Influence of Carbon Cap on Self-Diffusion in Silicon Carbide

Abstract

Self-diffusion of carbon (${}^{12}$C and ${}^{13}$C) and silicon (${}^{28}$Si and ${}^{30}$Si) in 4H silicon carbide has been investigated by utilizing a structure containing an isotope purified 4H-${}^{28}$Si${}^{12}$C epitaxial layer grown on an n-type (0001) 4H-SiC substrate, and finally covered by a carbon capping layer (C-cap). The ${}^{13}$C and ${}^{30}$Si isotope profiles were monitored using secondary ion mass spectrometry (SIMS) following successive heat treatments performed at $2300–2450{}^{\circ }\mathrm{C}$ in Ar atmosphere using an inductively heated furnace. The ${}^{30}$Si profiles show little redistribution within the studied temperature range, with the extracted diffusion lengths for Si being within the error bar for surface roughening during annealing, as determined by profilometer measurements. On the other hand, a significant diffusion of ${}^{13}$C was observed into the isotope purified layer from both the substrate and the C-cap. A diffusivity of $D=8.3\times {10}^6{e}^{-10.4/k_{\mathrm{B}}T}$ cm${}^2$/s for ${}^{13}$C was extracted, in contrast to previous findings that yielded lower both pre-factors and activation energies for C self-diffusion in SiC. The discrepancy between the present measurements and previous theoretical and experimental works is ascribed to the presence of the C-cap, which is responsible for continuous injection of C interstitials during annealing, and thereby suppressing the vacancy mediated diffusion.

1. Introduction

Self-diffusion of atomic species is a fundamental process in semiconductors, and vital for understanding both the thermally induced melting process and diffusion of impurities which are introduced during growth and doping. In silicon carbide, self-diffusion has been investigated extensively using both experimental [1,2,3,4,5,6] and theoretical [7,8] methods, but with conflicting results. Initially, carbon self-diffusion was predicted to have an energy barrier of $E_{\mathrm{A}}=7.4$ eV in intrinsic 6H-SiC [2] using ${}^{14}$C radio-tracer techniques, with silicon diffusivities being more than 2 orders of magnitude lower than those for carbon [2,3]. Later, secondary ion mass spectrometry (SIMS) experiments on the 4H polytype yielded activation energies for ${}^{13}$C self-diffusion of $8.5$ eV [5] and $7.6$ eV [6]. Interestingly, the findings of Ref. [6] suggested that Si and C diffusivities are of comparable magnitudes, in contrast to that of Hong et al. [2] and Hon et al. [3]. Although the experiments provide hints as to the mediating species for C and Si self-diffusion in SiC, they are not conclusive, with Ref. [5] predicting vacancy-mediated C self-diffusion and Ref. [6] proposing an interstitial-mediated mechanism, both on the C sub-lattice. Moreover, reported diffusivity values for Si differ by several orders of magnitude. Note, however, that for high temperature annealing of SiC material and devices, a carbon cap is typically utilized in order to suppress the loss of surface atoms (in particular Si) and prevent surface roughening [9,10,11]. This may influence the migration mechanisms, further complicating the interpretation and understanding of self-diffusion in SiC.

The main intrinsic defects relevant for self-diffusion processes in SiC are the Si and C vacancies ($V_{\mathrm{Si}}$ and $V_{\mathrm{C}}$), interstitials (Si${}_i$ and C${}_i$), and antisites (Si${}_{\mathrm{C}}$ and C${}_{\mathrm{Si}}$). The antisite defects have proven challenging to detect experimentally in 4H-SiC, potentially due to their predicted lack of charge state transition levels in the upper portion of the band-gap [12]. The vacancies and interstitials, on the other hand, are all electrically active in 4H-SiC, and behave as deep electron or hole traps in n-type or p-type material, respectively [12,13,14]. Hence, their presence is strongly detrimental to power electronics applications, and $V_{\mathrm{C}}$ is in fact known to be a prominent minority lifetime killer in 4H-SiC [15]. Conversely, $V_{\mathrm{Si}}$ is a high-spin single-photon emitter, showing considerable promise for quantum technology applications [16].

