Stability and electronic structures of native defects ... - APS Link Manager

PHYSICAL REVIEW B 89, 205417 (2014)

Stability and electronic structures of native defects in single-layer MoS2 Ji-Young Noh,1,2 Hanchul Kim,2 and Yong-Sung Kim1,3,* 1

Korea Research Institute of Standards and Science, Yuseong, Daejeon 305-340, Korea Department of Nano Physics, Sookmyung Women’s University, Seoul 140-742, Korea 3 Department of Nano Science, Korea University of Science and Technology, Daejeon 305-350, Korea (Received 15 July 2013; revised manuscript received 17 March 2014; published 19 May 2014) 2

The atomic and electronic structures and stability of native defects in a single-layer MoS2 are investigated, based on density-functional theory calculations. Native defects such as a S vacancy (VS ), a S interstitial (Si ), a Mo vacancy (VMo ), and a Mo interstitial (Moi ) are considered. The Si is found to have S-adatom configuration on top of a host S atom, and the Moi has Mo-Moi split interstitial configuration along the c direction. The formation energies of the native defects in neutral and charged states are calculated. For the charged states, the artificial electrostatic interactions between image charges in supercells are eliminated by a supercell size scaling scheme and a correction scheme that uses a Gaussian model charge. It is found that the VS has a low formation energy of 1.3–1.5 eV in the Mo-rich limit condition, and the Si has 1.0 eV in the S-rich limit condition. The VS is found to be a deep single acceptor with the (0/−) transition level at 1.7 eV above the valence-band maximum (VBM). The Si is found to be an electrically neutral defect. The Mo-related native defects of VMo and Moi are found to be high in formation energy above 4 eV. The VMo is a deep single acceptor and the Moi is a deep single donor, of which the (0/−) acceptor and (+/0) donor transition levels are found at 1.1 and 0.3 eV above the VBM, respectively. DOI: 10.1103/PhysRevB.89.205417

PACS number(s): 73.61.Le, 71.55.Ht, 73.20.Hb, 91.60.Ed

I. INTRODUCTION

Molybdenite (MoS2 ) is an abundant mineral in the earth crust and has been widely used as a mechanical lubricant, a desulfurization catalysis in petrochemisty, and a principal ore for Mo extraction. The molybdenite crystal (2H-MoS2 ) has a layered atomic structure [Fig. 1(a)], from which a single-layer MoS2 can be fabricated by, for example, mechanical exfoliation [1–3]. There are a variety of interesting properties [4]. The MoS2 is a semiconductor electrically. The 2H-MoS2 bulk and thicker MoS2 than a single-layer have an indirect band gap, while the single-layer MoS2 has a direct band gap [5–7]. The two-dimensional nature of the single-layer MoS2 implicates a strong Coulomb interaction between charged species therein, which has remarkable consequences on the correlation of electrons and charged defects. The fundamental (GW ) band gap of a single-layer MoS2 is 2.8 eV [8,9], but the large excitonic binding energy of 0.9 eV renders the optical band gap measured at 1.9 eV in absorption and photoluminescence spectroscopy [5,6]. In n-type single-layer MoS2 thin-film transistors, the optically excited trions comprising two electrons and one hole have been also identified in presence of carrier electrons [10–13]. Besides the large electronic correlation effects, the intrinsic broken inversion symmetry in a single-layer MoS2 induces spin-orbit split bands at the K and −K points in the hexagonal Brillouin zone [14] and is now opening a new era for the spinand valley-controlled electronics [15–19]. The carrier mobility of electrons in a single-layer MoS2 is as high as ∼200 cm2 V−1 s−1 in HfO2 /MoS2 /SiO2 stacked (top-gate) thin-film transistors [1], while it lowers to about ∼1 cm2 V−1 s−1 in an air/MoS2 /SiO2 stacked bottom-gate structure [3]. The measured high electron mobility makes the single-layer MoS2 promising for next generation high-speed

*

[email protected]

1098-0121/2014/89(20)/205417(12)

thin-film electronics. The role of the dielectric encapsulation layers is to screen the Coulomb scattering interaction between the carrier electrons and charged impurities or defects [20]. A single-layer MoS2 exfoliated from a naturally grown MoS2 bulk crystal shows typically n-type electrical conductivity [1,3], which is doped unintentionally. A certain amount of defects and impurities are always present in single-crystalline materials in nature, and even with the small imperfections, the electrical and optical properties, such as the electrical conductivity, are largely affected. However, understanding of defects and impurities in single-layer MoS2 is still premature, even though an efficient and controllable n- and p-type doping technology as well as the mobility engineering is an essential constituent for the electronics application of a single-layer MoS2 [21–25]. In this study, we investigate the native defects such as a S vacancy (VS ), a S interstitial (Si ), a Mo vacancy (VMo ), and a Mo interstitial (Moi ) in a single-layer MoS2 , through density-functional theory (DFT) calculations. We find that VS and Si have a low formation energy, about 1 eV, in Mo- and S-rich conditions, respectively, and the Mo-related defects of VMo and Moi have a high formation energy above 4 eV. The atomic structure of Si is a S-adatom on top of a host S atom, and that of Moi is a Mo-Moi split interstitial along the c direction. The VS is found to be a deep single acceptor having the (0/−) transition level at 1.7 eV above the valence band maximum (VBM), and the Si is an electrically neutral defect. The VMo is a deep single acceptor with the (0/−) transition level at 1.1 eV above the VBM, and the Moi is a deep single donor, with the (+/0) transition level at 0.3 eV above the VBM. II. METHODS

The density-functional theory (DFT) calculations are performed as implemented in the Vienna ab initio simulation package (VASP) [26,27]. The projector augmented wave (PAW) pseudopotentials [28,29] and the local density approximation

205417-1

©2014 American Physical Society

JI-YOUNG NOH, HANCHUL KIM, AND YONG-SUNG KIM


FIG. 1. (Color online) (a) Atomic structure of the bulk crystalline 2H-MoS2 . The (b) 1 × 1 × 2, (c) 1 × 1 × 4, (d) 1 × 1 × 8, (e) 1 × 1 × 16, (f) 4 × 4 × 4, (g) 8 × 8 × 8, and (h) 12 × 12 × 12 supercell structures, in which a single-layer MoS2 is placed. In (a), (b), and (f), the top views (the c-axis projection) are shown at the top. The large (purple) and small (yellow) balls indicate Mo and S atoms, respectively.

