and the number of active nucleation sites to be constant, the diffusion of adsorbed species on the substrate then becomes the predominated factor. In this case, the surface temperature plays an important role in determining whether the adatoms are able to stay at the lowest free energy sites and configure into the orientation. Generally, a combination of a proper substrate temperature with an impinging flux is necessary for epitaxial nucleation.
Diamond heteroepitaxial nucleation usually occurs under the applied electric field in a CVD setup, which is called bias-enhanced nucleation (BEN) [81,82]. Thus, the whole stage is the ion bombardment process with high-energy atoms/ions, and the kinetic energies of the bombarding species and their flux rate toward the substrate are key parameters for the nucleation.

Inter-atomic interplay
Due to the strong interaction between the epitaxial film and substrate, chemical bonds are formed at the interface between the heteroepitaxy layer and substrate. This is also called the conventional heteroepitaxy. In the contrast, with the growth of two-dimensional materials attracting more and more attention, van der Waals epitaxy [83,84] is proposed referring to that the epitaxial layer interacting with the substrate by the weak van der Waals interaction, such as graphene growing on the Cu(111) surface [19,20].
The formation of chemical bonds is attributed to the strong interaction between dangling bonds of substrate surface and epitaxial layer, which is very common in the field of heteroepitaxy of semiconductors such as II-VI, III-V and IV-IV semiconductors [3]. The strong interaction of chemical bonds at the interface can force each atom of the epitaxial layer to match with the substrate atoms: the strong bonding is responsible for the crystalline ordering of the epitaxial layer, inducing the epitaxial layer to mimic the crystalline symmetry of the substrate [85][86][87]. Generally, these chemical bonds are strong enough to modify the lattice parameters of the epitaxial layer in the interface area. As a result, the crystal deformation at the beginning growth stage of the epitaxial layer is frequently observed, with the strain energy stored in the film. The strain energy accumulates with the thickness of the epitaxial layer increasing. According to the theory proposed by Matthews and Blakeslee [88], once the thickness of the thin film exceeds the critical value, defects such as misfit dislocations, are easily formed to relax the strain energy partially [89].
For diamond heteroepitaxy, chemical bonds are formed between diamond and the hetero-substrate due to the dangling bonds of the clean substrate surface and diamond layer. Arnault et al. [90] reported a diamond nanocrystal with a lateral size of 7 nm and a height of 2.7 nm formed at a single crystalline Ir surface. Diamond nanocrystal with the lattice of d 220 = 0.136 nm and d 002 = 0.192 nm, which gives a unit cell parameter of 0.381 nm. This is very close to the unit cell parameter of iridium (0.383 nm). Wang et al. [91] also observed that the diamond (111) interplanar spacing decreased from 0.222 to 0.206 nm with the increase of the distance from the interface, which means with the increase in the thickness of diamond film, the strain is gradually relaxed.
From the distribution of charge density and charge density difference between Ir atoms and carbon atoms, the formation of strong C-Ir ionic and covalent bonds is clearly shown [92].

MPCVD
According to Wang et al. [101] and Chavanne et al. [114], the substrate holder is negative and electrically insulated from the reactor walls which is grounded. During the BEN step, a negative bias voltage is applied between the sample holder and the walls. Schreck and Stritzker [115] described a second setup where a circular electrode inserted in the plasma and placed about 20 mm above the substrate is positively biased while the substrate remains grounded. Delchevalrie et al. [116] reported a similar configuration where a positive counter electrode is installed on a translator and electrically insulated from the reactor wall while the substrate holder is kept grounded. Yaita et al. [106][107][108] invented an antenna-edge CVD setup where the microwave electric field can be concentrated at the tip of the antenna (also as an electrode in the BEN process), and plasma density is greatly increased. As a result, the reactive gas is effectively decomposed, the amount of diamond nucleation precursors is increased, and the diamond crystal quality and growth rate are improved. These reported configurations are described in Figure 2(a-c).

DCCVD
Except in the MPCVD setup, Gsell et al. [117] described a second setup in their work (Figure 2(d)). The Ir substrate surface was irradiated with ions produced by a DC discharge in a CH 4 /H 2 gas mixture. A copper cylinder with a diameter of 5 cm was placed 2 mm above the substrate and a bias voltage is applied. Ohtsuka et al. [118] and Sawabe et al. [105] invented a three-electrode DCCVD setup to conduct the ion irradiation pretreatment. Only 1 mm above the substrate is a circular electrode, and the cylinder electrode is 2.5 cm away from the substrate. By controlling the potential between circular electrode, the cylinder electrode and the substrate, the growth or the ion irradiation can be realized, respectively.