Density functional theory (DFT) has been used to investigate the mechanism for Si and C self-diffusion in SiC. When assessing the predicted activation energy for self-diffusion using DFT, we proceed by considering each defect species individually. First, a Fermi level range relevant for the annealing temperature being used is chosen, then the relevant defect charge state is identified, and finally the corresponding formation energy and energy barrier for migration are combined to form the activation energy for self-diffusion. This was done for several candidate defects in SiC in, e.g., Ref. [7]. Note that defect migration is anisotropic in 4H-SiC, often with higher barriers for migration between than within lattice planes. For the mid-gap Fermi levels ($E_{\mathrm{F}}$) which manifest during the high-temperature heat treatments needed to induce migration in n-type 4H-SiC, $V_{\mathrm{C}}$ is expected to occupy the neutral or positive charge state [12,13], while $V_{\mathrm{Si}}$ will be negatively charged [14]. Both antisites are likely neutral, which is the case for the carbon interstitial as well, while Si${}_i$ defects reside in either the $q=2+$ or $q=4+$ charge state depending on whether they occupy a hexagonal (h) or pseudo-cubic (k) lattice plane in 4H-SiC, respectively [12].

Carbon vacancy migration was studied in-depth and for different migration paths using density functional theory in Ref. [17] in combination with the Heyd–Scuseria–Ernzerhof (HSE06) screened hybrid functional [18], the nudged elastic band [19,20] and dimer [21] methods, and comparing to experimental data, resulting in energy barriers for $V_{\mathrm{C}}^0$ migration of $3.7–4.2$ eV. In Ref. [22] Monte Carlo methods were instead employed, yielding migration barriers of $4.2–4.4$ eV for the neutral charge state, while studies using the Perdew–Burke–Ernzerhof (PBE) functional [23] yield lower values of $3.3–3.5$ eV for the same charge state of $V_{\mathrm{C}}$ [7,24]. Using the PBE functional is known to produce lower energies than, e.g., HSE06 as a result of underestimating the SiC band gap, which likely explains the lower values found in Refs. [7,24] as compared to Refs. [17,22]. Moreover, the calculations of Ref. [24] were not able to capture the full energetics of $V_{\mathrm{C}}$, as no negative-U characteristics were shown for the $V_{\mathrm{C}}$($0/2-$) transition [25], and Ref. [7] focused on the 3C polytype only. The HSE06-level values of Ref. [17] are therefore quoted for $V_{\mathrm{C}}^0$ migration herein, using the axial barriers for $V_{\mathrm{C}}$ diffusion along the c-direction.

Migration of $V_{\mathrm{Si}}$ is an emerging popular focus [22,24,26] due to the qubit potential of the Si vacancy [16]. Under intrinsic conditions, $V_{\mathrm{Si}}$ inhabits the negative charge state, with predicted inter-planar migration barriers of $3.2–4.2$ eV [22,24,26]. Interstitial defects are somewhat less studied than vacancies in 4H-SiC, perhaps on account of their greater complexity compared to the vacancies with more fixed lattice positioning. Migration barriers ($E_{\mathrm{A}}$ for migration) for both Si and C interstitials were estimated at $\sim 1$ eV in Ref. [27], but lattice configuration and Fermi level dependence were not discussed. Refs. [7,24] incorporated the charge state dependence of $E_{\mathrm{A}}$ for migration of Si${}_i$ and C${}_i$, yielding barriers between $0.1–2$ eV for both defect species and different migration paths. Using PBE and combining the formation and migration energies, Ref. [7] estimated activation energies for C self-diffusion in 3C-SiC (for $q=0$) at around 7 eV via both C vacancies and interstitials, while that for Si was accompanied by higher barriers in the $8$–12 eV range. An even more complete picture was painted in Ref. [8], where self-diffusion of C and Si via their respective self-interstitials was studied while incorporating both temperature and entropy effects, although for 3C-SiC only. Here, activation energies for self-diffusion of carbon via C${}_i$ in 3C-SiC were estimated at $7.1$ eV and $7.8$ eV under C rich conditions for the neutral and doubly positive charge state, respectively. Si self-diffusion via Si${}_i$ was accompanied by much higher barriers, in the $10$–11 eV range [8].