(LDA) [30] for the exchange-correlation energy of electrons are employed. A kinetic energy cutoff of 350 eV for the plane-wave basis set expansion is used. The atomic structures are relaxed until the Hellmann-Feynman forces are less than ˚ For the single-layer MoS2 , the calculated band gap 0.02 eV/A. is direct at the K point in the hexagonal Brillouin zone with the value of 1.851 eV within the LDA. The atomic structure of the bulk crystalline 2H-MoS2 is shown in Fig. 1(a). The calculated lattice constants and internal structural parameters of the bulk and single-layer MoS2 are compared in Table I. The in-plane lattice constant (a) is ˚ both in the bulk and single-layer calculated to be 3.124 A MoS2 . The distance between the top and bottom S layers ˚ in bulk and (2ζ ) in a S-Mo-S layer [see Fig. 1(a)] is 3.115 A ˚ 3.112 A in an isolated single-layer MoS2 . In the bulk 2H-MoS2 , the lattice constant (c) perpendicular to the S-Mo-S layers is ˚ and the interlayer spacing [(c − 4ζ )/2] between 12.066 A, ˚ The adjacent two S-Mo-S layers [see Fig. 1(a)] is 2.918 A. thickness (t) of a S-Mo-S layer including the interlayer spacing ˚ and the thickness t [t = c/2 = 2ζ + (c − 4ζ )/2] is 6.033 A,

TABLE I. Calculated lattice and internal structural parameters (in ˚ of bulk 2H-MoS2 and single-layer MoS2 in LDA. † The thickness A) (t) of a single-layer MoS2 is chosen for convention to be the same as the t in bulk 2H-MoS2 , which includes the distance between the top and bottom S layers (2ζ ) in a S-Mo-S layer and the interlayer spacing [(c − 4ζ )/2] between adjacent two S-Mo-S layers. Parameters

2H-MoS2

Single-layer MoS2

a c 2ζ t (c/2) (c − 4ζ )/2

3.124 12.066 3.115 6.033 2.918

3.124 – 3.112 6.033† –

of a single-layer MoS2 is arbitrary chosen to be the same as the t of bulk 2H-MoS2 . Some of the supercell structures used in our calculations are shown in Figs. 1(b)–1(h) for a single-layer MoS2 . We represent the supercell structures as l × m × n, as indicated in Fig. 1(f), where the l × m is the in-plane periodicity and n is the out-of-plane supercell periodicity along the c direction perpendicular to the S-Mo-S layers. The in-plane ˚ of a unit length is the in-plane lattice constant (a = 3.124 A) single-layer MoS2 , and the out-of-plane unit length (a⊥ ) along ˚ The the c direction is chosen to be the same: a⊥ = 3.124 A. supercell length along the c direction (Lz ) is then Lz = n × a⊥ , and the supercell volume () is = la × ma × na⊥ . The choice of a⊥ = a is for the purpose of nearly uniform scaling of the supercell sizes in all the directions. We are interested in the properties of an isolated defect, which is placed in an (in-plane) infinite-size single-layer MoS2 in between the (out-of-plane) infinite-size vacuum. They can be effectively extrapolated from the finite-size supercell calculations by the (nearly) uniform scaling scheme, which is especially useful for charged defect calculations [31]. The static dielectric constant tensor (ε) of a single-layer MoS2 is calculated by density-functional perturbation theory (DFPT) with the 1 × 1 × n (n = 2, 4, 8, 16) supercells [Figs. 1(b)–1(e)] [32,33]. The DFT calculations for the native defects in single-layer MoS2 are performed with the α(1 × 1 × 1) (α = 4, 6, 8, and 10) supercells. Electrostatics calculations that solve the Poisson equations are done with the α = 4–80 supercells. Figures 1(f)–1(h) show the supercell structures of α = 4, 8, and 12, representatively. In the DFPT calculations, the 8 × 8 × 1 k-point mesh is used for the 1×1 × n supercells. In the DFT calculations, the 2 × 2 × 1 k-point mesh is used for the α = 4 and 6 supercells, and the point is used for the α = 8 and 10 supercells. In the α = 6 supercell, the single-layer MoS2 contains 108 host atoms and the vacuum ˚ thickness (Lz − t) is 12.711 A.

205417-2

STABILITY AND ELECTRONIC STRUCTURES OF NATIVE . . .


iso The formation energy (Eform ) of an isolated native defect is calculated by

−

15

Ni μi

i

pristine + q VBM + EFermi + Ecorr ,

(1)

pristine

is the total energy per a supercell of a pristine where Etot defect single-layer MoS2 without a defect, Etot is the total energy of a single-layer MoS2 containing a defect in the supercell, Ni is the number of i element (Mo and S) added (or removed with minus sign) in the supercell for the native defect, μi is the chemical potential of the element i, q is the charge pristine state of the defect, VBM is the VBM level of the single-layer MoS2 without a defect, EFermi is the Fermi level with reference pristine to the VBM level, and Ecorr is the total energy correction term including the level alignment and removing the spurious electrostatic interaction between image charges in supercells for charged defects. The calculation of Ecorr is described in Sec. III C in detail. We consider the EFermi in the range from the reference VBM level (EVBM = 0 eV) to the conduction-band minimum (CBM) level calculated in LDA (corresponding to the LDA band gap at the K point in the Brillouin zone) [0 eV (VBM) < EFermi 4) [the (black) filled squares in Fig. 4(a)], the Eform (α) 1− of the (1−) charged VS is found to strongly depend on the α [the (black) filled circles in Fig. 4(a)].

super

iso Ecorr = EM − EM

V = [VM (r) − VDFT (r)]r=far , defect(q=0)

VDFT (r) = VH

(r).