HFCVD
In a hot-filament CVD (HFCVD) setup, Wang et al. [119] designed the distance of about 8 mm between the filament and the substrate. The negative bias relative to the filament is applied to the substrate through a graphite holder, and the resistance of the bias circuit is larger than 10 MΩ to avoid current leakage. Janischowsky et al. [120] designed an additional tungsten electrode (grid) for extraction and acceleration of electrons from the hot filaments, placed at a distance of 6 mm behind the filaments with a potential of +60 V with respect to the filaments. This generated a glow discharge between the filament and the grid. Positive ions within the discharge are accelerated through the filament openings toward the substrate, placed on a negative potential of typically −140 V with respect to the filaments. Arnault et al. [121] applied a negative bias voltage between an anode grid located above the upper filaments and a cathode grid placed between the two pairs of tungsten filaments. Positive ions are further accelerated by an extraction voltage produced between the cathode and the sample surface.

Bias-process window
Bias process parameters include the reactor pressure, reactive gas content, bias voltage, bias time, substrate temperature, etc. There is a parameter space where the high-density epitaxial nucleation can be attained. This parameter space is called the bias-process window. However, it should be noted that the bias process is indeed reactor-dependent. The process window for different reactors or research teams is always different.
Schreck et al. [109] and Thürer et al. [110] proposed this concept of process window when they studied the BEN on Si in Figure 3(a). Heteroepitaxial orientation is achieved over a wide range of different parameters provided that the bias time is within a definite time interval (time window). The width of the time window, and the bias time for optimal azimuthal alignment, strongly decrease with the absolute value increase of the bias voltage. Yaita et al. [106] found the bias current decreased at the beginning of the BEN process and then increase during diamond formation shown in Figure 3(b). An increase in the bias current of 10% leads to epitaxial diamond nucleation with a high nucleation density and is the optimum condition for the diamond nucleation on 3C-SiC in terms of the density and epitaxial nucleation. This technique is based on the difference between the secondary electron emission coefficient of diamond and that of the material underneath. Regmi et al. [122] presented a comprehensive study of BEN parameter space for high density epitaxial nucleation of diamond on Ir substrates in Figure 4(c,d). A high nucleation density exceeding 10 11 cm −2 occurs only in a narrow bias voltage range from 125 to 175 V and a narrow CH 4 content range from 1.5 to 3%. At bias voltages and methane concentrations outside these windows epitaxial diamond nucleation densities fall abruptly to near zero.   [109]; (b) time-current-voltage process window for 3c-Sic substrate [106]; (c) voltage window for ir substrate [122]; (d) methane content window for ir substrate [122]. Reprinted from Journal of Applied Physics and Diamond and Related Materials with permission from aiP Publishing and Elsevier, respectively.  [127]. (b) Fractural domains and modified ir surface observed after BEn [129]. (c, d) Fractural domains and respective diamond growth for the same region [129]. Reprinted from Diamond and Related Materials with permission from Elsevier.