The available defect species for mediating self-diffusion in 4H-SiC, their charge state(s) when $E_{\mathrm{F}}$ resides at mid-gap, and the corresponding formation energy and migration barrier as collected from the literature are summarized in

Table 1. Calculated activation energies ($E_{\mathrm{A}}$) for self-diffusion collected from the literature, defined as the sum of the formation energy and energy barrier for migration, via the various defect species available in 4H-SiC. During high-temperature annealing, the Fermi level ($E_{\mathrm{F}}$) is located near mid-gap in the 4H-SiC samples, and the formation energy is quoted for this value of $E_{\mathrm{F}}$. All formation energies are, wherever possible, quoted for Si poor and C rich conditions. Formation energies are quoted for HSE06 calculations, while the migration barriers are mostly deduced at PBE-level.

Defect Type	q When ${\mathit{E}}_{\mathbf{F}}=$ Mid-Gap	Formation Energy (eV)	Migration Barrier (eV)	${\mathit{E}}_{\mathbf{A}}$ For Self-Diffusion
$V_{\mathrm{C}}$	+ or 0	$4.8$ eV [13,17]	$4.2$ eV [17]	$9.0$ eV, $7.9$ eV [7] (3C)
C${}_i$	0	$6.3$ eV [12], $6.5$ eV [24]	∼1 eV [27], $0.9$ eV [24]	$7.5$ eV, $7.1$ eV [8] (3C), $6.6$ eV [7] (3C)
C${}_{\mathrm{Si}}$	0	3 eV [24]	∼10 eV [24]	13 eV
$V_{\mathrm{Si}}$	−	$7.0$ eV [14]	$4.2$ eV [26], 3 eV [22]	$10–11$ eV
Si${}_i$(h)	$2+$	$8.5$ eV [12]	∼1 eV [27]	∼9.5 eV, $10.2$ eV (3C) [8]
Si${}_i$(k)	$4+$	$9.5$ eV [12]	∼1 eV [27], $1.8$ eV [24]	∼10.5 eV, ∼11 eV (3C) [8]
Si${}_{\mathrm{C}}$	0	5 eV [24]	∼10 eV [24]	15 eV

. C self-diffusion via C interstitials and vacancies appears to be separated by an energy difference of up to $1.5$ eV, while $E_{\mathrm{A}}$ for Si self-diffusion is predicted in the $10–11$ eV range regardless of whether Si${}_i$ or $V_{\mathrm{Si}}$ is responsible for mediation. Notice also that antisite defects are accompanied by large activation energies for self-diffusion around $13–15$ eV, which translates into the interstitial and vacancy defects being more likely candidates for mediating thermally induced Si and C redistribution from their lattice sites in SiC.

In the present work, we revisit the concept of self-diffusion in the 4H SiC polytype. By studying the motion of both C and Si within the same sample, we attempt to establish whether Si self-diffusion enthalpies are comparable to those of C or not, as the literature is not conclusive in this regard. An important feature of the present study is the use of a carbon cap (C-cap) covering the 4H-SiC surface during annealing to reduce the loss of Si from the surface and thereby suppressing surface roughening. However, the C-cap also presents a continuous injection channel for C interstitials from the C-cap/4H-SiC interface towards the 4H-SiC bulk. By comparing the present and previous experiments, we find that the mechanism for self-diffusion is significantly affected by the presence of the C-cap. In fact, the pre-factors for diffusion are several orders of magnitude higher when the C-cap is present as compared to the cap-free case, indicating a mechanism involving Frenkel pair formation in the sample bulk for self-diffusion.

