Computational study of Mg insertion into amorphous silicon: advantageous energetics over

crystalline silicon for Mg storage

Fleur Legrain1, Oleksandr I. Malyi

1, Teck L. Tan

2, and Sergei Manzhos

1

1 Department of Mechanical Engineering, National University of Singapore, Block EA #07-08, 9

Engineering Drive 1, 117576, Singapore

2 Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis, 138632,

Singapore

Email: mpemanzh@nus.edu.sg

ABSTRACT

We show in a theoretical density functional theory study that amorphous Si (a-Si) has

more favorable energetics for Mg storage compared to crystalline Si (c-Si). Specifically, Mg and

Li insertion is compared in a model a-Si simulation cell. Multiple sites for Mg insertion with a

wide range of binding energies are identified. For many sites, Mg defect formation energies are

negative, whereas they are positive in c-Si. Moreover, while clustering in c-Si destabilizes the

insertion sites (by about 0.1/0.2 eV per atom for nearest-neighbor Li/Mg), it is found to stabilize

some of the insertion sites for both Li (by up to 0.27 eV) and Mg (by up to 0.35 eV) in a-Si. This

could have significant implications on the performance of Si anodes in Mg batteries.

INTRODUCTION

The development of high-power energy density and high-rate electrochemical batteries is

key to sustainable development. They will enable large-scale storage of electricity derived from

intermittent sources (such as wind and solar) as well as their use to directly power machinery and

electronics. Li-ion batteries provide today the highest energy density among commercial

batteries (up to 200 Wh/kg at rates of fractions of 1C) at hundreds or thousands of

charge/discharge cycles [1], but even higher energy density and/or charge/discharge rates will be

needed to achieve requirements for all-electric vehicles and bulk storage. For high energy density

applications, Mg is a promising alternative to Li. Mg is divalent and cheaper/more abundant than

Li, of which the deposits and their geographic distribution may not be compatible with massive-

scale battery applications [2]. Commercialization of Mg-ion batteries is believed to be possible in

the near future [3], but their good cyclability remains a problem. Reactions between anode and

electrolyte are one of reasons for this. While development of new electrolytes is one way to

alleviate this issue [4], the development of alternatives, specifically insertion type anodes, is also

important for making Mg-ion batteries a viable commercial technology [5-7]. Computational

studies are necessary for the rational design of prospective anode materials.

Among prospective anode materials for metal-ion batteries, Si has emerged as a very high

capacity anode for Li-ion batteries [8] and has been considered for Na and Mg-ion batteries [7,

9-11]. It has been shown that the theoretical capacity of Si for Mg-ion batteries (3817 mAh/g) [7]

is comparable to that of Li-ion batteries (4200 mAh/g) [8]. Most theoretical studies to date

focused on c-Si (diamond structure) [7, 9-14]. In c-Si, metal atoms occupy tetrahedral (T) sites

with computed defect formation energies (vs. vacuum reference states) ranging -1.52...-0.94 eV

for Li and +0.32…+0.94 eV for Mg [7, 9-14]. This is of significant interest for practical

applications, as it was reported that despite the negative formation heat of SiMg2, the positive

defect formation energy of Mg defects in c-Si results into limited insertion of Mg atoms and

makes theoretical capacity unreachable under realistic battery operating conditions [9]. Previous

work has also highlighted that clustering (occupation of neighboring T sites) is unfavorable but

leads to a significant reduction of diffusion barriers [7, 10, 11, 13-15]. Specifically, for Mg

insertion, the cost of clustering was found to be the highest among Li, Na, Mg at about 0.2 eV [7,

10].

Li insertion into a-Si has been studied in Refs. [16-21] and much higher diffusivity

compared to c-Si was predicted [17, 21]. In Ref. [17], a number of stable insertion sites (by

“stable sites” we mean local minima on the potential energy surface located between Si atoms)

with widely varying formation energies, ranging about 1.5 eV, were identified together with a

complex diffusion path passing through low-as well as higher-energy sites. There are no studies

of Mg insertion into a-Si. Here, we therefore address the following questions: what are the

insertion sites for Mg insertion into a-Si and how does their energetics compare to Li insertion

and to c-Si? What are the implications for the possible use of a-Si as the anode in Mg batteries?