Epitaxial nucleation phenomenology
Considering the largest single crystal diamond wafer is obtained on Ir substrates and the length of this review, the epitaxial nucleation phenomenon and process on Ir substrates are carefully illustrated in this part. Because a large-area single crystalline iridium is not available, the iridium heteroepitaxy on foreign substrates is necessary. Epitaxial Ir films are often deposited on MgO [123], SrTiO 3 [91], Al 2 O 3 [87], KTaO 3 [124], Pd/Al 2 O 3 [125], YSZ/Si [126] and SrTiO 3 /Si [90] substrates by e-beam evaporation, pulsed laser deposition, molecular beam epitaxy and magnetron sputtering. Currently, the synthesis of 4-in. epitaxial Ir film with a good crystallinity on foreign substrates is already not a difficult task.
Domain formation and surface modification are both most common phenomenon. Schreck et al. [117,127] investigated the domain formation on Ir(001) surfaces after BEN in Figure 4(a). Bright regions (called domains) with lateral dimensions up to a few micrometers are observed by scanning electron microscopy (SEM). Two round domains can merge when they meet each other, and no obvious boundaries are observed. From AFM morphologies [128], the relative height of inside a domain is about 1 nm lower than that outside a domain. In Figure 4(b), Vaissiere et al. [129] observed different nanostructures underneath domains, including Ir balls, furrows and ridges. In Figure 4(c,d), when the growth step is applied after the BEN, domains develop into islands of the same shape, composed of epitaxial diamond and with a high density of oriented grains. Surface modifications refer to the Ir surface modified by BEN and covered by microstructures (furrows and ridges) distributed along with some preferential orientations, which can be seen in Figure 4(b,c). In situ characterizations for domains is efficient to understand the nucleation. Meanwhile, it can be used to adjust BEN parameters. Delchevalrie et al. [116] developed the spectroscopic ellipsometry method, which may be a sensitive tool to monitor domains formation on Ir substrates because of its sensitivity to the optical index differences at interfaces.
To understand whether there exist diamond nuclei in domains, and further study the domain structure and composition, a variety of characterization methods are adopted to analyze domains. Some characterization methods prove a failure in confirming the existence of diamond nuclei due to the resolution. For instance, high resolution transmission electron microscopy (HRTEM) is used to confirm the interface between the Ir substrate and the domain layer, and there exist no diamond grains in the domain layer at the interface [130]. (Here, it should be noted the nanodiamond has been observed from HRTEM by Arnault et al. [90].) Electron backscatter diffraction patterns inside and outside domains never show any significant difference [127]. In RHEED and low energy electron diffraction, domains do not give any sign of crystalline diamond, either [131]. Some methods can distinguish the difference between inside and outside domains. The friction force-load curves inside and outside the domain are tested by lateral force microscopy, and the change curves inside and outside the domain are different from that in the Ir substrate reference showing a linear change trend. The difference may be due to the different plastic deformation of the metal layer under a low load [132]. Meanwhile, some methods can directly confirm the existence of diamond nuclei inside domains. In Figure 5(a,b), C KLL and Ir MNN Figure 5. Small spot aES spectra of the carbon Kll and ir mnn lines taken (a-1, a-2) outside and (b-1, b-2) insides the domain areas [132]. (c-1, c-2) XPD images for ir reference and diamond reference [131]. (d-1, d-2) XPD images of the surface with domains after BEn [131]. Reprinted from Diamond and Related Materials with permission from Elsevier. spectral lines measured by spatially resolved Auger electron spectrum (SR-AES) show a diamond characteristic peak at 262 eV inside domains and a weak characteristic peak of graphitic structure outside domains. SR-AES also shows that the C-phase density in domains is 30-50% higher than that outside domains. The thicknesses of the surface layer inside and outside a domain are calculated to be 1.73 ± 0.16 and 2.08 ± 0.07 nm, respectively [132]. As shown in Figure 5(c,d), X-ray photoelectron diffraction (XPD) shows a clear C 1s pattern in domains, indicating that carbon atoms in domains are arranged in the structure of a crystalline diamond. Finally, Bernhard et al. [133] used X-ray absorption spectroscopy to study the composition of domains, and most of these carbon atoms exist in the diamond structure. However, the true structure of the BEN layer is more complex than the pure composition of perfect diamond crystallites embedded in an amorphous matrix [129,131]. Figure 6 shows SEM, AFM, RHEED, and TEM characterization results of diamond grains with different growth times after BEN [128]. With the increase in the growth time, diamond grains can grow from domains ( Figure 6(a,b)), and the height difference inside and outside the domain gradually increases from 1.5 nm (5 s) to 6 nm (60 s) (Figure 6(c,d)). This means that the diamond grains in the domain begin to grow vertically or a certain etching phenomenon occurs in the area outside the domain. When the growth starts, the Ir substrate surface gradually transitions from a single Ir structure to a mixed structure of Ir and diamond (Figure 6(e)). From the cross-sectional TEM results (Figure 6(f)), as the growth progresses, the amorphous carbon layer on the surface of the Ir substrate in domains has completely disappeared, while diamond grains begin to appear in the crystal domain and grow laterally and vertically.

