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Abstract
The critical condition and mechanism of the insulator-to-metal transition (IMT) for the black diamond were studied by the molecular-dynamics-Landauer method. The IMT will occur at sufficiently high contents of vacancies in the diamond. The critical concentration of vacancies for the IMT might be between V:C143 (0.69%) and V:C127 (0.78%). At a low concentration of vacancies (below 0.69%), the intermediate band (IB) consists of a filled band and a separate empty band, which makes the material to be an insulator. The IMT of the black diamond is due to the mergence between the two isolated IBs when the concentration of vacancies is high, and the merged IB is partially filled by electrons. The distribution of vacancies also influences the IMT of the black diamond.
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1. Introduction
A novel and attractive nano-photoelectronic material called “black diamond”, which is prepared by irradiating the chemical vapor deposition (CVD) diamond with ultra-fast pulsed laser, has been proposed in recent years [1–5]. The black diamond exhibits amazing optical and photoelectronic properties, e.g., it can absorb the visible and near-infrared sunlight strongly and exhibit noticeable sub-bandgap responsivity [1–5]. The unique properties of the black diamond will greatly extend the application of traditional diamond, which has a wide bandgap of 5.47 eV and is optical transparent to the visible and near-infrared wavelength range [6–8].
According to the calculational results, the unique optical property of black diamond is attributed to the introduction of high concentration of vacancy defects in the diamond during the preparation [9]. The high concentration of vacancies in the diamond can form a partially filled intermediate-band (IB) in the wide bandgap of diamond, which can absorb sub-bandgap energy photons by exciting electrons from valence band (VB) to the IB and from the IB to conduction band (CB) [9]. The electronic structure of black diamond is similar to that of IB solar cell proposed by Luque et al., which can break the limiting efficiency of 41% for single-gap solar cells and has an especially high theoretical efficiency of 63% [10–13]. Based on these reports, the black diamond will have remarkable application potential in photoelectronic detectors and solar cells.
The experimental results indicate that, compared with the remarkable sub-bandgap absorption, the responsivity of black diamond is not outstanding [3], the reason of which is not clear yet. We presume that one of the main reasons should be that the introduction of defect states in the diamond bandgap might cause the Shockley-Read-Hall (SRH) non-radiative recombination and limit the photoelectric conversion. According to the reports of Luque et al., the SRH non-radiative recombination can be suppressed when the impurity states in the bandgap are extended and the Mott insulator-to-metal transition (IMT) occurs [10], which was confirmed by experiment [14]. Therefore, for the black diamond, IMT should be a key to further improve its photoelectric conversion efficiency and it has important significance to study the critical condition of the IMT.
So far, the critical condition of the IMT in black diamond is still unclear and has not been investigated in both of theory and experiment. In this work, we studied the critical condition of IMT in black diamond in detail by using molecular dynamics (MD) simulations in combination with standard Landauer transmission calculations. Based on the temperature dependent transmission results, the temperature dependent conductance of the configurations with different content of vacancies were obtained from the Landauer formula. Combined with the reliable electronic structure calculations, the critical condition and physical mechanism of IMT in black diamond were investigated.