2. Methods

An epitaxial layer (epi-layer) of 4H-${}^{28}$Si${}^{12}$C silicon carbide was grown by chemical vapor deposition (CVD) on a n-type (0001) 4H-SiC substrate at the University of Linköping. The CVD-grown layer is ∼7 $\mathsf{\mu }$m thick, and the isotope purity is estimated to be ${}^{28}$Si $>99.85\%$ and ${}^{12}$C $>99.98\%$. A C-cap was applied to all samples prior to annealing by graphitizing a photoresist film at 900 ${}^{\circ }$C on the sample surface, in order to avoid surface roughening during the elevated heat treatments. The samples were subjected to high temperature treatments employing an inductively heated furnace (RFA) holding Ar atmosphere in the temperature range $2300–2450$ ${}^{\circ }$C. A Cameca IMS7f Secondary Ion Mass Spectrometer (SIMS) equipped with an O${}_2$ primary ion beam source was used to measure the concentration versus depth profiles of Si and C. Absolute concentration values were obtained by assuming natural abundance in the substrate. For depth calibration, the sputtered crater depths were measured by a Dektak 8 stylus profilometer and a constant erosion rate was assumed.

The diffusion profiles were fitted according to the error function

$$c=c_2+\frac{c_1-c_2}{2}\mathrm{erfc}\left\{\frac{x-d}{2\sqrt{Dt}}\right\}$$

(1)

to extract the ${}^{13}$C diffusivities (D) at each temperature, where c is the concentration, t the annealing time, x denotes the depth and d the interface position. Moreover, $c_1$ and $c_2$ define the boundary conditions at $t=0$ for $x-d\le 0$ and $x-d>0$, respectively. Finally, the diffusion parameters for the migrating species were determined according to an Arrhenius model

$$D=D_0{e}^{-E_{\mathrm{A}}/k_{\mathrm{B}}T},$$

(2)

with $D_0$ being the exponential pre-factor for diffusion, $E_{\mathrm{A}}$ the activation energy for self-diffusion, $k_{\mathrm{B}}$ the Boltzmann constant, and T the annealing temperature.

Surface roughness will broaden the measured SIMS profiles. To account for this extra broadening, the surface irregularities have been included as root mean square (rms) values and the measured diffusion coefficients have been corrected according to

$${\sigma }_D=\sqrt{{\sigma }_M^2-{\mathrm{rms}}^2},$$

(3)

where $\sigma =\sqrt{2Dt}$. M and D stand for measured and corrected, respectively. Note that extraction of diffusivity values was performed for the samples annealed for the longest times only, in order to ensure steady-state conditions.

3. Results

An illustration of the sample geometry prior to heat treatments, by means of ${}^{13}$C concentration profiles as detected by SIMS, is depicted in Figure 1. The figure reveals a ${}^{13}$C-lean epi-layer with both the C-cap and substrate acting as ${}^{13}$C sources. Please note that the SIMS signal in the C-cap layer has not been normalized, and that no correction was performed for the lower carbon ionization yield in the pure carbon (C-cap) matrix relative to the 4H-SiC matrix. The lower ionization yield in the graphitized layer compared to 4H-SiC explains the differences in carbon concentration between the C-cap and 4H-SiC layers in Figure 1. Moreover, one should expect slightly different sputtering rates in the SIMS measurements for the C-cap and 4H-SiC material layers.