Epitaxial nucleation mechanism
When mentioning diamond nucleation and growth during CVD, the phase transformation of graphite into diamond is usually considered. Since graphite is more stable than diamond, a large amount of energy needs transferring to the system, inducing the lattice rehybridization and shift the thermodynamic balance from sp 2 -graphite to sp 3 -diamond. Moreover, the kinetic barrier hindering the phase transformation has to be overcome, too. For instance, even for the switch of diamond into the more thermodynamically favorable graphite, it usually takes a long time to occur spontaneously [134]. Thus, a fundamental understanding of the nucleation, thermodynamics and kinetics of diamond phase transitions is necessary to enable the application of gas phase-diamond transitions. Yugo et al. [135] firstly applied the bias voltage to enhance the nucleation, and several questions have to be addressed to study the epitaxial nucleation under the ion bombardment: (1) Why is diamond the more stable phase than graphite in the CVD atmosphere? (2) How is the diamond nucleation enhanced during BEN? (3) How is the epitaxial orientation determined?
For question (1), the role of hydrogen atom in the CVD atmosphere has been well illustrated. A high content of atomic hydrogen is crucial for a number of main processes [136], and CH 3 is considered the dominant reactive hydrocarbon radical. It is hydrogen that makes diamond a more stable phase than graphite [20,[136][137][138]. Though diamond nucleation and growth can be realized from the thermodynamics, the kinetic barrier has also to be overcome so that nucleation can occur. Due to the large surface energy difference between diamond and a heterogeneous substrate the nucleation density is so low that some enhancement methods need adopting. This is a core question.
For question (2), the key is related with the role of the ion bombardment and highlighted based on diamond epitaxial nucleation on Ir substrates. When the electric field is applied in the plasma, the charged species (ions and electrons) move directionally. The positively charged ions bombard the substrate while the negatively species move toward the anode. The process of positive ions moving toward the negative substrate is called the ion bombardment [29]. The sub-plantation, preferential etching, and secondary-electron emission are three main consequences under the ion bombardment. The widely used sub-plantation model is proposed by Lifshitz et al. [139,140]. They carefully studied the mechanism of the film growth from hyper-thermal species. The model involves a shallow sub-plantation process, energy loss, preferential displacement of atoms with low displacement energy, leaving the atoms with the high displacement energy intact, sputtering of substrate material, and inclusion of a new phase due to incorporation of a high density of interstitials in a host matrix. More specifically [141][142][143], a dense amorphous hydrogenated carbon (a-C:H) layer is firstly formed, and then pure sp 3 carbon clusters containing dozens of atoms are spontaneously precipitated in the a-C:H layer, caused by the "thermal spike" of the impinging energetic species [144,145]. By converting amorphous carbon to diamond at the amorphous matrix-diamond interface, diamond clusters grow to a few nanometers. This transition is caused by a "preferential displacement" mechanism mainly caused by the influence of high-energy hydrogen atoms. On this basis, Schreck et al. [29] proposed the ion bombardment induced buried lateral growth mechanism on Ir substrates in Figure 7. In their model, there are at least five stages in the nucleation process, and Figure 7(a-1-a-5) represents the exposed Ir surface before BEN, a-C:H layer formation, primary nucleation from the a-C:H layer at the interface or the domain formation, a great many secondary nuclei formation from highly defective crystalline matrix or the domain spreading, nuclei growth after BEN. By comparing the height difference across the domain (Figure 7(b-1,b-2)), a mechanism of the ion bombardment induced buried lateral growth is carefully described in Figure 7(c). The a-C:H layer is firstly formed (region III), and then the highly defective matrix including a large amount of sp 3 C-C bonds is formed (region II). Diamond nuclei with the crystallinity structure are further formed from this defective matrix (region I). The preferential etching [146,147] was once thought to be the key mechanism for epitaxial nucleation. However, it just refers to the difference of the ability of epitaxial or non-epitaxial nuclei resisting the ion bombardment. The secondary emission [113,115] refers to that diamond has the higher electron emission ability than the substrate. Thus, the bias current representing the number of moving species can increase once diamond is deposited. This has been shown in Section 3.2, but not thought to be so important in diamond epitaxial nucleation. Overall, the sub-plantation is more acceptable to researchers with the further research.
Moreover, the ion bombardment can lead to a further ionization and dissociation of the reactive gas [148][149][150], and an increase in the substrate temperature [91]. According to the classical nucleation theory, the specie concentration, the substrate temperature, and the substrate surface roughening by furrows and ridges can efficiently enhance the nucleation ratio and then the nucleation density.
For question (3), what determines the heteroepitaxy on a foreign substrate is a mystery. In recent years, with the development of 2D materials, a great many studies focus on the epitaxy of 2D materials. Dong et al. [151] proposed a general theoretical model for the epitaxial growth of a 2D material on an arbitrary substrate. From DFT calculations, the interplay between the 2D material and the substrate plays a critical role in the epitaxial growth of the 2D material. When the material is epitaxially grown, the binding energy between 2D material and the substrate is strongest. Through the calculation, the difference between the maximum and minimum binding energy is so large that 2D material can grow in a well-aligned configuration on the substrate. From these studies, we may explore the epitaxial mechanism of 3D materials (including diamond). It is inferred that only when diamond is grown epitaxially the binding energy between diamond and a foreign substrate is the largest and the following growth is the most stable.