2. Computational methods
All of the results calculated in this work were performed in Atomistix Toolkit. The band structures and density of states (DOS) were calculated by GGA-1/2 method with the Perdew, Burke, and Ernzerhof (PBE) functional and tight-binding (Slater-Koster) method [15–17]. The GGA-1/2 method is a semi-empirical method, which is fast and can give accurate electronic structures for the IB material [15,18]. For the GGA-1/2 calculations, the linear combinations of atomic orbitals (LCAO) were used to expand the wave functions of the valence states and the SG15 pseudopotential was used for the C element in the basis set [19,20]. To construct configurations of black diamond with different concentrations of vacancies, we constructed 2 × 2 × 2, 2 × 2 × 3, 3 × 3 × 2, and 3 × 3 × 3 supercells of the conventional C8 unit cell and the 4 × 4 × 4 supercell of the C2 primary cell, and removed one of the C atoms as vacancy. For the 2 × 2 × 2, 2 × 2 × 3, 4 × 4 × 4, 3 × 3 × 2, and 3 × 3 × 3 supercells, the concentration of vacancies is V:C63 (1.56%), V:C95 (1.04%), V:C127 (0.78%), V:C143 (0.69%), and V:C215 (0.46%), respectively. The configurations were optimized by the traditional GGA-PBE method. The force tolerance is 0.01 eV/Å and the stress error tolerance is 0.02 GPa. In the electronic structure calculations, the density mesh cut-off is 185 Ha and 20 Ha for the GGA-1/2 and Slater-Koster calculations, respectively. The k-point samplings are 4 × 4 × 4, 4 × 4 × 2, 2 × 2 × 4, and 2 × 2 × 2 for the 2 × 2 × 2, 2 × 2 × 3, 3 × 3 × 2, and 3 × 3 × 3 supercells of the C8 unit cell, and it is 3 × 3 × 3 for the 4 × 4 × 4 supercell of the C2 primary cell.
Based on the electronic structure calculations, we constructed three device configurations based on the 2 × 2 × 3, 3 × 3 × 2, and 4 × 4 × 4 supercells, respectively. The temperature dependent conductance of the configurations was obtained from the MD-Landauer method, which has been proved reliable for calculating the temperature dependent conductance of metals and semiconductors. In the MD calculations, a classical force field potential of Tersoff_C_2010 [21,22] with a langevin thermostat [23] were used, and run for 5 ps with a time step of 1 fs. After the MD simulations, a snapshot of the configuration was taken and the electronic transmission was calculated by standard Green’s function. The Slater-Koster method was used for the electronic transmission calculations. Since the calculations of MD and transmission are rapid, we performed 10 different MD simulations for each configuration to get an average transmission. The density mesh cut-off is 20 Ha for all of the device configurations. The k-point samplings are 4 × 2 × 260, 2 × 2 × 260, and 2 × 2 × 120 for the 2 × 2 × 3, 3 × 3 × 2, and 4 × 4 × 4 supercells.
3. Results and discussion
The electronic structures of the configuration in different supercells are shown in Fig. 1. The calculations were performed by GGA-1/2 and Slater-Koster methods. By comparison, it can be found that there is significant difference in the total bandgaps (VB-CB) and sub-bandgaps (VB-IB and IB-CB) calculated by the two methods. From our previous studies [18], the GGA-1/2 method can give accurate electronic structure results for the IB material, and thus results calculated by GGA-1/2 can be used as criteria. Therefore, for the Slater-Koster methods, the total bandgap and sub-bandgap from IB to CB are overestimated, while the sub-bandgap from VB to IB is underestimated. Even so, the two methods exhibit similar electronic structure results at around Fermi energy, i.e., the Slater-Koster methods can provide reliable results at around the Fermi energy and will be well used in the conductance calculations in this work. From the calculational results, for the configuration with higher content of vacancies (V:C63), an isolate and partially filled IB is formed in the middle of the bandgap and the configuration shows typical metallic properties. With the decrease of the content of vacancies, the IB begins to split into two narrow bands. At the concentration of V:C143, the two narrow bands just separate from each other and the Fermi energy locates at the top of the lower one, i.e., the configuration may exhibit insulator properties. The details of the bandgap between the IBs are shown in Table 1. Therefore, the IMT will occur with the variation of the content of vacancies. In the following, to investigate the critical condition of the IMT for the black diamond, the temperature-dependent transmissions and conductance were calculated.
Fig. 1. (a) and (b) Band structures of the configurations in 2 × 2 × 2 supercell (calculated by GGA-1/2 and Slater-Koster methods, respectively. (c) and (d) Band structures of the configurations in 3 × 3 × 3 supercell calculated by GGA-1/2 and Slater-Koster methods, respectively. (e)∼(g) Density of states calculated by GGA-1/2 method of the configurations in 2 × 2 × 3, 4 × 4 × 4, and 3 × 3 × 2 supercells, respectively. (h)∼(j) Density of states calculated by Slater-Koster method of the configurations in 2 × 2 × 3, 4 × 4 × 4, and 3 × 3 × 2 supercells, respectively. The inset figures in (e)∼(j) show the enlarged views of the DOS near the Fermi energy.
