Structural and energetic properties of vacancy defects in MXene surfaces

Density functional theory calculations were employed to investigate the structure and energetics of vacancy defects on 84 MXene surfaces, composed of different M (Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, or W), X (carbon or nitrogen), T (oxygen of fluorine), and atomic layer stacking (ABC or ABA) combinations. Structurally, we found that, in most cases, the effect of the presence of the vacancy on the surrounding atoms is to push them away from the vacant site due to the missing screening charge and the reinforcement of the remaining interatomic bonds. The reconstruction energy associated to this surface relaxation was found to range from a few tenths of eV to a few eV. On the bare ABC-stacked carbide MXenes, we found M and X vacancy formation energies between 2 and 3 eV. On nitrides, M vacancy formation energies are decreased, while X ones seem to increase. By terminating the MXenes with an O layer, M formation energies are significantly increased, reaching over 9 eV in some cases, in agreement with theoretical and experimental results of the literature, whereas X vacancy formation energies become smaller. The F termination was found to have the same effect as O on X vacancy formation energies with respect to clean MXenes, but to decrease M vacancy formation energies. The F termination allows more easily created M and N vacancies than the O termination. The creation of vacancies on some ABC-stacked ${\mathrm{M}}_2{\mathrm{XF}}_2$ MXenes led to extreme lattice deformation which suggests that the experimental synthesis of ${\mathrm{M}}_2{\mathrm{NF}}_2$ with metals from groups 5 and 6 of the Periodic Table, and ${\mathrm{M}}_2{\mathrm{CF}}_2$ with metals from group 6, through etching using a hydrogen fluoride aqueous solution [HF (aq)] will lead to ABA-stacked MXenes. The formation energies of the MT double vacancies were found to correlate approximately linearly with the sum of the corresponding M and T single-atom vacancy formation energies. All MXenes studied in this work fit in the same correlation line, implying that one can predict the formation energy of double vacancies on MXenes, made of any M, X, and T elements, and with atomic layers stacked in an ABC or ABA fashion, if the formation energies of the corresponding single vacancies are known. The method used by theoretical works, such as this one, to calculate formation energies has two crucial limitations: the values thus obtained depend on the chosen single-atom energy references, and kinetic effects such as energy barriers are not included. Here, we cautiously discuss these limitations, compare formation energies calculated using different energy references, and propose mechanisms for the formation of vacancies that include atomic migration barriers.

Figure 1

Top (top images) and side (bottom images) views of MXene $p$(4 × 4) supercells. On the left, the ${\mathrm{Ti}}_2{\mathrm{CO}}_2$ MXene is shown, as an example of an MXene that displays ABC atomic layer stacking, and whose O surface termination aligns with the farthest metal layer, as displayed by the letters ABC next to the side view (letters in parentheses refer to the alignment of the termination layer). At the center and on the right, the ${\mathrm{Mo}}_2\mathrm{C}$ MXene is shown, terminated by O and F, respectively, and displaying ABA atomic layer stacking, seen by the vertical alignment of its two metal layers, and identified by the letters ABA next to the side view. Color code for spheres: C in black, O in red, Ti in grey, Mo in blue, and F in green.

Figure 2

Formation energies of vacancies in ${\mathrm{M}}_2\mathrm{X}$ MXenes (values taken from Table S1 [24]). Carbides are shown on the left side, and nitrides on the right. The grey lines denote metal vacancies, and the black and blue lines denote carbon or nitrogen vacancies, respectively. The dashed line is placed at $E_{\mathrm{form}}=0$. Note that different reference species were considered in the calculation of $E_{\mathrm{form}}$ for different types of vacancies and direct comparison must consider this premise.

Figure 3

Top views of the ${\mathrm{Ti}}_2\mathrm{C}$ MXene when defect-free (left panel), with a Ti vacancy (center panel), or with a C vacancy (right panel), after relaxation of the positions of the atoms. Color code for spheres: C in black and Ti in grey. The numbers on the spheres are the calculated charges on the respective atoms, in electron charge units, and the green arrows point in the direction in which the associated atoms moved when the defect was created.