Large-scale nucleation
There are two approaches to realize the large-scale nucleation. One is to develop the high-power CVD setup with the large-area plasma; fundamentally, each diamond research group has been trying to enlarge the plasma area (for instance, designing a 915 MHz MPCVD instead of 2.45 GHz) [29,[152][153][154]. Meanwhile, a second method is to adjust the plasma shape in situ by changing the electrical-magnetic field in the chamber. For example, Yoshikawa et al. [155] introduced a simple and effective method to extend the area of BEN for heteroepitaxial diamond growth by metal-covered half-ring Si plate right outside the substrates. It is claimed that this method has the ability to enlarge the nucleation region up to 2-in.  -1-a-5) represent before BEn, amorphous hydrogenated carbon layer formation, primary nucleation from the a-c:H layer at the interface, highly defective crystalline matrix formation and a great many secondary nuclei formation, nuclei growth after BEn [29]. (b) aFm image for a round domain (b-1) and the line scan across the domain boundary (b-2) [29]. (c) the ion bombardment induced buried lateral growth schematic [29].

Growth of individual grain
When a nucleus with the size exceeding the critical size is formed at the substrate surface, it can continue to grow. For any grain, its growth meets the fundamental principle. Usually, the crystal plane with a lower surface energy is more stable. Thus, if the growth is not restricted by the substrate, grains are going to show the crystal plane with the lowest surface energy [156]. However, for heteroepitaxy on foreign substrates, the growth can be influenced by the substrate, too.
Growth parameters α, β, and γ based on the Bravais-Friedel-Donnay-Harker model [157] that the growth rate of a crystal plane is inversely proportional to the crystal plane spacing were proposed by Silva et al. [48,158], 100 113 . Figure 8 shows the grain shape when α, β, and γ parameters are in different regimes, where Figure  8(a,b) shows the facets coexistence domains are surrounded by topological boundaries that depend on α/β parameters, α/γ parameters, respectively. When β or γ parameter is very small, the grain shape mainly depends on α parameter. When α is smaller than 1, the grain is a cube; when α is larger than 3, the grain is an octahedron; when α is at the range from 1 to 3, the shape is cuboctahedron. To obtain the (001)-preferred diamond grains, Chavanne et al. [159] proposed that α close to 3 promoted a faster growth rate of <001> directions and that lower than 1.5 led to a preferential growth in <111> direction. For the former, pyramid or octahedral grains are attained while other non-epitaxial grains are also overgrown. For the latter, those epitaxial grains grow and smooth (001) surfaces are observed. CVD parameters such as methane fraction and substrate temperature can be adjusted to get both α cases. What should be noted is that the model above never takes the effect of the substrate on the grain growth into account [160].
Finally, the growth process is also studied using the in situ reflectance interferometry. Aida et al. [161] found the difference of the substrate temperature and the reflectance profile when polycrystalline or heteroepitaxial diamond film is grown on Ir/MgO substrates. In contrast with polycrystalline growth, when the heteroepitaxial growth is conducted, the reflectance profile shows a dynamic interference pattern. Oscillations in the reflectance profile are able to provide the information on the grown thickness and the growth rate in real-time. Overall, this method can be used to monitor the normal growth process of diamond.

Merging of neighboring grains
Once grains meet each other, the condition is a little different and the interaction between grains has to be considered. This merging process can be at least described from two perspectives [162][163][164][165][166].
The elastic energies associated with complete low-angle grain boundaries and with partial wedge disclinations is given by Michler et al. [165] and Schreck et al. [162]. Equation (7) is the derived equation describing the relationship between the grain size and the twist angle, where b is the Burgers vector b = a/2 < 110> and a = 0.35 nm, R is the grain radii, ω crit is the critical grain-boundary angle.
In their opinion, when grains meet, the grain size and the twist angle play a key role. Only if the grain and angle are small enough, can the merging with the low-angle grain boundary replaced by the disclination. Based on this, a very high epitaxial nucleation density is necessary; otherwise, a mosaic crystal (diamond on Si substrate) is easily formed instead of a single crystal (diamond on Ir substrate). Figure 9 is the schematic of grains merging and the disclination formation when they meet each other, where the blue, red, orange and yellow balls represent Ir, H, C and selected atoms, respectively. Before two diamond grains meet each other (a), more atoms are added with they grow laterally (b). If both grains have no tilt or twist, they will merge into a larger grain (c). If both grains have a small twist or tilt angle, disclination can form instead of the small-angle grain boundary (d, e). In contrast, if the twist or tilt angle is too large, the grain boundary can be observed.
In the meanwhile, DFT is also used to describe the merging process. Currently, in contrast with diamond, 2D graphene merging process has been studied systematically [164,166]. Dong et al. [164] calculated the formation energy of different grain boundary angles when studying the seamless stitching of graphene domains. In their opinion, the rotation and merging of graphene domains can minimize the formation energy and contribute to the formation of a larger domain. Though this work is applied in 2D graphene growth on liquid Cu substrate, this principle may also be applied in explaining the merging process of diamond grains.

