Introduction

Two-dimensional (2D) semiconductors beyond graphene, such as transition metal dichalcogenides,1,2,3 boron nitride,2,4,5 black phosphorus,6,7,8 group-III chalcogenides,9,10 and thin layers of topologically insulating Bi2Te3, Bi2Se3, and Sb2Te311 have attracted considerable attention in recent years. Their unique advantages in scaling semiconductors into atomic layers have raised the prospect of possible continuation of the Moore’s Law.12,13 One such example is the recently proposed 2D MoS2-based metal-oxide-semiconductor field-effect transistor which features a gate length of only 1 nm but excellent switching characteristics and an on–off current ratio of 106.14 To broaden the applications in electronics, however, it is desirable to dope these 2D semiconductors by impurities,12,13,15 which is a standard procedure for carrier transport in three-dimensional (3D) semiconductors. For instance, to fabricate silicon-based complementary-metal-oxide-semiconductor integrated circuits, silicon requires both p- and n-type doping.

Developing electronics based on 2D semiconductors can be complicated. First, a significant reduction of the dielectric constant ε from 3D to 2D can be expected, which will adversely impact carrier ionization from dopants. Second, several theoretical studies have predicted that ionization energy in 2D semiconductors can be deeper than that in their 3D counterparts, well beyond the reduction of ε.16,17,18,19 As such, the difficulty in ionizing dopants can make it particularly challenging to yield a reasonable carrier concentration. In spite of these difficulties, experiments have consistently shown a certain degree of n- or p-type conductivity in 2D semiconductors.20,21,22,23,24 This apparent contradiction hints that defect physics in 2D semiconductors could be qualitative different from that in 3D.

In this paper, we identify a unique ionization process of impurities for carrier transport in 2D semiconductors. While our systematic examination by first-principles calculations of defects and impurities in a prototypical 2D semiconductor MoS2 indeed reveals very deep ionization energies IE∞, due to the reduced dimensionality and reduced electronic screening, there is also a substantial Coulomb binding Edb(n) between charge carriers and ionized defects, where n is a quantum number in qualitative analogy to the hydrogenic model.25 Hence, besides the fully ionized n = ∞ state, carriers in the ground state of defects can also be excited to a set of defect-bound band edge (DBBE) states with an excitation energy IEn = IE∞ − Edb(n). In particular, the n = 1 state is spatially localized when compared to the n = ∞ state but noticeably more delocalized when compared to the ground state of the defect. Note that the hydrogenic model here is only a rather crude approximation. In the case of ReMo, for instance, the n = 1 DBBE state is above the conduction band minimum (CBM), which pushes the CBM down to form a new band whose effective mass at K is only 30% larger than that in defect-free system. This non-intuitive result may help reconcile theory with experiment on doping MoS2 and pave the way for eventual applications of all the 2D materials for electronics.

Results and discussion

Formation energies and density of states

To evaluate defect ionization, we calculate both IE∞ and the density of states (DOS) of charge neutral state (q = 0). Here, defects include native defects, i.e., vacancies: VS, VMo, and V2S, and antisites: MoS, SMo, and Mo2S with the (−2), (−1), (0), (+1), and (+2) charge states, and extrinsic impurities, i.e., NbMo and ReMo with the (−1), (0), and (+1) charge states, respectively. Among the native defects, VS has the lowest neutral formation energy as shown in Fig. 1a, which is in line with the relative ease to observe such defects in MoS2.26,27 Other native defects have higher energies by at least a factor of 2, as can be seen in Figs. S1 and S2 in the Supplemental Materials (SM). According to Fig. 1b, extrinsic impurities ReMo/NbMo have relatively low neutral formation energies too. Although both VS and ReMo/NbMo are low in energy, their electronic behaviors are qualitatively different. For example, VS prefers to be in the charged states of (0) and (−1) with a transition level at ε(0/−1) = EVBM + 1.55 eV, where VBM denotes the valence band maximum (while CBM stands for the conduction band minimum). Hence, as an acceptor, its IE∞ for holes is very large. The same deepness is quantitatively reflected by the deep gap states in the DOS. In contrast, IE∞’s for ReMo (donor) and NbMo (acceptor) are 0.45 and 0.55 eV, respectively, which are high enough to prevent a full carrier ionization at room temperature. Here, we also used the revised Tao–Perdew–Staroverov–Scuseria meta-generalized gradient approximation (RTPSS meta-generalized gradient approximation (GGA)),28 which is above the GGA rung in the “Jacob’s ladder” of approximations, to calculate IE∞ for ReMo. The RTPSS results are similar to those of Perdew–Burke–Ernzerhof (PBE), e.g., with a vacuum region of 20 Å, the IE∞ is 0.41 eV (PBE) and 0.39 eV (RTPSS), respectively. Note that RTPSS still underestimates the bandgap compared with many-body perturbation theory GW calculations and experiment.29,30 This suggests that the actual ionization energy may be even larger than what has been predicted here. However, according to the DOS computed for a finite cell size, both appear as delocalized states and are shallow with respect to the respective band edges, which suggests that there could be a delocalized-to-localized (or shallow-to-deep) transition for ReMo and NbMo with the concentration of dopants.