(9)

The VDFT (r) can be decomposed as q

0 VDFT (r) = VDFT (r) + VDFT (r),

(10)

where

(b)

0.7

defect(q=0)

defect(q=0)

(r) − VH

defect(q=0)

0 VDFT (r) = VH

3

Ecorr (eV)

0.6 1-

VS

2.5 2

0

(r),

(11)

1.5

1/α

0.2

0.4 0.3

0

ρDFT ρM

-5 (c) q

VM (e)

VDFT q

VM-VDFT

-3 -2 -1 0 1 x (a)

0

0.3

2

3

(r).

(12)

The represents the DFT Hartree potential that is induced only by the excess charge (q) of the defect, and 0 VDFT (r) is only by the defect (VS ) in the charge neutral. Under 0 (r) is short-ranged, Eq. (8) can be the assumption that the VDFT written as

0.5 1-

VS

0.1 0.1

pristine

(r) − VH

q VDFT (r)

0.2

VS

Potential (V) Charge (ea)

Formation energy (eV)

(a)

1 0 Potential (V) Charge (ea)

pristine

(r) − VH

0.8

4

0.1 0 -0.1

(8)

where

q

-10

(7)

iso where EM − is the electrostatic energy difference between the isolated and supercell configurations for the model charge system as described in Sec. III B, q is the charge state of the defect, and V is the potential difference between the electrostatic potential [VM (r)] in the model charge calculation (as described in Sec. III B) and the DFT Hartree potential [VDFT (r)] that is induced only by the charged defect, far from the defect site:

VDFT (r) = VH 3.5

+ qV ,

super EM

0

0.1

1/α

0.2

0 -5

ρM

-10

ρDFT

0.4 0.2 0 -0.2 -0.4

VDFT V M

(d)

q

0

1

2

(f)

3 4 z (a⊥)

V = V q + V 0 0 q = VM (r) − VDFT (r) r=far − VDFT (r) r=far .

0.3

5

6

FIG. 4. (Color online) (a) Calculated formation energies of VS0 (black filled squares) and VS1− without (black filled circles) and with (gray filled circles) Ecorr as a function of 1/α. The formation energies of the isolated defects are indicated by the open symbols at 1/α = 0. super iso − Eform (α) in DFT (black filled circles) and Ecorr (b) Calculated Eform super (gray filled circles). The polynomial fit curves for Eform (α) in (a) and super iso Eform − Eform (α) in (b) are shown by the red thick solid lines. The model charge contribution to Ecorr is shown by the red dashed line in (b). [(c) and (d)] Plane-averaged charge profiles of ρM (r) (red thick solid line) and ρDFT (r) (black thin solid line) for the α = 6 supercell along the (c) in-plane x and (d) out-of-plane z direction. [(e) and (f)] Plane-averaged potential profiles of VM (r) (red thick solid line) and q VDFT (r) (black thin solid line) for the α = 6 supercell along the (e) in-plane x and (f) out-of-plane z direction. The plane average of the q difference of VM (r)-VDFT (r) is shown by the blue dashed line in (e).

(13)

0 The −[VDFT (r)]r=far term is simply the level alignment term between the pristine single-layer MoS2 and the supercell containing a neutral defect. The calculated VBM level of defect(q=0) ] is the the supercell containing a neutral defect [VBM pristine pristine defect(q=0) defect(q=0) sum of VBM and VBM = VBM − VBM . The defect(q=0) VBM is the VBM shift by the presence of the neutral defect in the supercell, which originates from the DFT convention that the average of VH (r) over a supercell volume defect(q=0) is the Hartree potential () is set to zero. Thus the VBM difference between the pristine single-layer MoS2 and the supercell containing the neutral defect far from the defect: defect(q=0) 0 VBM = −[VDFT (r)]r=far . We would like to emphasize 0 that the calculation of −[VDFT (r)]r=far should be done at the location far from the defect in the xz- or yz-plane averaged 0 0 V DFT (y) or V DFT (x). When a neutral defect has a dipole moment perpendicular to the slab, the level alignment calcu0 lation of −[VDFT (r)]r=far can be done unambiguously, because the potential plane average is done along the perpendicular 0 direction. [In this case, the long-range VDFT (r) along the z direction in the supercell is corrected by a counter dipole moment inside the vacuum, and the DFT total energy itself is corrected without the Ecorr term.] However, if a defect has a 0 dipole moment along an in-plane direction [long-range VDFT (r) 0 through inside the slab], the calculation of −[VDFT (r)]r=far can

205417-5



+

5

si /α + t3 /α , i

3

(14)

i=1

where si ’s are as already fit for the model charge system [found iso in Sec. III B], and Eform and t3 are the only fitting parameters iso without an here. Thus we can extrapolate the Eform (α) to Eform explicit calculation of the qV term nor Ecorr . The calculated Eform (α) for the four α = 4, 6, 8, and 10 supercell sizes [the (black) filled circles in Fig. 4(a)] are fit by Eq. (14), and the fitting curve [the (red) thick line in Fig. 4(a)] with the obtained iso = 3.21 eV and t3 = −6.84 agrees very well with the Eform DFT values. This approach extends the linear extrapolation scheme, which can be applied only to the perfect uniform scaling scheme (such as for homogeneous bulk, surface, and interface systems) [31,38,39], to a more general case such as for one- or two-dimensional or complex inhomogeneous systems. The explicit calculation of the qV term and thus Ecorr is described here. The reference ρM (r) for VS1− is chosen as described in Sec. III B [Fig. 3(b)], and compared with ρDFT (r) in Figs. 4(c) and 4(d) (for α = 6). The center of the (1−) Gaussian model charge is placed at the center of the slab (a Mo atomic site in a single-layer MoS2 ). Although the atomic-structural location of the VS is at a S atomic site, the defect charge of the VS1− mainly comes from the