Origin
When diamond is grown on foreign substrates via heteroepitaxy, two types of dislocations are introduced from two aspects: misfit dislocations and threading dislocations [89,167,168]. The threading dislocation is at a particular angle with respect to the growth interface and extends to the diamond epitaxial layer, and there are two sources for it. One is inherited from the heterogeneous substrates, and it propagates through the diamond epitaxial layer and terminates at the diamond surface. The other is generated with the movement of misfit dislocations. Therefore, misfit and threading dislocations are both inevitable during the growth of the heteroepitaxial diamond film. In contrast with misfit dislocation, the threading dislocation generation cannot reduce the strain. However, it is generally believed to primarily affect the quality of diamond epitaxial films and its bending also leads to the change of the intrinsic stress [169].
The residual stress of non-freestanding diamond films is a sum of the external stress and the intrinsic stress [165,170]. The former is caused by a difference in coefficients of thermal expansion (CTE) between the film and the substrate and builds up upon cooling down from the deposition temperature to room temperature [171][172][173], so it is also called the thermal stress. The latter is the cumulative result of chemical and microstructural defects incorporated during deposition [126,[174][175][176], the stress correlating with process parameters such as the substrate temperature or methane content in the gas phase. The microstructure, which involves coherency strain, surface energy effects, and disclinations, can contribute substantially to the stress in thin layers [172]. Moreover, the threading dislocation can also contribute to the stress formation. When dislocations move out of the glide plane spanned by the line direction and the Burgers vector, an effective climb is performed and increases the number of lattice cells per surface area at the growth stage, consequently, the stress is generated [175,176].

Reduction technologies
The reduction of dislocations and stress is an important issue for heteroepitaxial diamond growth. Wang et al. [89] summarized the dislocation reduction technologies from several perspectives. Generally, the threading dislocation density can be reduced to below 10 8 cm −2 from above 10 10 cm −2 with the Raman line width decreases from >10 cm −1 to 1.86 cm −1 when the film thickness increased to 1 mm. This is mainly due to the enhanced reactions between neighboring dislocations [177]. Moreover, adopting the off-axis substrate can also reduce the dislocation density by changing the dislocation propagation direction [30,175,178]. Epitaxial lateral growth is a very useful method to control the dislocation density, including the conventional epitaxial lateral growth [160,[179][180][181], pendeo-epitaxial lateral growth [180,181], and the patterned nucleation growth method [31,182,183]. Typically, Mehmel et al. [184] found the use of micrometric laser-pierced hole-arrays could lead to a reduction in dislocation density by two orders of magnitude to 6 × 10 5 cm −2 in the hole region where the lateral growth occurred. This value is equivalent to that typically measured for commercial type Ib single crystal diamonds. Kim et al. [31] adopted a similar method with the microneedle formation and then dislocation density of a 1-in. diamond wafer with the thickness of 500-600 μm was only 1.4 × 10 7 cm −2 .
Except the classical methods mentioned above, there are several novel methods reported that can effectively reduce the dislocation density. Ohmagari et al. [185,186] designed a method to control the dislocation propagation by incorporating W in a hot-filament CVD setup. A large reduction in the dislocation density from 10 6 cm −2 to 10 4 cm −2 was demonstrated. One possible reason for these phenomena is that dislocation propagation is suppressed by W impurities inadvertently provided from the heated filament wire. At the intersection of the dislocation and W impurity reactions, it is possible for the local compressive expansion strain to disappear complementarily, in which case the annihilation of the dislocation is more energetically stable.
The reduction in intrinsic stress can cause the continuous growth of diamond thick films without the crack. Kim et al. [30] proposed that the off-axis substrate allows step-flow growth of diamond films, and tensile stress is released in the diamond layer. Consequently, the 2-in. diamond layer delaminates naturally from the substrate without cracking. For the diamond grown on the Ir/Al 2 0 3 (11 20) substrate misoriented by 7° toward the [1 1 00] direction, the widths of the (004) and (311) X-ray rocking curves were 98.35 and 175.3 arc sec, respectively, which prove a better crystallinity of diamond films than the diamond wafer by Schreck et al. [29].