THEORETICAL AND COMPUTATIONAL METHODS

The electronic structure was computed using DFT (density functional theory) [22] and

the SIESTA code [23]. The PBE exchange-correlation functional [24] and the DZP basis set

(double- polarized orbitals) were used. We used a standard basis set as generated by SIESTA,

but the cut-off radii were increased from the default values by choosing Eshift = 0.01 Ry. A cutoff

of 100 Ry was used for the Fourier expansion of the density, and Brillouin-zone integrations

were done with a 3×3×3 k-point Monkhorst-Pack mesh [25]. Core electrons were treated within

the effective core approximation with Troullier-Martins pseudopotentials [26] (provided with

SIESTA). Geometries were optimized until forces on all atoms were below 0.02 eV/Å.

The a-Si structure was modeled by a 64-atom simulation cell with periodic boundary

conditions, same as in Ref. [17] (see Fig. 1a and Table I). The 32 Li insertion sites identified in

Ref. [17] with a similar computational setup served as initial guesses for Li and Mg-inserted

structures. All structures were fully relaxed until the stresses were below 0.1 GPa. The defect

formation energies (Ef) were computed as

Ef = (ESi/nX – ESi – nEX)/n, (1)

where ESi/nX is the total energy of the simulation cell with n dopants of type X, X = Li, Mg; ESi is

the energy of a pure Si cell, and EX is the energy of atom X in vacuum (modeled as a cubic cell of

size 11x11x11 Å). Spin-polarized calculations were performed for all systems and spin

configurations with the lowest energies were used. However, in most cases, spin polarization was

found insignificant. Reported energies do not include basis set superposition error (BSSE)

corrections, which likely balance each other out in the comparison between c-Si and a-Si, but

may affect clustering results.

RESULTS

Starting from the 32 sites reported previously [17] and shown in Fig. 1b, we predicted the

defect formation energies (Ef) for Li and Mg. Ef of a-Si varies for different insertion sites (see

Fig. 1c). However, for both Li and Mg, the lowest Ef of a-Si is significantly lower than the value

obtained for T sites of c-Si with the same computational setup (-1.66 eV vs. -0.94 eV [13] for Li

and -0.90 eV vs. +0.94 eV [13] for Mg).

Table I. Computed lattice vectors (Å) of a-Si with comparison to those from Ref. [17].

a-Si

This work

a 11.067 -0.175 0.187

b -0.175 11.180 -0.077

c 0.188 -0.077 11.075

Ref. [17]

a 11.096 -0.172 0.189

b -0.172 11.217 -0.077

c 0.189 -0.077 11.109

Figure 1. (a) The a-Si simulation cell. (b) Illustration of initial insertion sites in a-Si. (c) Defect

formation energies for Li and Mg defects in a-Si. Color scheme here and elsewhere: yellow – Si

atoms, pink – initial Li/Mg sites. Corresponding values for c-Si are shown as black squares.

Some insertion sites are close enough to each other so that their double occupation can be

influenced by Li-Li or Mg-Mg interactions. The distances between these sites are comparable to

free Li and Mg dimers, 2.67 Å and 3.89 Å, respectively [27, 28]. We have computed the

formation energy per metal atom of dimers (Eq. 1) where the interatomic distance is less than

4.0 Å and compared it with the average formation energy of both sites when only one of them is

occupied:

Ecl = Efn=2

– 0.5 (Efsite 1

+ Efsite 2

) (2)

We found a number of pairs of sites for which Ecl < 0, i.e. clustering is favored and helps

stabilize the sites. They are shown in Fig. 2, and information about the structures and energetics

of these dimers is given in Table II. The lowest Ecl reaches a significant -0.35 eV per atom for

Mg, while it is -0.27 eV for Li. In contrast, at low concentrations, neither Li nor Mg clustering is

favored in c-Si [7, 10].

Figure 2. Pairs of neighboring insertion sites for Li (left) and Mg (right) with negative

(favorable) clustering energy (Eq. 2). For Mg clusters, two simulation cells are shown as dimers

involve atoms from repeating cells.