Fig. 1
figure 1

Density states and formation energies of defects. Density of states and formation energies of a a typical intrinsic defect, VS and b typical extrinsic dopants, ReMo and NbMo in 2D MoS2. Acceptor and donor states are labeled with A and D in DOS, respectively. The local DOS for individual Re and Nb are also shown with a factor of 10 (red lines). Fermi level is set to 0 eV. The possible valences (noted by red number) of defects are fully considered for their formation energies under Mo- and S-rich condition

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Transition levels and Coulomb binding energies

As mentioned earlier, the binding energy of DBBE states can be rather significant. Figure 2 shows Edb (n = 1), as well as the ground-state energy or transition level, with respect to the band edges (CBM and VBM) for donors and acceptors, respectively. We see that, in 2D MoS2, Edb = 0.25–0.56 eV can be quite large. The details can be also found in Table S1 (Supplemental Materials). Therefore, in 2D semiconductors most charge carriers are bound carriers, oppose to be almost free in 3D semiconductors.

Fig. 2
figure 2

Transition levels and binding energies. Binding energy between ionized carrier (at DBBE state) and its oppositely charged defect Edb (n = 1) in monolayer MoS2. The transition levels according to the IE∞ (solid line) of acceptors (0/−1) and donors (+1/0) within the calculated band gap. The Edb are marked with color coding next to band edge: green for acceptors and orange for donors

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To understand why Edb can be so large, we again resort to the hydrogenic model. In a 3D system, the dielectric screening ε is usually large so the binding is weak as shown in Fig. 3.25,31,32 This leads to \(E_{\mathrm {db}}(n,{\mathrm {3D}}) = \frac{1}{{n^2\varepsilon ^2}}R_y\), where Ry is the Rydberg energy and n runs from 1 to ∞. In analogy, for our 2D systems, the model25,33-34 yields \(E_{{\mathrm {db}}}(n,{\mathrm {2D}}) = \frac{1}{{(n - 1/2)^2\varepsilon ^2}}R_y\). Numerical results are schematically shown in Fig. 3a for 3D and in Fig. 3b for 2D. Note that holes are mirror images of electrons with different effective masses. They are schematically illustrated in Fig. 3c for 3D and in Fig. 3d for 2D. The ratio of the binding energy between 2D and 3D = Edb (n = 1, 2D)/Edb (n = 1, 3D) can be very important, which is enhanced by a factor of 4[ε (3D)/ε(2D)]2 for both electrons and holes. Here, we stress that the hydrogenic model is just used to qualitatively illustrate the fundamental difference between 2D and 3D in terms of Coulomb binding. The model has, however, not been used in any quantitative evaluation of the physical properties.

Fig. 3
figure 3

Schematic illustration of charge-carrier ionizations in a, c 3D and b, d 2D semiconductors. Emphases have been placed on the 2D cases where IE∞ is the energy of an electron or a hole completely removed from their respective defect states (labeled as i). Due to the Coulomb attraction by the ionized defect, there should be a set of bound states (derived from the band edge states), namely, the DBBE states discussed in the main text. IE1 corresponds to moving a hole or an electron from the defect state i to n = 1 DBBE state, i.e., IE1 = IE∞ − Edb (n = 1). Here, we have used the hydrogenic model for illustration purpose with the understanding that the model needs not represent a real defect. As such, the principal quantum number n here only indicates the order of the energy levels

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Spatial distribution of DBBE states

As mentioned earlier, the spatial distribution of n = 1 DBBE state can be considerably different from that of defect ground state. Taking acceptors as an example, Fig. 4 shows the spatial distribution of the n = 1 hole state for a number of native defects and NbMo. Note that to study the spatial distribution, the supercell needs to be large enough. For this reason, we show in Fig. 4 two sets of results calculated with a small 147-atom supercell and a larger 507-atom cell. The 147-atom cell has been used to calculate Edb. The same trend in the localization of the DBBE states can be clearly seen in these two sets of calculations, and the holes become more and more localized with increasing Edb from 0.0 (no defect) to 0.44 eV.

Fig. 4
figure 4

Real-space distributions of n = 1 hole states for various acceptor-like defects in 2D MoS2, calculated by using a small 147-atom cell and a larger 507-atom cell. Green isosurface of density is set to 4 × 10−4 and 1.2 × 10−4e/a0−3 for the small and larger cells, respectively, where ɑ0 is the Bohr radius. Circles at the center of the supercells denote the positions of the defects. Histograms at the bottom of the figure show the corresponding binding energies (Edb)

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The presence of a large Edb associated with the DBBE states resolves both of the aforementioned perplexing issues. Firstly, it explains the deep levels associated with ReMo and NbMo that were found in the calculation of the ionization energy IE∞. This can be traced back to the poor screening in the case of 2D semiconductors. As the ionization energy is the difference between the charged and neutral formation energies, \(\varepsilon (q/0) = \frac{{\Delta H(D^q) \,-\, \Delta H(D^0)}}{q}\), the lack of screening means that the charged defect is closer to a bare charge, substantially increasing ΔH(Dq) relative to the 3D counterpart. Increasing the formation energy of the charged defects relative to the neutral defect naturally leads to an increase in the ionization energies. This, in turn, is linked with the large DBBE binding as the reduced screening opens the door for the ionized carriers to effectively screen the charged defects, substantially lowering the energy of the ionized carriers when in the DBBE states.