Figure 2 displays the temperature-dependent ${}^{13}$C profiles as deduced from SIMS measurements after sequential annealing steps. The ${}^{13}$C profiles are shown after isothermal annealing at 2300 ${}^{\circ }$C for $15–230$ min at the interface between the 4H-SiC substrate, which has a natural abundance of ${}^{13}$C, and the ${}^{12}$C isotope purified epitaxial layer in Figure 2a, at approximately 7 $\mathsf{\mu }$m from the 4H-SiC epi-layer surface. The same interface is studied in Figure 2b, but for 15 min annealing steps in the temperature range $2300–2450$ ${}^{\circ }$C. Figure 2c showcases the ${}^{13}$C profiles at the C-cap/4H-SiC interface for isochronal annealing at temperatures between $2300–2450$ ${}^{\circ }$C for 15 min. Indeed, Figure 2 demonstrates that C diffuses readily at both the internal 7 $\mathsf{\mu }$m interface (4H-SiC epi-layer/substrate) and at the C-cap/4H-SiC interface above a temperature of ∼2300 ${}^{\circ }$C. The profiles are fitted reliably by Equation (1), as indicated by the dotted data points and the solid fit lines in Figure 2. Due to the different ionization levels in the C-cap and 4H-SiC layers and accompanying challenges with normalization, the fitted concentration profiles for ${}^{13}$C are only shown on the 4H-SiC epi-layer side of the interface depicted in Figure 2c. Note that the placing of the C-cap/4H-SiC interface adds some additional uncertainty to the extracted diffusivities for C moving from the C-cap layer to the isotope purified 4H-SiC material.

The corresponding ${}^{30}$Si depth profiles are shown in Figure 3 at the 4H-SiC substrate/isotope purified epi-layer interface. The time dependence of Si motion at 2300 ${}^{\circ }$C is highlighted in Figure 3a, and temperature dependence in Figure 3b for 15 min isochronal annealing between $2300–2450$ ${}^{\circ }$C. In contrast to the ${}^{13}$C case, the ${}^{30}$Si profiles exhibit a small redistribution only, i.e., any self-diffusion observed for Si within the studied temperature range is significantly smaller than that found for C. Indeed, the redistribution of ${}^{30}$Si observed for, e.g., the 2300 ${}^{\circ }$C sample is similar to that measured at a reference sample (not annealed) with a C-cap (not shown in Figure 3). Note that the shoulder in the ${}^{30}$Si profiles extending from the interface towards the surface, e.g., exhibiting a ${}^{30}$Si concentration of $0.3$ at $-0.2\mathsf{\mu }\mathrm{m}$ for the 230 min sample in Figure 3a, likely arises due to pinholes manifesting in the C-cap at longer annealing times and high temperatures. The pinholes were unavoidable in our SIMS experiment, but are not considered with respect to the diffusion behavior. An alternative explanation for the Si shoulder in Figure 3a related to transient diffusion is discarded, as the same profile shapes are obtained for a reference sample with a C-cap and samples annealed during 15 and 60 min. Most likely, the observed ${}^{30}$Si profile broadening is caused by variations in the thickness and surface structure of the C-cap film. Unfortunately, for long annealing times (230 min), the C-cap starts to degrade.

Estimated diffusivities (D) for ${}^{13}$C (at the two interfaces) and ${}^{30}$Si (at the internal 4H-SiC interface) are collected in

Table 2. Estimated diffusivities for ${}^{13}$C and ${}^{30}$Si migration in 4H-SiC samples under intrinsic conditions and at various temperatures. ${}^{13}$C profiles are collected both at the interface 7 $\mathsf{\mu }$m into the epi-layer, and at the C/4H-SiC surface, while ${}^{30}$Si diffusivities are from the internal 4H-SiC interface only.

T (${}^{\circ }$C)	t (min)	D (cm${}^2$/s)	D (cm${}^2$/s)	D (cm${}^2$/s)
		${}^{13}$C	${}^{13}$C	${}^{30}$Si
		7 $\mathsf{\mu }$m into epi	SiC/C Interface	7 $\mathsf{\mu }$m into epi
2300	15	$6.63\times {10}^{-14}$	$4.45\times {10}^{-14}$	$6.3\times {10}^{-16}$
2300	60	$4.45\times {10}^{-14}$	$3.15\times {10}^{-14}$	$1.6\times {10}^{-16}$
2300	230	$4.45\times {10}^{-14}$	$3.02\times {10}^{-14}$	$2.0\times {10}^{-16}$
2350	15	$9.68\times {10}^{-14}$	$8.90\times {10}^{-14}$	$2.6\times {10}^{-15}$
2400	15	$4.59\times {10}^{-13}$	$3.94\times {10}^{-13}$	$7.9\times {10}^{-16}$
2400	101	$1.97\times {10}^{-13}$	$1.30\times {10}^{-13}$	$4.2\times {10}^{-15}$
2450	15	$6.18\times {10}^{-13}$	$3.80\times {10}^{-13}$	$4.2\times {10}^{-15}$