Table 1. The bandgaps between the IBs calculated by GGA-1/2 and Slater-Koster methods. The values are given in eV
From the electronic structural results, the critical concentration of vacancies for the IMT should be at around V:C127 (4 × 4 × 4 supercell). Therefore, the 2 × 2 × 3 (V:C95), 4 × 4 × 4 (V:C127), and 3 × 3 × 2 (V:C143) supercells were used to be constructed into devices. Figures 2(a)∼2(c) show the transmission directions of the V:C95, V:C127, and V:C143 configurations, respectively. Figures 2(d)∼2(f) show the device configurations based on the 2 × 2 × 3, 3 × 3 × 2, and 4 × 4 × 4 supercell, respectively. The device configuration contains two semi-infinite electrodes, a MD region, and two electrode extensions on the either side of the MD region. The incoming electrons interact with phonons in the MD region. For the 2 × 2 × 3 and 3 × 3 × 2 supercells, the length of the MD region is 7.134 Å and the cross sections perpendicular to the transmission direction are 7.134 Å × 10.7 Å and 10.7 Å × 10.7 Å, respectively. For the 4 × 4 × 4 supercell, the length of the MD region is 14.267 Å and the cross section is 10.088 Å × 10.088 Å.
Fig. 2. (a)∼(c) Transmission directions of the 2 × 2 × 3 (V:C95), 4 × 4 × 4 (V:C127), and 3 × 3 × 2 (V:C143) configurations, respectively. (d)∼(f) Device configurations based on the 2 × 2 × 3, 3 × 3 × 2, and 4 × 4 × 4 supercells, respectively.
After the MD simulation, the temperature-dependent transmission functions were calculated by standard Green’s function method:
In the equation, Gr[E, x(T), L] and Gr[E, x(T), L] are the retarted and advanced Green’s function, x(T) is the set of displacement coordinates of atoms, L is length of MD region, and ΓL(E) and ΓR(E) are the left and right electrode width. To obtain a good sample averaging, 10 MD simulations and transmissions were performed and the average transmissions are shown in Fig. 3. It can be found that the transmission spectra match well with the corresponding electronic structures. The results indicate that the transmission spectra are less affected by temperature as a whole. Considering that the transmission function at around the Fermi energy is crucial for the conductance computation, the results at the Fermi energy are listed in Table 2. It can be found that, at the Fermi energy, the transmission functions of the V:C95 configuration are slightly larger than that of the V:C127 configuration. For the V:C143 configuration, the transmission functions at the Fermi energy are zero, which is due to the completely splitting of the two IBs. The transmission results are consistent with the electronic structure results that the IB begins to split with the decrease of vacancy contents.
Fig. 3. Average temperature-dependent electronic transmission functions of the configurations with concentration of (a) V:C95, (b) V:C127, and (c) V:C143.
Table 2. Details of the transmission functions at the Fermi energy under different temperature
The temperature-dependent conductance of the configurations was calculated by Landauer formula based on the transmission function results:
In the equation, ${T_L}({E,T} )\; $ is the average transmission function, which only depends on the energy and temperature, and f(E, µ, T) is the Fermi-Dirac distribution function at the chemical potential µ.
The temperature-dependent conductance is shown in Fig. 4. Based on the conductance and lattice parameters of the device configurations, the conductivity can be rough estimated by using σ=GlS-1. In the equation, σ is the conductivity, G is the conductance, l is length of the configuration in the transmission direction (MD region), and S is the cross section of the configuration perpendicular to the transmission direction. The details of the conductance and conductivity of the V:C95, V:C127, and V:C143 configurations are listed in Table 3. It should be noted that the rigorous calculation for the conductivity should be performed like in Ref. [24]. Nevertheless, the conductivity cannot be well defined in this work, because the conductance does not vary linearly with the length of MD region due to the limitation of the calculation. Even so, the conductance in this work can give qualitative explanation for the properties of the material. In addition, the variation of the resistance of black diamond with the change of temperature has been studied previously by experiment [25]. Nevertheless, the due to the uncertain of the thickness of the conductive layer in the experiment [25], the conductance and conductivity in this work cannot be compared with the experiment results directly.
