Transition state redox during dynamical processes in semiconductors and insulators

Activation barriers associated with ion diffusion and chemical reactions are vital to understand and predict a wide range of phenomena, such as material growth, ion transport, and catalysis. In the calculation of activation barriers for non-redox processes in semiconductors and insulators, it has been widely assumed that the charge state remains fixed to that of the initial electronic ground state throughout a dynamical process. In this work, we demonstrate that this assumption is generally inaccurate and that a rate-limiting transition state can have a different charge state from the initial ground state. This phenomenon can significantly lower the activation barrier of dynamical process that depends strongly on charge state, such as carbon vacancy diffusion in 4H-SiC. With inclusion of such transition state redox, the activation barrier varies continuously with Fermi level, in contrast to the step-line feature predicted by the traditional fixed-charge assumption. In this study, a straightforward approach to include the transition state redox effect is provided, the typical situations where the effect plays a significant role are identified, and the relevant electron dynamics are discussed.


INTRODUCTION
Macroscopic dynamical phenomena ranging from ion diffusion to chemical reactions are frequently understood and predicted by analyzing the transition states (TSs) of elementary dynamical processes occurring in extended systems, such as in bulk materials or on bulk material surfaces. A widely-used assumption [1][2][3][4][5][6][7] in TS modeling is that the charge state remains fixed during a non-redox process, such as the ion diffusion. Unlike metallic systems, the Fermi level (E F ) in the ab-initio modeling of an extended semiconducting or insulating system is typically not at the value of the real system, so electron exchange with the bulk states associated with E F must be included by explicit postprocessing. Therefore, the fixed-charge assumption prohibits potential electron exchange between the bulk states and the local region where a dynamical process takes place.
In this article, we remove the fixed-charge assumption and relax the TS charge state to obtain the lowest activation barrier for defect/impurity diffusion in semiconductors and insulators as concrete illustrations. Our density functional theory (DFT) computations confirm that it is energetically favorable for the TS to exchange electrons with the bulk states in several diffusion processes. By allowing such TS redox, the activation barrier is lowered and the E F dependence of activation barrier becomes continuous. We compare these computational results with available diffusion experiments, analyze the magnitude of the correction associated with the method proposed in this work, and discuss the electron dynamics during the TS redox.

MODELS AND METHODS
Our ab initio calculations are carried out using DFT as implemented in the Vienna ab initio Simulation Package (VASP). 8 An energy cutoff of 400, 450, and 300 eV is set to the plane-wave basis sets for GaAs, 4H-SiC, and Si systems, respectively, and the following projector-augmented wave potentials are utilized: Ga_GW(4s 2 4p 1 ) for Ga, As_GW(4s 2 4p 3 ) for As, Si_GW(3s 2 3p 2 ) for Si, C_GW (2s 2 2p 2 ) for C, and Li_GW (2s 1 ) for Li. The HSE06 9 hybrid functional is employed, which predicts the band gaps of GaAs, 4H-SiC, and Si to be 1.38, 3.16, 1.15 eV, in good agreement with experimental values 10, 11 of 1.42, 3.26, 1.12 eV at 300 K, respectively. The k-point sampling is a 3 × 3 × 3 Monkhorst-Pack grid for the GaAs, 4H-SiC, and Si supercells with a = b = c = 11.20 Å, a = b = 9.21 Å and c = 10.04 Å, and a = b = c = 10.87 Å, respectively. For defective systems, the defect content is one defect per supercell. The ab initio method proposed by Freysoldt, Neugebauer and Van de Walle (FNV) 12 is adopted to remove the image charge interaction and adjust the potential alignment between the perfect and defected structures.
The defect formation energy E f of a defect D with charge state q is defined as Eqn. 1, 13 Second, we identify the lowest formation energy curve among these different charges of the TS over the entire range of E F in the band gap following the same procedure of determining the lowest defect formation energy curve for stable states. Third, we take the difference between the lowest TS energy curve and the initial state energy curve as the E F dependence of the lowest activation barrier. Two notable consequences of the above relaxed-charge approach are that (1) the activation barrier must be continuous as a function of E F , because the energy curves of TS and initial state are both continuous; (2) the activation barrier must be equal or lower than that obtained from the fixed-charge assumption, because the TS here has the lowest energy in the entire range of E F .