0.8 0.7

4.5 (a) 4

Si

1-

0.6 Ecorr (eV)


5

3.5 3 0

2.5 2

0.4 0.3 0.1

0

0.1

0

1/α

0.2

0.3

ρDFT ρM

-5 -10

(b)

0.5

0.2

Si

(c)

0.04 q VDFT VM 0.02 0 q -0.02 VM-VDFT (e) -0.04 -3 -2 -1 0 1 2 3 x (a)


Eform (α) =

iso Eform

surrounding Mo atoms (will be shown in Sec. III D). For the dielectric environment displayed in Fig. 3(a) (for α = 6), the q VM (r) is calculated and compared with VDFT (r) in Figs. 4(e) and 4(f). By the excess charge of q = −1, both the potentials q vary in a long range. The difference of [VM (r) − VDFT (r)] q removes the long-range part of VDFT (r), which depends on the supercell sizes, and approaches to a constant value far from the defect site. Thus, we can obtain a constant value of q V q = [VM (r) − VDFT (r)]r=far , and in combined with V 0 , the qV correction term is determined according to Eq. 13. The calculated Ecorr for the α = 4, 6, 8, and 10 supercells is super iso compared with the Eform − Eform (α) in Fig. 4(b), which shows a good agreement with each other especially for α 6. With iso of VS1− is obtained, as including the calculated Ecorr , the Eform shown by the (gray) filled circles in Fig. 4(a), and it is found to have only a weak dependence on the supercell size α and iso converge fast to the extrapolated Eform value of 3.21 eV. The formation energies of Si (the on-top configuration, which will be discussed in Sec. III E) in the neutral (S0i ) and (1−) charged (S1− i ) states are plotted as a function of the inverse of the supercell size (1/α) in Fig. 5(a) [in the


be problematic, but for the considered native point defects of VS , Si , VMo , and Moi in a single-layer MoS2 , such a situation pristine defect(q=0) and VBM is not encountered. Even though the VBM iso are different, its effect on the Eform is none for neutral defects defect(q=0) (q = 0 and Ecorr = 0), but the VBM contributes the Ecorr for charged defects (q = 0). The V q term is a level alignmentlike term by the charge q and has its origin not at the potential average over a supercell volume but at the difference between the model [ρM (r)] and DFT [ρDFT (r)] charge distributions, where ρDFT (r) = ρ defect(q=0) (r) − ρ defect(q=0) (r) [31,38,39]. The V q term essentially contains the contribution of ρDFT (r) to Ecorr except that of ρM (r). The dependence of the choice of ρM (r) on Ecorr is thus eliminated by adopting the V q term. The V q is a constant value, insofar as the center of ρM (r) is appropriately chosen [making ρM (r)-ρDFT (r) have no net dipole], the ρM (r) and ρDFT (r) are distributed well inside the supercell, and equivalently the supercell is larger than q the short-range potential [VDFT (r) − VM (r) + V q , where the long-range counterpart is VM (r) − V q ] extent. In our inhomogeneous dielectric system, the V q is found to have slightly different constants between obtained from the xz- or q q yz-plane averaged V DFT (y) or V DFT (x) and from the xy-plane q averaged V DFT (z), far from the defect site. Although the reason q is not very clear, the V q obtained from the V DFT (y) or q V DFT (x) far from the defect site (as in the case of V 0 ) is found to give better Ecorr values. Both the level-alignment qV 0 and level-alignment-like qV q terms are scaled by α −3 with the supercell size α (proportional to the reciprocal supercell volume 1/ ), and then the supercell-size-dependent Eform (α) of a charged defect can be fit by the same functional to that [EM (α)] for the model charge system [Eq. (6)] but additionally with including the level alignmentlike correction term (t3 /α 3 ):

0

1-

Si

0

0.1

1/α

0.2

0.3

0 -5

ρM ρDFT

-10 0.4 0.2 0 -0.2 -0.4

(d)

VM q

VDFT 0

1

(f) 2

3 4 z (a⊥)

5

6

FIG. 5. (Color online) (a) Calculated formation energies of S0i (black filled squares) and S1− i without (black filled circles) and with (gray filled circles) Ecorr as a function of 1/α. The formation energies of the isolated defects are indicated by the open symbols at 1/α = 0. super iso − Eform (α) in DFT (black filled circles) and Ecorr (b) Calculated Eform super (gray filled circles). The polynomial fit curves for Eform (α) in (a) and super iso Eform − Eform (α) in (b) are shown by the red thick solid lines. The model charge contribution to Ecorr is shown by the red dashed line in (b). [(c) and (d)] Plane-averaged charge profiles of ρM (r) (red thick solid line) and ρDFT (r) (black thin solid line) for the α = 6 supercell along the (c) in-plane x and (d) out-of-plane z direction. [(e) and (f)] Plane-averaged potential profiles of VM (r) (red thick solid line) q and VDFT (r) (black thin solid line) for the α = 6 supercell along the (e) in-plane x and (f) out-of-plane z direction. The plane average q of the difference of VM (r)-VDFT (r) is shown by the blue dashed line in (e).