Figure 4

Formation energies of vacancies in ${\mathrm{M}}_2{\mathrm{XO}}_2$ MXenes (values taken from Table S3 [24]). Carbides are shown on the left side, and nitrides on the right. The grey lines denote metal vacancies, the black and blue lines denote carbon or nitrogen vacancies, respectively, the full red lines denote oxygen vacancies, and the purple lines correspond to metal-oxygen double vacancies. The dashed red lines correspond to the formation energy of an oxygen vacancy neighboring a metal vacancy. The dashed grey line is placed at $E_{\mathrm{form}}=0$. Note that different reference species were considered in the calculation of $E_{\mathrm{form}}$ for different types of vacancies and direct comparison must consider this premise.

Figure 5

Top views of the ${\mathrm{Ti}}_2{\mathrm{CO}}_2$ MXene when defect-free (top left panel), with a Ti vacancy (top right panel), with a C vacancy (bottom left panel), or with an O vacancy (bottom right panel), after relaxation of the positions of the atoms. Color code for spheres: C in black, Ti in grey, and O in red. The numbers on the spheres are the calculated charges on the respective atoms, in electron charge units, and the green arrows point in the direction in which the associated atoms moved when the defect was created.

Figure 6

Formation energies of vacancies in ${\mathrm{M}}_2{\mathrm{XF}}_2$ MXenes (values taken from Table S6). Carbides are shown on the left side, and nitrides on the right. Data for ${\mathrm{W}}_2{\mathrm{NF}}_2$ are shown in Table S6 but omitted here to allow a clearer visualization of the other points. The grey lines denote metal vacancies, the black and blue lines denote carbon or nitrogen vacancies, respectively, the full green lines denote fluorine vacancies, and the purple lines correspond to metal-fluorine double vacancies. The dashed green lines correspond to the formation energy of a fluorine vacancy, assuming the energy of a fluorine atom is $E\left({\mathrm{F}}_2\right)/2$. The dashed grey line is placed at $E_{\mathrm{form}}=0$. Note that different reference species were considered in the calculation of $E_{\mathrm{form}}$ for different types of vacancies and direct comparison must consider this premise.

Figure 7

${\mathrm{W}}_2{\mathrm{CF}}_2$ (top diagrams) and ${\mathrm{W}}_2{\mathrm{NF}}_2$ (bottom diagrams) MXene surfaces, before (left) and after (right) the creation of a W vacancy, by removing the atom marked by the red cross on the left images. In each rectangle, the top and bottom schematics represent top and side views of the surface, respectively. Color code for spheres: C in black, N in blue, F in green, and W in pink.

Figure 8

Formation energies of vacancies in ${\mathrm{M}}_2\mathrm{X}$ MXenes (values taken from Table S9) assuming ABA atomic layer alignment. Carbides are shown on the left side, and nitrides on the right. The grey lines denote metal vacancies, and the black and blue lines denote carbon or nitrogen vacancies, respectively. Note that different reference species were considered in the calculation of $E_{\mathrm{form}}$ for different types of vacancies and direct comparison must consider this premise.

Figure 9

Formation energies of vacancies in ${\mathrm{M}}_2{\mathrm{XO}}_2$ MXenes (values taken from Table S10) assuming ABA atomic layer alignment. Carbides are shown on the left side, and nitrides on the right. The grey lines denote metal vacancies, the black and blue lines denote carbon or nitrogen vacancies, respectively, the full red lines denote oxygen vacancies, and the purple lines correspond to metal-oxygen double vacancies. The dashed grey line is placed at $E_{\mathrm{form}}=0$. Note that different reference species were considered in the calculation of $E_{\mathrm{form}}$ for different types of vacancies and direct comparison must consider this premise.

Figure 10

Formation energies of vacancies in ${\mathrm{M}}_2{\mathrm{XF}}_2$ MXenes (values taken from Table S11) assuming ABA atomic layer alignment. Carbides are shown on the left side, and nitrides on the right. The grey lines denote metal vacancies, the black and blue lines denote carbon or nitrogen vacancies, respectively, the green lines denote fluorine vacancies, and the purple lines correspond to metal-fluorine double vacancies. The dashed grey line is placed at $E_{\mathrm{form}}=0$. Note that different reference species were considered in the calculation of $E_{\mathrm{form}}$ for different types of vacancies and direct comparison must consider this premise.