RESULTS
We computationally examine three diffusion processes: gallium vacancy, V Ga , diffusion in GaAs; carbon vacancy, V C , diffusion in 4H-SiC; and lithium interstitial, Li i , diffusion in silicon. These systems are chosen because they have been extensively studied and thus serve as ideal models to verify our predictions. Additionally, they exemplify three different cases how a TS charge state can differ from that of the initial state, which include electron loss, electron gain, or no electron exchange.
the region of 0 ≤ E F ≤ 0.28 eV according to a previous study 18

and the complex diffusion in this range
is not investigated here. The TS curve by the relaxed-charge method shows different charge states from those of the stable state in certain regions, indicating that electron exchange of TS with the bulk will occur in these regions. We predict that the electron exchange varies with the E F : specifically, the TS state loses one, then two, and one electron relative to the initial state in the ranges of 0. 38   Li i diffusion in Si. Figure 3a shows that Li i has two stable charge states in bulk Si, + and 0, with a shallow defect level 0.07 eV below the CBM, which is in excellent agreement with the experimental value of 0.03 eV below the CBM 31 . In contrast to the V Ga in GaAs and V C in 4H-SiC, Li i and its TS possess almost the same charge states (Fig. 3a) and therefore the fixed-and relaxed-charge approaches yield almost the same activation barriers. Figure 3b shows that the hopping barriers of

Magnitude of correction by the relaxed-charge method. To gain insights into when the TS redox
is likely to play a significant (minor) role, we compare the correction of hopping barrier by the relaxed-charge method with the discontinuity of hopping barrier predicted by the fixed-charge method. Figure 4 shows that the maximum correction of hopping barrier generally increases with the Li i discontinuity. In other words, the relaxed-charge method plays a significant (minor) role when the fixed-charged method predicts a strong (weak) charge dependence. Physically, this correlation arises from the fact that a discontinuity is caused by forcing the TS into an incorrect charge state, so a larger discontinuity corresponds to a larger error in the TS energy and more reduction when this error is corrected.
It is unclear what combination of migrating species and host material would exhibit strong charge dependence, although it is known that a charge state can impact activation energy through the change of ionic size and bonding characteristics. A previous study 34  , an additional mechanism that involves the variation from initial state q 1 to the TS q 2 and back to final state q 1 will be invoked, which can be realized through electron emission/absorption during the ionic hop. Both mechanisms predict the same collective diffusion barrier as that based solely on thermodynamic equilibrium. Note that the electron emission/absorption during a hop is not equivalent to the charge redistribution caused by rehybridization in a traditional TS calculation. The rehybridization may play a significant role in determining the activation energy, such as for oxygen vacancy hopping in some perovskite oxides, 42 but it does not involve electron emission/absorption and does not require the special treatments described in this work.     Here, we show that the TS is insensitive to the charge states of the initial and final states for at least our examined defects (V Ga in GaAs, V C and V Si in 4H-SiC, and Li i in Si). As shown in Table S1, TSs obtained from different charge states of initial states differ by energies less than 8 meV. As a result, the total number of calculations can be reduced from m×n to roughly m+n by doing only one calculation for each TS charge state. Table S1. Maximum energy differences, δ (meV), among TSs calculated using different initial states. E F is set to valance band maximum; q TS is the charge state of TS and q initial is the examined range of charge sates of initial states.
V Ga hopping in GaAs V C hopping in 4H-SiC V Si hopping in 4H-SiC Li i hopping in Si  Figure S1 shows that for all the examined initial states, the largest atomic displacement induced by different charge states is less than 0.09 Å. These small structural differences also lead to relatively small energy differences. For example, the total energy of V Ga -(V Ga 2-) using the relaxed geometrical structure of V Ga 2-(V Ga -) is only 70 (90) meV higher than that obtained from the relaxed structure of V Ga -(V Ga 2-).
Therefore, a defect with different charge states are in the same energy basin, and different initial/final states are expected to lead to the same TS, at least in theory.  Atomic displacement (Å)

Results of V Si in 4H-SiC
V Si is predicted to possess three defect levels at 1.28, 2.39, and 2.89 eV above the valance band maximum (Fig.   S2a). The hopping barriers in Fig. S2b show that the fixed-charge method predicts a step-line curve with three jumps of 0.63, 0.46, and 0.35 eV at the boundaries of 0/−, −/2− and 2−/3−, respectively. By contrast, the relaxedcharge approach lowers the hopping barrier and smoothens the curve in the entire region. We predict that the TS gains one electron relative to the initial state in three ranges: 0.65 < E F ≤ 1.28 eV, 1.93 < E F ≤ 2.39 eV, and 2.54 < E F ≤ 2.89 eV. Based on the well-known relationship of Eqn. S1 1 , the effective density-of-state masses, and the band gap relationship with temperature, the calculated E F at particular temperatures for intrinsic GaAs, 4H-SiC, and Si are listed in Table S2. * Table S3. Chemical potentials µ (eV) of Ga, C and Li under As-rich, C-rich, and Li-rich conditions for the energy calculations of V Ga in GaAs, V C and V Si in 4H-SiC, and Li i in Si. µ Ga = E tot (per formula of GaAs bulk) -E tot (per atom of hexagonal As bulk); µ C = E tot (per atom of graphite); µ Si = E tot (per formula of 4H-SiC bulk) -E tot (per atom of graphite); µ Li = E tot (per atom of body-centered cubic Li bulk).  Figure S3. Relationship between activation barrier change, δE b , and defect level shift, δE l , in two charge states. The activation barrier of charge state q1 (q2) is E b q1 (E b q2 ) and the angle between the segment q1 (q2) relative to the horizontal axis is θ q1 (θ q2 ).