205417-6


The calculated atomic and electronic structures of a S vacancy (VS ) in a single-layer MoS2 are shown in Fig. 6. One S atom in a single-layer MoS2 is removed [Figs. 6(a) and 6(b)] in a supercell, and the three Mo atoms surrounding the VS is ˚ in the neutral and 0.076 A ˚ in found to be relaxed by 0.101 A the (1−) charge state toward the inward direction. Three defect states are generated inside the band gap. Since the VS has trigonal symmetry [Figs. 6(a) and 6(b)], the defect states can be classified as a singlet a1 state and doubly degenerate e states. Due to the absence of an anion in the single-layer MoS2 , in the neutral charge state (VS0 ), two excess electrons occupy the a1 state, and the two e states are empty, as shown in Fig. 6(c), where the Fermi level is located in between the a1 and e states, which agrees with the previous DFT calculation [25] and scanning tunneling spectroscopy [24]. The charge density of the a1 state is shown in Fig. 6(a), and that of the e states are in Fig. 6(b). The Kohn-Sham (KS) level of the a1 state is found to be near the VBM, and that of the empty e degenerate states is located inside the band gap, as shown in Fig. 6(c). With including the Ecorr for the α = 6 supercell, we iso calculate the formation energies (Eform ) of VS in the various charge states of (2+), (1+), (1−), (2−), (3−), and (4−) (considering possible occupancy of the KS a1 and e defect levels). The formation energies are plotted as a function of the Fermi level in the Mo-rich and S-rich limit conditions in

Fig. 7. The only stable charge states of VS are found to be (0) and (1−), when the Fermi level is inside the band gap. The (0/−) transition level is found at (0/−) = EVBM + 1.7 eV. Therefore VS is a deep single acceptor in a single-layer MoS2 . The formation energy of an isolated VS in a single-layer MoS2 is as low as 1.3–1.5 eV depending on the Fermi level in the Mo-rich limit condition, as shown in Fig. 7(a). In this case, the estimated concentration of VS is 0.6–4.7×109 cm−2 (or 1.0–7.8×1016 cm−3 with supposing the same formation energy of VS in MoS2 bulk) at 1200 K (growth temperature) in thermodynamic equilibrium. Therefore the VS is an important (abundant) defect in single-layer MoS2 , whereas it acts as an electron trap center with ionized into VS1− in a typical n-type MoS2 . The positively charged states of VS are not stable in 6

6 (a) Mo-rich

5

3+

3-

4 2+

3 2 1 0

1+

0

VS

1VS

2-

0.5 1 1.5 EFermi-EVBM (eV)


D. S vacancy

FIG. 6. (Color online) Electronic orbitals of the (a) singlet a1 and (b) doublet e states of VS0 in a single-layer MoS2 . The charge isosurface is 0.003e per a supercell (α = 6). The total (black) and local (red) density-of-states for (c) VS0 and (d) VS1− in the α = 6 supercell are shown, and the a1 and e defect states inside the band gap are indicated. The Fermi levels are shown by the (red) dashed lines in (c) and (d).


Mo-rich limit condition and EFermi = 0 eV (VBM)]. While super the calculated Eform (α) of the neutral S0i is nearly independent of the supercell size α (for α > 4) [the (black) filled squares super in Fig. 5(a)], the Eform (α) of the S1− strongly depends on i the α [the (black) filled circles in Fig. 5(a)], as in the case of VS . By using the extrapolation scheme of Eq. (14), we fit the super line in Eform (α) of S1− i , and the fitting curve [the (red) thick super Fig. 5(a)] is found to be in a good agreement with the Eform (α) iso = 4.61 eV and t3 = 4.26]. [the fitting parameters are Eform Thus the formation energy of an isolated S1− i in a single-layer iso = 4.61 eV in the Mo-rich limit MoS2 is obtained to be Eform condition, when the EFermi is at the VBM. We explicitly calculate the qV term and thus Ecorr by comparing the calculated model [VM (r)] and DFT [VDFT (r)] potentials. The center of the (1-) Gaussian model charge in this ˚ shifted site (toward the Si site) case of Si is placed at the 0.26 A from the center of the slab [Fig. 5(d)]. This choice removes the dipole moment of the ρM (r)−ρDFT (r) charge distribution, as q can be seen in the VM (r) − VDFT (r) profile along the z direction super in the vacuum [Fig. 5(f)]. The electrostatic energies [EM (α)] of the off-center Gaussian model charge are different only by less than 3 meV from those of the Gaussian model charge, of which the center is exactly at the center of the slab, and thus super we use the same EM (α) as shown in Figs. 3(d) and 3(e). The 4, 6, 8, and 10 supercells calculated Ecorr for S1− i in the α = super iso is compared with the Eform − Eform (α) in Fig. 5(b), which shows a reasonable agreement with each other. The calculated iso Eform of S1− i with including the calculated Ecorr in the α = 4, 6, 8, and 10 supercells is shown in Fig. 5(a) by the (gray) filled circles, which shows a weak dependence on the supercell sizes iso and fast convergence to the extrapolated Eform value of 4.61 eV.


(b) S-rich 5 4 3

1+

1-

0 VS

2-

VS

2 1 0


FIG. 7. Calculated formation energies of an isolated VS in a single-layer MoS2 in various charge states as a function of the Fermi level inside the band gap in the (a) Mo-rich and (b) S-rich limit conditions. The stable charge states are shown by the solid lines.