Fig. 4. Conductance at different temperature (2 K, 8 K, 14 K, 20 K, 30 K, and 40 K) of the V:C95, V:C127, and V:C143 configurations. For the V:C143 configuration, the conductance at 2 K is 6.06 × 10−35 S.
Table 3. Details of the conductance and conductivity of the V:C95, V:C127, and V:C143 configurations under different temperature. The values of conductance are given in 10−6 S and the values of conductivity are given in (Ω · cm)-1
From the calculational results, we can find that the conductance of V:C95 and V:C127 configurations keeps constant with the variation of temperature. The result indicates that the V:C95 and V:C127 configurations show metallic state. While for the V:C143 configuration, the conductance is very low and it increases sharply with the increase of temperature, which indicates that the V:C143 configuration exhibits insulator state. The critical concentration of vacancies for the IMT might be between V:C143 and V:C127, which is similar to that of the hyper-doped silicon [18]. The conductivity at the critical condition of the IMT is below 30 (Ω · cm)-1, which is also similar to experiment value of the hyper-doped silicon [26]. Nevertheless, the mechanism of the IMT for the black diamond is different from that of the hyper-doped silicon. For the hyper-doped silicon, with the increase of the dopant contents, the IMT will occur when the impurity band is merged with the CB (S-hyperdoped silicon) or the impurity band is expanded and lead to the weakness of the carrier localization (N-hyperdoped silicon) [18]. For the black diamond at low concentration of vacancies, the separation of the filled and empty IBs makes it to be an insulator. The IMT of the black diamond is due to the mergence between the two isolated IBs when the concentration of vacancies is high, and the merged IB is partially filled by electrons. The partially filled IB is wide enough to lead to the delocalization of the carriers in the IB and make the material to exhibit metallic property. From Table 3, the conductivity of V:C127 is slightly larger than that of the V:C95 configuration. The abnormal phenomenon should be attributed to the different distribution of vacancies in the two configurations. For the V:C127 configuration, which is constructed based on the 4 × 4 × 4 supercell of the C2 primary cell, the distribution of vacancies is more even than that in a 2 × 2 × 3 supercell of the conventional C8 unit cell.
It should be noted that the quantitative analysis in this work is in an ideal case and it might be different from the reality. In this work, the vacancies were distributed regularly in the models. Nevertheless, the reality is more complicated, e.g., the distribution of vacancies should be random. In addition, besides the vacancies, the material should contain other defects, such as self-interstitial defects. Therefore, the critical condition of the IMT should be more complicated and needs further investigation in both theory and experiment. In addition, the computation method used in this work does not properly count strong electron correlations. Therefore, the IMT mechanism in this work might be different from the Mott transition theory [27,28] and also needs further investigation.
4. Conclusion
We first studied the IMT of black diamond by using MD simulations in combination with standard Landauer transmission calculations. Based on the electronic transmission, the temperature-dependent conductance was calculated and the critical condition of the IMT of black diamond was studied. The results indicate that the content of vacancies has great influence on the electronic and transport properties of black diamond. When the concentration of vacancies is below 0.69%, the impurity band is composed by a filled band and a separate empty band, which makes the material to be an insulator. As the content of vacancies increases, the two isolate IBs merge together and form a partially filled IB and the IMT occurs. The critical concentration of vacancies might be between V:C143 (0.69%) and V:C127 (0.78%), which is similar to that of the hyperdoped silicon. Besides the concentration of vacancies, the distribution of vacancies also influences the IMT of the black diamond, and it still needs to be studied further.
Funding
National Natural Science Foundation of China (62275074, 12274117); Program for Innovative Research Team (in Science and Technology) in University of Henan Province (24IRTSTHN025).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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