DISCUSSION AND CONCLUSIONS

We have studied the structures and energetics of Mg insertion into a-Si as compared to

c-Si and to Li insertion into a-Si. In contrast to c-Si, where Mg defect formation energies are

positive, we discover negative formation energies of up to -0.90 eV for some sites in a-Si. The

highest energy of +1.11 eV is comparable to that of c-Si (for which a value of +0.94 eV is

obtained with the same computational setup [13]). a-Si is metastable under normal conditions.

Computed results indicate that the cohesive energy of a-Si is 0.13 eV per atom larger (a-Si is less

stable) than that of c-Si. This observation indicates that an a-Si anode will have larger average

anode voltages compared to those of c-Si. For instance, if we assume that the charging of the

anode materials goes according to Eq. 3

xSiMxMSi (3)

Then, using the well-defined methodology [29], the average anode voltages (V ) vs. metallic

reference states can be calculated according to Eq. 4

zxe

xEEEV

MSiSiM X

, (4)

where E is the corresponding DFT energy; z and e are the charge (in electrons) and absolute

value of the electron charge. Assuming that fully charged Si anodes can be reached (in this work,

we considered that the final charge states are Li3.75Si (Li15Si4) and Mg2Si) and taking into

account the number of valence electrons for the considered metal atoms (1 for Li and 2 for Mg),

we estimated the average voltage of c-Si and a-Si anodes. For Li-ion batteries, the computed

average voltages are 0.23 V and 0.27 V for c-Si and a-Si anodes, respectively. For Mg-ion

batteries, the average voltages are 0.09 V and 0.12 V for c-Si and a-Si anodes, respectively. On

the one hand, the calculations predict that a metal-ion battery with an a-Si anode will have a

lower average battery voltage (higher average anode voltage) than the same battery with a c-Si

anode. On the other hand, since it is well known that many electrolytes are not stable within the

battery operating voltage [30], the larger anode voltage (lower battery voltage) may palliate the

electrolyte stability problem and improve cycle life and safety of the battery.

Table II. Li and Mg dimer structures and energetics.

Dimer Sites involved

(see Fig 1b)

Dopant-dopant

distance, Å Ecl, eV

Li2 8, 23 2.92 -0.05

Li2 2, 13 2.98 -0.01

Li2 11, 13 3.44 -0.27

Mg2 5, 11 2.87 -0.35

Mg2 4, 21 3.10 -0.16

Mg2 5, 16 3.16 -0.15

Mg2 12, 13 3.41 -0.23

Mg2 3, 21 3.58 -0.05

Mg2 4, 12 3.64 -0.09

For c-Si, it has been shown that while a single-dopant diffusion barrier is much higher for

Mg (~1.0 eV) than for Li (~0.6 eV) [7, 9, 10], interactions between Mg atoms at neighboring T

sites significantly reduce the diffusion barrier (to about 0.6 eV), in fact making Mg diffusion

competitive with that of single Li. The energy cost of about 0.2 eV to form nearest neighbors in

c-Si will however work against this scenario unless a certain Mg concentration is reached (for

example, after 1/4th

of the T sites are homogenously filled in a well-dispersed way, diffusion of

any additional dopant atoms has to involve neighboring sites). Increased diffusion rates with Li

concentrations have been computed with molecular dynamics for anodes and cathodes [21, 31,

32], including a-Si [21]. Dimer formation and cooperative diffusion of Li clusters were reported

in cathode materials [31, 32].

The energetics of Mg-Mg interactions is therefore expected to have an important

influence on migration and thereby battery performance in a-Si. While in c-Si, clustering of Li

and Mg is energetically unfavorable, in a-Si, we identified a number of neighboring insertion

sites where Li-Li and Mg-Mg interactions stabilize the sites rather than destabilize them as in

c-Si, and to a larger extent for Mg than for Li. Can Mg-Mg interactions reduce the migration

barrier also in a-Si? If that were the case, then the much more favorable energetics of occupation

of neighboring sites must mean that such barrier reduction will be more likely to occur, in which

case a-Si would look a more attractive anode material. Of course, should dopant-dopant

interactions lead to a higher barrier, the same argument would hold; for example, the energy

required to break the dimer would contribute to increase the barrier. The final confirmation has

to come from the calculations of diffusion barriers and paths themselves, which are work in

progress.

ACKNOWLEDGMENTS

This work was supported by Tier 1 AcRF grant R-265-000-430-133 by the Ministry of

Education of Singapore.

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