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a single-layer MoS2 , and it cannot be a source of electron carriers. The electronic density of states of VS1− is shown in Fig. 6(d). The VS1− state, which is stable in the n-type single-layer MoS2 , is spin-polarized having the spin moment of μ = 1/2. The e states are singly occupied, and the Fermi level crosses the doubly degenerate e levels of the majority spin. Whether the spin moments of VS1− ’s in a single-layer MoS2 have a magnetic ordering is not clear yet and beyond our scope. The VS2− state can have a μ = 1 spin moment, but it is unstable. When the electrons occupy the e states, the strong on-site electron-electron (repulsive) interaction at the VS site destabilizes the q < −1 charged states, as can be seen from the higher e levels in VS1− than those in VS0 [Figs. 6(c) and 6(d)]. Thus, the independent electron picture as expected from the KS levels of VS0 [Fig. 6(c)] does not correctly describe the electron excitations in a single-layer MoS2 . We also check the formation energies and acceptor transition level of the VS in a single-layer MoS2 by using the hybrid functional of Heyd-Scuseria-Ernzerhof (HSE) with a mixing parameter ˚ −1 [40,41]. The of 0.25 and a screening parameter of 0.2 A VS is found to be a deep single acceptor in HSE as well as in LDA. The (0/−) transition level is calculated to be (0/−) = EVBM + 1.9 eV, where the band gap is 2.3 eV in HSE. The acceptor transition level obtained in the HSE is only slightly deeper than in the LDA. The formation energies of VS are calculated to be 1.4–1.8 eV in HSE, which is slightly higher than in LDA. E. S interstitial

The atomic and electronic structures of a S interstitial (Si ) in a single-layer MoS2 are shown in Fig. 8. The most stable atomic configuration of Si in single-layer MoS2 is found to be the S adatom structure on top of a host S atom, as shown in Figs. 8(a)–8(c). We find two other meta-stable atomic configurations for Si : the bridge configuration of Si bonding with two host S atoms on the surface (2.8 eV higher in energy in the neutral charge state) and the hexagonal interstitial in the

Mo layer bonding with three host Mo atoms (6.4 eV higher in energy in the neutral charge state). The S-Si bond length in the ˚ S adatom configuration is 1.912 A. The electronic density-of-states of the on-top Si in a single-layer MoS2 in the neutral charge state is shown in Fig. 8(d). Three defect states are found to be generated inside the band gap (or near the band edges) by the Si : doubly degenerated occupied px px π ∗ and py py π ∗ states near the VBM and an empty pz pz σ ∗ state near the CBM, where the Fermi level is located in between the ppπ ∗ and ppσ ∗ levels in the neutral S0i state [Fig. 8(d)]. The corresponding electronic orbitals are shown in Figs. 8(a)–8(c). The electronic structure of Si can be easily understood based on the diatomic p-p orbital hybridization. In the Si on-top configuration, the Si atom forms a Si -S bond with a host S atom. The host S atom is an anion that is in the S2− oxidation state in the single-layer MoS2 , while the Si is in the neutral S0i state. The one ppσ and two ppπ bonding states are fully occupied by the six p electrons [located deep inside the valence bands, not shown in Fig. 8(d)] and the remaining four p electrons occupy the two ppπ ∗ anti-bonding state [located near the VBM, as shown in Fig. 8(d)]. Then, the one ppσ ∗ state remains empty [located near the CBM, as shown in Fig. 8(d)] among the diatomic p-p hybridizations. The fully occupied three bonding (1 ppσ and 2 ppπ ) levels and the (only) one empty antibonding (1 ppσ ∗ ) level contribute to the S-Si linear diatomic chemical bonding nature. The electronic configuration of the S2− -S0i reminds us the well-known peroxide (O2− 2 ) configuration in oxides [42]. Since the occupied ppπ ∗ and unoccupied ppσ ∗ levels are located at both the ends of the band gap [Fig. 8(d)] in S0i , neither the excitation of the ppπ ∗ electrons nor the occupation of the ppσ ∗ level is expected to be likely, unless the Fermi level is beyond the VBM or CBM. The calculated formation energies iso ) of an isolated Si in a single-layer MoS2 in various (Eform charge states are plotted as a function of the Fermi level in the Mo-rich and S-rich limit conditions in Fig. 9 (for the α = 6 supercell with including the Ecorr ). The only stable charge state of Si is found to be neutral, when the Fermi level is inside the band gap. Therefore Si is an electrically neutral defect in a single-layer MoS2 . The formation energy of S0i is 1.0 eV in the S-rich limit condition, from which the estimated

FIG. 8. (Color online) Electronic orbitals of the [(a) and (b)] (doubly degenerated) ppπ ∗ and (c) ppσ ∗ states of S0i in a single-layer MoS2 . The charge isosurface is 0.003e per a supercell (α = 6). (d) The total (black) and local (red) electronic density-of-states for S0i in the α = 6 supercell is shown, and the ppπ ∗ and ppσ ∗ defect states near the band gap edges are indicated. The Fermi level is shown by the (red) dashed line.

6 (a) Mo-rich

5

2+

2-

4 1+

3

0

1-

Si

2 1 0

0




6

(b) S-rich

5

3+

4 2+

3 2

1+

1 0

0

2-

0

1-

Si


FIG. 9. Calculated formation energies of an isolated Si in a singlelayer MoS2 in various charge states as a function of the Fermi level inside the band gap in the (a) Mo-rich and (b) S-rich limit conditions. The stable (neutral) charge state is shown by the solid line.