. The D-values are extracted from fits to the experimental data presented in Figure 2 and Figure 3 using Equation (1). Indeed, if one assumes that Si migration is observed in Figure 3, the Si diffusivities are at least a factor $100–1000$ times lower than those for C, which agrees reasonably well with previous studies [2,5,6,28,29]. However, the high-temperature heat treatments induce surface roughening effects that complicate measurements of the Si and C depth distributions. For the 2300 ${}^{\circ }$C samples, the peak-to-peak roughness is about 20 nm, which is further increasing to around 30 nm for the 2400 ${}^{\circ }$C samples. From the estimated diffusivities extracted for Si from Figure 3 (see Table 2), we infer diffusion lengths for Si in the $20–30$ nm range. Hence, the surface roughness is in the same range as the observed ${}^{30}$Si diffusion lengths, meaning that the estimated ${}^{30}$Si diffusivities must necessarily be of the same order as the uncertainty arising from the surface roughening. Therefore, the ${}^{30}$Si D-values and redistribution depicted in Table 2 and Figure 3 must be considered as an upper bound. Moreover, due to the overlap between ${}^{30}$Si diffusion lengths and surface roughening, the estimates for $D_0$ and $E_{\mathrm{A}}$ for Si self-diffusion are highly uncertain and will therefore not be quoted herein. The redistribution of ${}^{13}$C, on the other hand, is above the surface roughness and therefore adequately represents the carbon migration. From Figure 2 and Figure 3 combined with the above discussion, we thus infer that carbon migrates readily in 4H-SiC within a $2300–2450$ ${}^{\circ }$C temperature interval while silicon does not.

Figure 4 displays the Arrhenius behavior of the carbon and silicon diffusivity values as summarized in Table 2. The diffusivities extracted for SiC samples employed both herein (No. 5 and 11, brown data points and fit) and in other works [2,5,6,28,29] are included in Figure 4, with the corresponding temperature range, migrating species, $D_0$, $E_{\mathrm{A}}$ value and reference collected in

Table 3. Collection of Arrhenius data for C and Si self-diffusion from the literature and the present work (No. 5 and 11), see also Figure 4.

No.	T Interval (${}^{\circ }$C)	Conditions	Isotope	${\mathit{D}}_0$ (cm${}^2$B)	${\mathit{E}}_{\mathbf{A}}$ (eV)	Reference
1	$1850–2180$	Si-rich, 6H	${}^{14}\mathrm{C}$	$8.6\times {10}^5$	$7.41$	[2]
2	$1850–2180$	Si-rich, 6H	${}^{14}\mathrm{C}$	$3.3\times {10}^7$	$8.20$	[2]
3	$2000–2290$	Si-rich, 6H	${}^{30}\mathrm{Si}$	$1.5\times {10}^5$	$8.18$	[2]
4	$2000–2290$	Si-rich, 6H	${}^{30}\mathrm{Si}$	$5.0\times {10}^2$	$7.22$	[2]
5	$2300–2450$	4H	${}^{13}\mathrm{C}$	$8.3\times {10}^6$	$10.4$	Present work
6	$2100–2350$	4H	${}^{13}\mathrm{C}$	$8.4\times {10}^2$	$8.50$	[5]
7	$2000–2450$	4H	${}^{13}\mathrm{C}$	$4.8\times {10}^{-2}$	$6.7$	[28]
8	$2000–2200$	4H	${}^{13}\mathrm{C}$ and ${}^{30}\mathrm{Si}$	$4.8$	$7.6$	[6] Deduced in combination with B-diffusion
9	$1950–2200$	Probably 6H	Si	$7.3\times {10}^{-5}$	$5.4$	[29] Deduced from B-diffusion
10	$2000–2450$	4H	${}^{30}\mathrm{Si}$	$3.8\times {10}^4$	$10.6$	[28]
11	$2300–2450$	4H	${}^{30}\mathrm{Si}$	Uncertain	Uncertain	Present work