Carrier transport mechanism

Secondly, the large Edb points toward an experimental mechanism for conduction despite the large ionization energies associated with these defects. ReMo constitutes such an example with a relative small IE∞ = 0.45 eV. Yet, this IE∞ is large enough to prevent significant carrier excitation at room temperature. Most surprisingly, however, the lack of IE1 in ReMo can be seen in Fig. 5 where the donor is spontaneously ionized. As a result, the pink ReMo-derived states in Fig. 5b are in fact above the CBM. It pushes down the bulk conduction band states. We propose that with a reasonable defect density, such as 1013 cm−2 in Qiu’s study,35 the pushed-down states form a band ideally suited for carrier conduction. To elaborate, Fig. 5c, d show the push-down states where a certain degree of charge overlap has been witnessed. Due to the presence of push-down states, electrons could transport through the band with an effective mass that is only 30% larger than that of the free electron at the CBM, as shown in Table S2 (Supplemental Materials).

Fig. 5
figure 5

Defect bound band edge transport with increasing defect density. Band structure of a no-defect and b Re-doped monolayer MoS2 with a defect density of about 1013 cm−2. The spatial distribution of the defect bound band c and its corresponding linear charge density d. The isosurface in c is 3 × 10−4e/a0−3, where a0 is the Bohr radius. Dashed circles denote the positions of the defects

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In summary, our first-principles calculations identify a set of DBBE states, which could have pronounced effects on the defect ionization in 2D semiconductors. The reason for the formation of the DBBE states is the strong binding between charged ions and ionized charge carriers, leading to an extra channel for low-energy excitation. The strong binding is a result of the spatial confinement as well as the reduced screening in 2D materials. The experimentally observed conductivity in Re-doped MoS2, on the other hand, could be a special case of DBBE-state transport where the n = 1 DBBE state enters the conduction band to push down the CBM. The present investigations suggest a unique picture of carrier ionization from defects and its implication to carrier transport in 2D semiconductors towards emerging nanoelectronic devices.

Methods

Calculation of ionization energies IE∞

All the calculations are performed within the density-functional theory (DFT)36,37 as implemented in the Vienna ab-initio simulation package (VASP).38,39 The projector-augmented plane wave basis and GGA with the PBE functional form are employed.40 The cutoff energy for the plane wave basis is 520 eV and a 3 × 3 × 1 Monkhorst-Pack mesh grid is used for k-point sampling. Spin polarization is included. The calculated lattice parameter is 3.185 Å and the PBE bandgap is 1.66 eV for 2D MoS2, which agree well with previous calculation.17 To obtain the ionization energy (IE∞) and the corresponding formation energy of the charge defect in 2D system, we use the extrapolation method in our previous work,18

$${\mathrm {IE}}_\infty (S,L_z) = {\mathrm {IE}}_\infty + \frac{\alpha }{{\sqrt S }} + \frac{{\beta L_z}}{S}$$
(1)

where IE∞ (S, Lz) is the size-dependent ionization energy, IE∞ is the true, size-independent ionization energy. S and LZ are lateral size and vacuum size, respectively. α is the defect-specific Madelung constant and \(\beta = \frac{{e^2}}{{24\varepsilon _0}}\). Note that this expression neglects higher order terms, which may not be negligible for thicker slabs.16,41,42 However, the error here for monolayer MoS2 (0.31 nm thickness) should be substantially less than 0.1 eV according to our previous studies, for example, the error for monolayer black phosphorus (0.21 nm thickness) is 0.04 eV and the error for thicker bilayer black phosphorus (0.77 nm thickness) is 0.13 eV.19 Here IE∞ is obtained at a fixed LZ = 40 Å with Lx × Ly ranging from 5 × 5 to 7 × 7. We have also tested the convergence of LZ, for example, for VS where IE∞ = 1.52, 1.52, and 1.55 eV, respectively, for LZ = 20, 30, and 40 Å.

Calculation of Coulomb binding energies E db

The Coulomb binding of the charge carrier with the corresponding charged defect is calculated by a fixed occupation method43 where we perform a constrained DFT calculation to treat the excited states. In the case when the defect state is degenerate, for example, the Sulphur vacancy acceptor (VS), we consider different electronic configurations for the excited states: e.g., exciting one spin-up electron, one spin-down electron, or half spin-up and half spin-down electrons. The results show that the binding energies are unchanged to within a couple of tens meV.