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9

F. Mo vacancy

32-

8 1+ 0

VMo

7

1VMo

6 5 4

The atomic structure of a VMo in a single-layer MoS2 is shown in Figs. 10(a)–10(d). The formation of a VMo accompanies breaking six Mo-S bonds where the Mo is eliminated, and the six S dangling bonds remain in the VMo 0 structure. In the neutral VMo , the surrounding six S atoms are ˚ and by 0.062 A ˚ in the slightly outward relaxed by 0.034 A, 1− . (1−) charged VMo The electronic density of states of VMo in a single-layer MoS2 in the neutral charge state is shown in Fig. 10(e) (for the α = 6 supercell). Since the VMo has trigonal symmetry and mirror symmetry between the two S layers out of the Mo layer, two singlet (a1 ) and two doublet (e) states are expected to be induced. One group of the a1 and e states comes from the six S-p orbitals mainly, as shown in Figs. 10(a) and 10(b) (denoted as a1 -p and e-p, respectively), and the other group of the a1 and e states comes from the six S-p and six (second nearestneighbor from the VMo ) Mo-d orbitals, as shown in Figs. 10(c) and 10(d) (denoted as e-pd and a1 -pd, respectively). The lowest a1 -p level is found inside the valence bands [0.4 eV below the VBM, as shown in Fig. 10(e)], and the other five defect levels are found inside the band gap. The

10 (a) Mo-rich

0



10 Formation energy (eV)

equilibrium concentration of Si is 1.1×1011 cm−2 at 1200 K (growth temperature) in a single-layer MoS2 . Although the Si is an electrically inactive defect, its concentration is expected to be rich in a single-layer MoS2 in S-rich conditions. Inside MoS2 bulk, the atomic configuration of Si is different from the on-top configuration, because there is no enough space to accommodate a Si in the interlayer spacing, having a higher formation energy.


(b) S-rich

9

3+

8

4-

7 2+

3-

6 1+

5 4

20

VMo

0

1-

VMo


FIG. 11. Calculated formation energies of an isolated VMo in a single-layer MoS2 in various charge states as a function of the Fermi level inside the band gap in the (a) Mo-rich and (b) S-rich limit conditions. The stable charge states are shown by the solid lines.

e-p doublet is occupied, and the e-pd and a1 -pd states are 0 unoccupied in the neutral VMo . The e-pd doublet level is found to be lower than the a1 -pd singlet level [see Fig. 10(e)], which is considered to be a pd-hybridization effect. All the six defect states are induced from the valence bands by the perturbation of the formation of VMo , and the absence of a Mo atom, which has six valence electrons, remains the top three defect levels (the a1 -pd singlet and e-pd doublet) empty, and thus the Fermi level is located in between the e-p and e-pd levels in the neutral 0 . charge state of VMo iso The calculated formation energies (Eform ) of an isolated VMo in a single-layer MoS2 in various charge states are plotted in Fig. 11, as a function of the Fermi level in the Mo-rich and S-rich limit conditions (for the α = 6 supercell with including the Ecorr ). The only stable charge states of VMo are found to be (0) and (1−), when the Fermi level is inside the band gap. The (0/−) transition level is found at (0/−) = EVBM + 1.1 eV. Therefore VMo is a deep single acceptor in a single-layer iso of VMo is found to be very high, as expected MoS2 . The Eform from the large number of S dangling bonds around the VMo . It is about 7–8 eV in the Mo-rich limit condition and about 4–5 eV in the S-rich limit condition, as shown in Fig. 11. Thus, the concentration of VMo in a single-layer MoS2 is expected to be extremely low in thermodynamic equilibrium. However, by applying for examples particle irradiation or atomic manipulation techniques, the VMo can be artificially 1− generated in a single-layer MoS2 experimentally. The VMo , which is stable in n-type single-layer MoS2 , is found to be spin-polarized, as shown in Fig. 10(f), with the spin having the e-pd electronic orbital. G. Mo interstitial

FIG. 10. (Color online) Electronic orbitals of the (a) a1 -p singlet, 0 (b) e-p doublet, (c) e-pd doublet, and (d) a1 -pd singlet states of VMo in a single-layer MoS2 . The charge isosurface is 0.003e per a supercell (α = 6). The total (black) and local (red) electronic density-of-states 1− 0 and (f) VMo in the α = 6 supercell are shown, and the for (e) VMo a1 -p inside the valence bands and e-p, e-pd, and a1 -pd defect states inside the band gap are indicated. The Fermi level is shown by the (red) dashed line.

We investigate various atomic configurations for Moi in a single-layer MoS2 , and find that the most stable one is the Mo-Moi spilt interstitial along the c direction, as shown in Fig. 12(a). We find four other meta-stable atomic configurations for Moi : the bridge configuration of Moi in between two surface S atoms (0.8 eV higher in energy in the neutral charge state), the Moi at a hexagonal hollow center on the surface (1.3 eV higher in energy), the Moi at a hexagonal hollow center in the Mo layer (2.9 eV higher in energy), and the

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PHYSICAL REVIEW B 89, 205417 (2014) 9 3-

(a) Mo-rich

8

7 3+ 2-

6 2+

5

1-

4 Mo 1+ i 3

0

0

Moi



9

1-

7

1+

Moi

0

Moi

6 5 4 3


(b) S-rich

8

0


FIG. 13. Calculated formation energies of an isolated Moi in a single-layer MoS2 in various charge states as a function of the Fermi level inside the band gap in the (a) Mo-rich and (b) S-rich limit conditions. The stable charge states are shown by the solid lines.

We plot the calculated formation energies of the isolated native defects of VS , Si , VMo , and Moi in a single-layer MoS2 for their stable charge states as a function of the Fermi level in Fig. 14. In the Mo-rich limit condition, the VS is found to be the most stable. Its formation energy is only about 1.5 eV for the VS0 and about 1.2 eV for VS1− when the Fermi level is at the CBM. Thus the concentration of VS is expected to be abundant in Mo-rich single-layer MoS2 in equilibrium. It acts as a single electron trap center (a deep single acceptor), which can reduce the electron carrier concentration in n-type single-layer MoS2 . The stable VS1− is a spin-polarized defect. On the other hand, the Si is found to be the most stable in S-rich single-layer MoS2 . Its formation energy is only about 1.0 eV. Only the neutral charge state is stable in Si , and thus it does not play any role electrically in a single-layer MoS2 , even though its concentration is expected to be high. The Mo-related native defects of VMo and Moi are very high in formation energy above 4 eV, independently of the stoichiometry. Therefore the Mo-related defects are expected to be rare in a single-layer MoS2 in thermodynamic equilibrium, although they can be generated artificially by atomic excitation techniques such as electron irradiation and atomic manipulation. The electrical property of 10 9 (a) Mo-rich (0/1-) 8 VMo 7 6 5 (1+/0) Moi 4 3 Si (0/1-) 2 VS 1 0 0 0.5 1 1.5 EFermi-EVBM (eV)