. Note that the data points used for the fit for No. 10 (data and fit collected from Ref. [28]) were chosen based on an upper limit for surface roughening of 15 nm.

Importantly, our results (Table 3 and Figure 4) exhibit a significantly higher activation energy ($E_{\mathrm{A}}^{\mathrm{C}}=10.4$ eV) and larger pre-exponential factor ($D_0^{\mathrm{C}}=8.3\times {10}^6$ cm${}^2$/s) for self-diffusion of C compared to that reported in the literature. In addition, from the extracted diffusivities collected in Table 2, we observe that the diffusivity of C decreases with time for the 2300 ${}^{\circ }$C annealing, and that D is lower at the C/4H-SiC surface than at the internal interface between the CVD-grown epi-layer and the substrate at 7 $\mathsf{\mu }$m depth. It should be noted that the data points used for representing the ${}^{13}$C diffusivities deduced in the present work, and presented as dark brown data points (No. 5) in Figure 4, were collected at the interface located $7\mathsf{\mu }$m into the 4H-SiC epi-layer (third column of Table 2), and not at the C-cap/4H-SiC surface.

4. Discussion

First, let us consider the mechanisms for self-diffusion of Si and C in SiC, that is, the sub-lattice on which migration occurs and the mediating species for migration. Diffusion via antisites (Si${}_{\mathrm{C}}$ and C${}_{\mathrm{Si}}$) is expected to be negligible due to the high activation energies for self-diffusion via antisites, as highlighted in Table 1. Thus, we anticipate that both Si and C self-diffuse strictly on their individual sub-lattice. Importantly, self-diffusion via interstitials would first necessitate the formation of Frenkel pairs, which has not been directly calculated in any of the cited works. Self-diffusion via vacancies, on the other hand, does not have to involve Frenkel pairs, as vacancies can be injected from the surface and thereby present a means of travel for the lattice atoms. We thus conjecture that injection of vacancies from the surface, followed by Si or C atoms from the lattice propagating via the vacancies throughout the lattice, is likely associated with a lower total activation energy than Frenkel pair formation in the semiconductor bulk.

The experimental findings of the present work show a strong discrepancy when comparing to previous reports, and provide an important hint towards illuminating the process of self-diffusion in SiC. Indeed, the barrier for C self-diffusion found herein ($E_{\mathrm{A}}^{\mathrm{C}}=10.4$ eV) shows agreement with neither previous experimental data (see Figure 4) nor theoretical predictions for migration via vacancies, interstitials and antisites on the C sublattice (see Table 1). Note that the high estimated pre-factor for C self-diffusion ($D_0^{\mathrm{C}}=8.3\times {10}^6$ cm${}^2$/s) evidences that we are not, in fact, monitoring diffusion solely via fundamental nearest-neighbor atomic hops [30]. However, an intriguing difference between previous experiments and the present ones is the presence of a C-cap combined with a lower as-grown vacancy concentration, as the epitaxial growth process has improved over the years. Importantly, using a C-cap is at present a standard processing step in device fabrication, with obvious implications for the C diffusivity as carbon interstitials are injected into the material.