Diode
Diode known as rectifiers acts as a one-way switch for current, which allows current to flow easily in one direction, but severely restricts current from flowing in the opposite direction. Homoepitaxial diamond diodes have been widely studied in the last twenty years while the study for heteroepitaxial diamond diodes are mainly concentrated in the last few years [90,[187][188][189][190][191] in Table 1.
In 2015, Kawashima et al. [188] fabricated lateral Schottky barrier diodes (SBDs) using heteroepitaxial diamond films grown onto Ir/SrTiO 3 /Si substrates. High rectification ratio of 10 12 and forward current density of 10 A/cm 2 were achieved. A breakdown voltage of 52 V was observed corresponding to a breakdown electric field strength of ~1 MV/cm. Murooka et al. [189] demonstrated the device properties of lateral Schottky barrier diodes (SBDs) fabricated on heteroepitaxial diamond films grown onto 3C-SiC/Si substrates, and investigated the electrical properties of SBDs through temperature-dependent characterization. The fabricated SBDs exhibit clear diode properties with rectification ratios above 10 9 at ±5 V. The diode properties and rectification ratios are maintained above 10 8 even at 500 K. The current density of these diodes is around 1 A/cm 2 . Arnault et al. [90] fabricated and characterized the lateral SBDs on heteroepitaxial diamond film on Ir/SrTiO 3 /Si substrate. I-V characteristics evidence a yield of working diodes equal to 92%, close to that obtained for diodes on homoepitaxial films. Despite a higher expected defect density, when heteroepitaxial diamond is used as a substrate, the active layer doping and the device characteristics (serial resistance, Schottky barrier, and ideality) appear highly uniform. Kwak et al. [190] fabricated lateral Schottky barrier diodes on alpha-sapphire based heteroepitaxy diamond film. The ideality factor of 1.4 and maximum breakdown field of 1.1 MV/cm are measured by power device analyzer respectively. Most of threading dislocations in Diamond epilayer grown on  [189] lateral Schottky 3c-Sic/Si (001) 50 (mo)~1 10 9 - [90] lateral Schottky ir/Srtio 3 /Si (001) 200 (Zr/Pt/au)~10 −3 10 4 -10 5 - [190] lateral Schottky al 2 o 3 (112 0) 100 (al)~6 10 6 50 (1.1 mV/cm) [191] lateral Schottky commercially product (mo/au) (5 × 10 −5 )~10 6 375 heteroepitaxial diamond substrate reveal as 45° mixed type deteriorating the device performance that induced early breakdown or large leakage current. To eliminate the impact of defects on the electrical performance, Sittimart et al. [191] used metal-assisted termination (MAT) method to reduce the defect density of the heteroepitaxy diamond film. After the insertion of the MAT buffer layer, the leakage current was depressed and yield of working diodes was enhanced. A breakdown voltage of 375 V was measured, which is the highest breakdown value among SBDs on heteroepitaxy substrates. Heteroepitaxial diamond substrate is expected to be more suitable for commercialization and application in semiconductor industry with lower cost and larger area. Those attempts to make Schottky barrier diodes show the potential of the heteroepitaxial diamond in electronic devices. However, more technologies should be applied to reduce the on-state resistance as well as enhance the breakdown voltage. Lateral SBDs usually have higher series resistance thus it should be replaced with pseudo vertical or vertical SBDs. Techniques such as MAT are required to improve the crystal quality.

Field effect transistor
Field effect transistors (FETs) in diamond [34,[192][193][194][195][196][197][198][199][200][201] should outperform FET structures on other wide bandgap materials such as SiC and GaN in high power/high temperature applications due to the desirable properties of diamond materials, which have attracted researchers' interest in the last few years. Syamsul et al. [192] deposited a 500 nm undoped homoepitaxial diamond layer on N-doped heteroepitaxial diamond on 3C-SiC/Si substrate and fabricated hydrogen-terminated diamond metal-oxide-semiconductor field effect-transistors (C-H MOSFETs). A maximum current density of 80 mA/mm and a I ON /I OFF ratio of 10 9 are achieved. A high breakdown voltage of more than 1 kV was obtained. Kasu et al. [193] reported diamond modulation-doped C-H MOSFETs and obtained high hole mobility of 2465 cm 2 /Vs fabricated on the heteroepitaxial diamond. The breakdown voltage as high as 703 V was achieved corresponding to a Baliga's figure of merit (BFOM) of 179 MW/cm 2 . Through surface Al 2 O 3 passivation layer, they further improved the breakdown voltage of the C-H MOSFETs on the heteroepitaxial diamond to 2608 V and BFOM of 344.6 MW/cm 2 [194]. Saha et al. [197] fabricated diamond C-H MOSFETs on chemical mechanical planarized heteroepitaxial diamond and achieved high current density of −6.8 A/mm, high breakdown voltage of −2568 V and highest BFOM for diamond of 875 MW/cm 2 . At the same time, they reported a −3326 V breakdown voltage diamond MOSFET [198]. Chen [196] grown boron doped layer on the heteroepitaxial diamond, and fabricated diamond metal semiconductor FETs. A breakdown voltage of −2360 V was obtained at room temperature.
Several works reported diamond FETs using heteroepitaxy diamond substrates recently and showed gratifying results. MOSFETs using hydrogen terminated diamond as conductive channel are most promising as they exhibited high current density as well as high breakdown voltage. More in-depth research should be conducted on C-H devices to further improve device performance.

