Abstract
The ionization of dopants is a crucial process for electronics, yet it can be unexpectedly difficult in twodimensional materials due to reduced screening and dimensionality. Using firstprinciples calculations, here we propose a dopant ionization process for twodimensional semiconductors where charge carriers are only excited to a set of defectbound band edge states, rather than to the true band edge states, as is the case in threedimensions. These defectbound states have small enough ionization energies but large enough spatial delocalization. With a modest defect density, carriers can transport through band by such states.
Introduction
Twodimensional (2D) semiconductors beyond graphene, such as transition metal dichalcogenides,^{1,2,3} boron nitride,^{2,4,5} black phosphorus,^{6,7,8} groupIII chalcogenides,^{9,10} and thin layers of topologically insulating Bi_{2}Te_{3}, Bi_{2}Se_{3}, and Sb_{2}Te_{3}^{11} 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 MoS_{2}based metaloxidesemiconductor fieldeffect transistor which features a gate length of only 1 nm but excellent switching characteristics and an on–off current ratio of 10^{6}.^{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 threedimensional (3D) semiconductors. For instance, to fabricate siliconbased complementarymetaloxidesemiconductor integrated circuits, silicon requires both p and ntype 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 ptype 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 firstprinciples calculations of defects and impurities in a prototypical 2D semiconductor MoS_{2} indeed reveals very deep ionization energies IE_{∞}, due to the reduced dimensionality and reduced electronic screening, there is also a substantial Coulomb binding E_{db}(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 defectbound band edge (DBBE) states with an excitation energy IE_{n} = IE_{∞} − E_{db}(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 Re_{Mo}, 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 defectfree system. This nonintuitive result may help reconcile theory with experiment on doping MoS_{2} 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: V_{S}, V_{Mo}, and V_{2S}, and antisites: Mo_{S}, S_{Mo}, and Mo_{2S} with the (−2), (−1), (0), (+1), and (+2) charge states, and extrinsic impurities, i.e., Nb_{Mo} and Re_{Mo} with the (−1), (0), and (+1) charge states, respectively. Among the native defects, V_{S} has the lowest neutral formation energy as shown in Fig. 1a, which is in line with the relative ease to observe such defects in MoS_{2}.^{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 Re_{Mo}/Nb_{Mo} have relatively low neutral formation energies too. Although both V_{S} and Re_{Mo}/Nb_{Mo} are low in energy, their electronic behaviors are qualitatively different. For example, V_{S} prefers to be in the charged states of (0) and (−1) with a transition level at ε(0/−1) = E_{VBM} + 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 Re_{Mo} (donor) and Nb_{Mo} (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 metageneralized gradient approximation (RTPSS metageneralized gradient approximation (GGA)),^{28} which is above the GGA rung in the “Jacob’s ladder” of approximations, to calculate IE_{∞} for Re_{Mo}. 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 manybody 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 delocalizedtolocalized (or shallowtodeep) transition for Re_{Mo} and Nb_{Mo} with the concentration of dopants.
Transition levels and Coulomb binding energies
As mentioned earlier, the binding energy of DBBE states can be rather significant. Figure 2 shows E_{db} (n = 1), as well as the groundstate energy or transition level, with respect to the band edges (CBM and VBM) for donors and acceptors, respectively. We see that, in 2D MoS_{2}, E_{db} = 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.
To understand why E_{db} 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 R_{y} is the Rydberg energy and n runs from 1 to ∞. In analogy, for our 2D systems, the model^{25,3334} 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 = E_{db} (n = 1, 2D)/E_{db} (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.
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 Nb_{Mo}. 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 147atom supercell and a larger 507atom cell. The 147atom cell has been used to calculate E_{db}. 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 E_{db} from 0.0 (no defect) to 0.44 eV.
The presence of a large E_{db} associated with the DBBE states resolves both of the aforementioned perplexing issues. Firstly, it explains the deep levels associated with Re_{Mo} and Nb_{Mo} 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(D_{q}) 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 E_{db} points toward an experimental mechanism for conduction despite the large ionization energies associated with these defects. Re_{Mo} 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 IE_{1} in Re_{Mo} can be seen in Fig. 5 where the donor is spontaneously ionized. As a result, the pink Re_{Mo}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 10^{13} cm^{−2} in Qiu’s study,^{35} the pusheddown states form a band ideally suited for carrier conduction. To elaborate, Fig. 5c, d show the pushdown states where a certain degree of charge overlap has been witnessed. Due to the presence of pushdown 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).
In summary, our firstprinciples 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 lowenergy 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 Redoped MoS_{2}, on the other hand, could be a special case of DBBEstate 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 densityfunctional theory (DFT)^{36,37} as implemented in the Vienna abinitio simulation package (VASP).^{38,39} The projectoraugmented 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 MonkhorstPack mesh grid is used for kpoint sampling. Spin polarization is included. The calculated lattice parameter is 3.185 Å and the PBE bandgap is 1.66 eV for 2D MoS_{2}, 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}
where IE_{∞} (S, L_{z}) is the sizedependent ionization energy, IE_{∞} is the true, sizeindependent ionization energy. S and L_{Z} are lateral size and vacuum size, respectively. α is the defectspecific 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 MoS_{2} (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 L_{Z} = 40 Å with L_{x} × L_{y} ranging from 5 × 5 to 7 × 7. We have also tested the convergence of L_{Z}, for example, for V_{S} where IE_{∞} = 1.52, 1.52, and 1.55 eV, respectively, for L_{Z} = 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 method^{43} 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 (V_{S}), we consider different electronic configurations for the excited states: e.g., exciting one spinup electron, one spindown electron, or half spinup and half spindown electrons. The results show that the binding energies are unchanged to within a couple of tens meV.
Data availability
The data that support the findings of this study are available from the corresponding author, Professor XianBin Li (email: lixianbin@jlu.edu.cn) upon reasonable request.
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Acknowledgements
Work in China was supported by National Natural Science Foundation of China (No. 11874171, No.11504368, No. 61775077, and No. 11704111). D.W. and S.Z. were supported by the Department of Energy under Grant No. DESC0002623. W.Q.T. thanks support from the Open Project of Key Laboratory of Polyoxometalate Science of Ministry of Education (NENU) and State Key laboratory of Supramolecular Structure and Materials (JLU) (No. SKLSSM201818). Also, we acknowledge the HighPerformance Computing Center (HPCC) at Jilin University for calculation resources.
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D. Wang, D.H., and D. West did the calculations. D. Wang, D. West, X.L., and S.B.Z. did the theoretical analyses. The paper is written by D. Wang, X. L., and S.B.Z. with the help from all the authors. D. West and S.B.Z. were actively engaged in the design and development of the theory, participated in all discussions, and draft of the manuscript. All the authors contributed to the interpretation of the results. X. L. proposed and initiated the project. D. Wang, D.H., and D. West contribute equally to this work.
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Correspondence to XianBin Li.
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