In Ref. [31] (on GaAs), it was proposed that self-diffusion processes with pre-factors in the ${10}^8–{10}^9$ cm${}^2$/s range indicate a self-diffusion mechanism related to Frenkel pair formation in the semiconductor bulk, which is expected to be dominant when vacancy formation is suppressed at the surface. Thus, with $D_0$ for C far above the ${10}^{-3}–{10}^{-2}$ cm${}^2$/s expected for fundamental hops [30], it appears evident that the present findings are not representative of undisturbed self-diffusion in SiC. Instead, our results illuminate the competition between Frenkel pair formation in the bulk, and vacancy injection from the surface, as mechanisms for self-diffusion. Previous works have shown that the C-cap is responsible for continuous injection of C interstitials during high-temperature annealing [32], creating a continuous route for recombination of C vacancies that might be generated at the 4H-SiC surface under the high-temperature conditions. Moreover, the idea that C interstitials are being injected from the surface is further supported by the difference in measured C diffusivity at the surface (C-cap/4H-SiC interface) and in the bulk (isotope-pure epi-layer/substrate interface). Therefore, when there is a C-cap covering the sample surface during annealing, self-diffusion via injected vacancies is suppressed, instead pointing towards Frenkel pair formation in the bulk as the dominant route for mediating atomic redistribution. Si self-diffusion may also be impacted by the C-cap, e.g., via formation of immobile antisites.

The present experimental approach is distinctly different from that of, e.g., studying diffusion using self-implantation. For example, in Ref. [17] we tentatively attributed the mediating species for C self-diffusion in SiC to C${}_i$. However, in samples without the C-cap, there is no permanent supply of C${}_i$ recombining with vacancy defects at the surface. Therefore, we can briefly consider both $V_{\mathrm{C}}$ and C${}_i$ as the responsible party for C self-diffusion, with $E_{\mathrm{A}}$ for self-diffusion via $V_{\mathrm{C}}$ being predicted at $9.0$ eV and $E_{\mathrm{A}}$ via C${}_i$ at $7.5$ eV (see Table 1). Interestingly, both values resonate well with experiment, albeit different ones; Ref. [5] predicted an $E_{\mathrm{A}}$ for ${}^{13}$C motion of $8.5$ eV, while Ref. [6] deduced that $E_{\mathrm{A}}=7.4$ eV. Indeed, the lower pre-factors for diffusion in Refs. [5,6] as compared to the present work indicate that Frenkel pair formation does not dominate in the un-capped samples, and instead we attribute C and Si self-diffusion to the formation and injection of Si and C vacancies at the surface. In fact, injection of C${}_i$ from the C-cap would delay C self-diffusion if it occurred via C vacancies, and one would expect a significant delay in C migration upon the recombination between the injected C${}_i$ and the thermally generated $V_{\mathrm{C}}$, which would thus prevent further motion of ${}^{13}$C atoms. A similar mechanism can be proposed for Si self-diffusion via $V_{\mathrm{Si}}$, if the injected C${}_i$ form nearly immobile C${}_{\mathrm{Si}}$ antisites. The injection of C${}_i$ from the C-cap therefore explains both the higher activation energies and the higher pre-factors for self-diffusion deduced for C (and Si) in the present work.

5. Concluding Remarks

We investigate the effect of C injection from a carbon cap during high-temperature annealing on self-diffusion of C and Si in 4H-SiC. The continuous excess of C interstitials results in recombination of the injected C interstitials with Si and C vacancies generated in the near-surface region. Hence, the presence of the C-cap promotes the formation of Frenkel pairs in the SiC bulk as opposed to the injection of vacancies from the semiconductor surface, as evidenced by the elevated exponential pre-factors for C self-diffusion found herein as compared to previous works. In conclusion, we tentatively attribute C self-diffusion in un-capped samples to the injection of carbon vacancies from the SiC surface. The case for Si, however, is less straightforward, with predicted activation energies for Si self-diffusion overlapping regardless of whether Si vacancies or interstitials act as the mediating species. Thus, resolving Si self-diffusion on the Si sub-lattice is challenging, but following the argument for the C case, injection of Si vacancies from the surface likely outcompetes Si Frenkel pair formation when a C-cap is not present.