Moi (Mo adatom) on top of a surface S atom (3.0 eV higher in energy). The Mo-Moi bond length in the split interstitial ˚ configuration is 2.079 A. The electronic density-of-states of the Moi (the spilt interstitial) in the neutral charge state is shown in Fig. 12(e). Five defect levels are found to be induced inside the band gap: an occupied doublet (e) [Fig. 12(b)], an occupied singlet (a1 ) [Fig. 12(c)] and an unoccupied doublet (e∗ ) [Fig. 12(d)] states, as indicated in Fig. 12(e). The inclusion of an additional Mo atom in a single-layer MoS2 (as a Moi ) provides six electrons, and the six electrons occupy the three defect levels from the lowest energy levels (the e and a1 levels), and thus the Fermi level is located in between the a1 and e∗ levels in the neutral Mo0i [Fig. 12(e)]. iso The calculated formation energies (Eform ) of an isolated Moi in a single-layer MoS2 in various charge states are plotted in Fig. 13, as a function of the Fermi level in the Mo-rich and S-rich limit conditions (for the α = 6 supercell with including the Ecorr ). The only stable charge states of Moi are found to be (1+) and (0), when the Fermi level is inside the band gap. The (+/0) transition level is found at (+/0) = EVBM + 0.3 eV. Therefore Moi is a deep single donor in a single-layer MoS2 . The formation energy of an isolated Moi is found to be very high. That of the neutral Mo0i is 4.3 eV in the Mo-rich limit condition and 7.2 eV in the S-rich limit condition, as shown in Fig. 13. Thus, the concentration of Moi in a single-layer MoS2 is expected to be extremely low in thermodynamic equilibrium. The Moi is the only native defect that has a donor character among the defects considered in this study. The Moi acts as a hole trap center in a single-layer MoS2 being ionized into 1+ Mo1+ is another spin-polarized defect, which is i . The Moi stable in p-type single-layer MoS2 , as shown in Fig. 12(f).

H. Stability


FIG. 12. (Color online) (a) Atomic structure of Mo0i (the MoMoi spilt interstitial configuration) in a single-layer MoS2 . Electronic orbitals of the (a) e doublet, (b) a1 singlet, and (c) e∗ doublet states of Mo0i in a single-layer MoS2 . The charge isosurface is 0.003e per a supercell (α = 6). The total (black) and local (red) electronic densityof-states for (d) Mo0i and (e) Mo1+ i in the α = 6 supercell are shown, and the e, a1 , and e∗ defect states are indicated. The Fermi level is shown by the (red) dashed line.

10 9 (b) S-rich 8 (1+/0) Moi 7 6 VMo (0/1-) 5 4 VS (0/1-) 3 2 Si 1 0 0 0.5 1 1.5 EFermi-EVBM (eV)

FIG. 14. (Color online) Calculated formation energies of the isolated native defects of VS , Si , VMo , and Moi in a single-layer MoS2 as a function of the Fermi level in the (a) Mo-rich and (b) S-rich limit conditions.

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VS 1.8

Si

VMo

Moi

1-

Energy level (eV)

1.6 1-

1.4 1.2 1 0.8 0.6

0 0

0 0

0.4 0.2

1+

0 FIG. 15. (Color online) Stable charge states and transition levels of the isolated native defects of VS , Si , VMo , and Moi in a single-layer MoS2 .

VMo is characterized by a single deep acceptor or an electron trap center, while that of Moi is by a single deep donor or a hole 1− and hole-trapped Mo1+ trap center. The electron-trapped VMo i are spin-polarized. The stable charge states and transition levels of the native defects are summarized in Fig. 15.


rigorously treat the finite-size supercell effects, by which the artificial electrostatic interaction energy between image charges are eliminated through a correction scheme that uses a gaussian model charge, and we obtain the formation energies of the isolated native defects in a single-layer MoS2 in various charge states. It is found that the VS and Si are low in formation energy, about ∼1 eV, in the Mo-rich and S-rich limit conditions, respectively, but their carrier doping ability is found to be poor, as the VS is a deep single acceptor [(0/−) = EVBM + 1.7 eV] and the Si is an electrically neutral defect. The Mo-related native defects of VMo and Moi are found to have high formation energies above 4 eV, and the electrical properties of VMo and Moi are characterized by a single deep acceptor [(0/−) = EVBM + 1.1 eV] and a single deep donor [(+/0) = EVBM + 0.3 eV], respectively. Based on the results, we conclude that the native defects cannot be efficient dopants in single-layer MoS2 , but act as electron (VS and VMo ) or hole (Moi ) trap centers, by which the electrical conductivity in single-layer MoS2 -based electronic devices can be suppressed in presence of the native defects.

ACKNOWLEDGMENTS

We investigate the native defects of VS , Si , VMo , and Moi in a single-layer MoS2 through density-functional theory calculations. For charged states of the native defects, we

This work was supported by the Nano R&D Program (No. 2013-0042633) and by the EDISON program (No. 2012M3C1A6035304) through the National Research Foundation (NRF) of Korea. HK acknowledges the use of the computing facilities through the Strategic Supercomputing Support Program from Korea Institute of Science and Technology Information (No. KSC-2008-S02-0010).

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IV. CONCLUSION

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