Detector
The study for heteroepitaxial diamond detectors is not so many [23,202,203]. The first one is reported by Berdermann et al. in 2010 [202]. They conducted diamond detectors used for hadron physics research. However, the charge collection efficiency for hole is quite low of 12% compared to that of the single crystal diamond detectors (95%). Recently, Berdermann et al. [23] prepared diamond detectors for particle detection and studied the performances for several kinds of particles including α and β sources, swift ions from the heavy-ion synchrotron, and relativistic protons. The charge collection efficiency for holes is improved to be 95% while that of electrons is still low (40%). Further improvement of the crystal quality and understanding the conducting mechanism of the heteroepitaxy substrates are needed.

Summary
(1) The fundamental of diamond heteroepitaxy is systematically introduced from three aspects including the nucleation and growth thermodynamics, nucleation at the atomic level, the interplay between substrate and film, which provide the theory basis for diamond heteroepitaxy. (2) BEN is the most important method to realize the high-density epitaxial nucleation. Several mainstream BEN configurations in the MPCVD, DCCVD and HFCVD setups are illustrated; the substrate holder kept a lower electric potential can lead to the directional bombardment of positively charged species. (3) Bias process is a combination of many process parameters including the substrate temperature, reactor pressure, methane concentration, bias voltage and bias time. The narrow process window involving the bias voltage and bias time, and methane content parameters are widely studied whether the foreign substrate is Si, 3C-SiC or Ir substrate. Outside the process window, diamond nucleation density decreases dramatically or the nucleus loses its epitaxial orientation. (4) Diamond epitaxial nucleation mechanism under the ion bombardment is studied when taking Ir substrate as an example. The ion bombardment causes several consequences: ion sub-plantation, substrate surface modification, further reactive gas ionization and dissociation, and substrate temperature increase. From the perspective of the nucleation process, two stages including a-C:H layer and diamond nucleation from a-C:H layer are observed and simulated. It is inferred that the epitaxial orientation with the substrate is determined by the strongest binding energy between the diamond and substrate. (5) Textured growth mechanism and process after BEN are illustrated, controlled by the growth parameters. When grains meet each other, the disclination can be introduced instead of the low-angle grain boundary when the elastic strain energy is the minimum. A relationship exists between the critical grain size and a critical grain boundary angle. Usually, the larger the grain size, the smaller the critical grain boundary angle. (6) With the dislocation and stress in the diamond film influencing the device performance considered, the efficient method to reduce the dislocation and stress includes the epitaxial lateral growth, off-axis substrate growth, the metal-assisted termination method, etc. The dislocation density can be reduced to 10 4 -10 6 cm −2 . (7) The electronic applications of heteroepitaxial diamond are introduced from three aspects including diode, FET, and detectors. The performance of these devices shows the potential of heteroepitaxial diamond in electronic devices.

Prospects
(1) Though Ir has proven the optimum substrate, exploring the novel and cheap substrate still needs a study. (2) The study for an interdependent relationship between bias process parameters (bias voltage, reactor pressure, etc.) is necessary to understand the narrow bias process window. (3) BEN mechanism can be further studied by experiment methods. For instance, the influence of bias process parameters on the a-C:H layer thickness and composition can be characterized by X-ray photoelectron spectrum, elastic recoil detection analysis, etc., which may provide new insights to understand the different nucleation pathways when bias process parameters change.
(4) With the development of DFT, the mechanism of diamond heteroepitaxial nucleation and growth needs studying in the future, when 2D material growth process and mechanism has been simulated carefully from several aspects. Especially, the molecular-dynamic simulation method shows the big value in understanding the specific process how diamond nucleates from gas or amorphous carbon, etc. (5) 1-3.5 in. diamond wafers have been fabricated by some researchers. Developing the technology for the stable, reproducible, uniform, high-density, and large-scale plasma is the premise to attain the diamond wafer with a large size.