Insulator-to-metal transition in intermediate-band materials: insights from temperature-dependent transmission computations

Insulator-to-metal (I–M) transition is crucial for an intermediate-band (IB) photovoltaic material to suppress the non-radiative recombinations and improve its efficiency. Nevertheless, the mechanism and critical condition of the I–M transition is not clear yet. In this work, the critical conditions of the I–M transition in two types of Si-based IB materials were studied by combining the molecular dynamics simulations and Landauer transmission calculations. It shows that the I–M transition of the substitutional configuration of S will occur when the filled IB is expanded and merged with the conduction band, which fulfills the Mott transition theory. But this type of IB material is not a standard IB material proposed by Luque. For the substitutional configuration of N, which can be regarded as a standard IB material, the I–M transition would be occurred when the partially-filled IB is expanded and the localization of the carriers in the IB is weakened. The metallic state of the IB material is different from typical metals and they still exhibit some semiconductor properties at low temperature.


Introduction
The intermediate-band solar cell (IBSC) has been proposed two decades ago, and it has attracted wide interest due to its special high detailed balance-limiting efficiency of 63% [1][2][3][4][5][6][7][8]. In the intermediate-band (IB) material, an isolated and partially-filled electronic band, which is called the IB, is placed in the bandgap of the host material [1][2][3][4][5][6][7][8]. The special band structure enables the material to absorb sub-bandgap energy photons and produce additional photogenerated carriers [2,3]. From Luque et al's report, the maximum efficiency can be achieved if the IB material has a total bandgap of 1.95 eV and two sub-bandgaps of 0.71 eV and 1.24 eV [2]. According to the traditional viewpoint, the impurity levels in the bandgap will result in the Shockley-Read-Hall (SRH) non-radiative recombinations. Nevertheless, the SRH recombinations would be suppressed if the concentration of the deep-level impurities is high enough to lead to Mott insulator-to-metal (I-M) transition, and meanwhile, the impurity states in the bandgap are extended to an impurity band [2]. Therefore, to obtain a high efficiency IBSC, besides finding appropriate IB candidate materials, explore the conditions of the I-M transition in-depth is also the focus of research for the IB material.
The research of the I-M transition of IB materials mainly focus on the chalcogen (S, Se, Te) hyperdoped silicon [9][10][11]. The results indicate that, at the critical concentration of dopants for the I-M transition, the impurity band is very close to or just touches the bottom of the conduction band (CB) [10,11]. In this case, the dopant electrons are delocalized and the charge transport can be realized without thermal activation [10]. These works give a general picture of the I-M transition for the silicon-based IB material. However, some key issues still exist and deserve further discussion. The main issue is that the chalcogen hyperdoped silicon is not a standard IB model, which should have a partially-filled IB in the middle of the bandgap. The IB formed by chalcogen is nearly filled and quite close to the CB [10,11]. Therefore, determining the critical condition and mechanism of I-M transition for the standard IB material is the key to optimize its efficiency.
In addition, the studies on the I-M transition of the hyperdoped silicon were performed by combining its electronic structure calculations and the measurement of its temperature-dependent conductivity [10,11]. Nevertheless, the theoretical model is quite different from the experimental sample, which makes the critical conditions difficult to be determined precisely.
This work, taking the silicon-based IB material as objects, first brought insight into the critical conditions of the I-M transition by computing the temperature-dependent conductivities based on the combination of molecular dynamics (MD) and standard Landauer transmission calculations. Firstly, we investigated the I-M transition of the S-hyperdoped silicon that has a nonstandard IB, and compared the results with previous experimental results to verify the accuracy and reliability of the method. Then, we further explored the I-M transition of the N-hyperdoped silicon, which can form a standard IB-partially-filled and isolated from the CB and valence band (VB), to illustrate the mechanism and critical condition of the I-M transition of an IB material.

Computational methods
All of the calculations in this work were carried out by combining nonequilibrium Green's function theory and density functional theory or tight-binding methods (Slater-Koster) in Atomistix Toolkit [12][13][14]. For the electronic structure and transmission spectra computations of the S-hyperdoped silicon, we used GGA-1/2 together with the Perdew, Burke, and Ernzerhof (PBE) functional to describe the exchange correlation [15,16], which can provide accurate electronic structures of the hyperdoped silicon systems. For the N-hyperdoped silicon, we used GGA-1/2 and Slater-Koster [14] for the electronic structure computations and performed only Slater-Koster method for the transmission spectra computations. For the GGA-1/2 method, the linear combinations of atomic orbitals were used to expand the wave functions of valence states and the SG15 was used as the pseudopotential for silicon, sulfur, and nitrogen elements [17,18]. All of the configurations were constructed in different supercells of the conventional Si 8 unit cell. For the S-hyperdoped silicon, the substitutional configuration of S (S S ) was selected as the focus due to its minimum energy configuration in silicon and key role in the I-M transition, which could be better compared with the experiment. Although the N-hyperdoped silicon may have complex dopant configurations, we also chose the substitutional one (N S ) as the research subject because it is a simple standard IB model.
The device configurations are shown in figure 1(a), which contains two semi-infinite electrodes and a central region. The central region also consists three parts: an MD region in which the incoming electrons interacts with phonons and the electrode copy on its either side. Classical potentials were used for the MD calculations, and subsequently the GGA-1/2 or Slater-Koster method was used for calculating the transmission coefficients by using a standard Green's function. In the MD calculations, we used classical force field potentials: ReaxFF_CHOCaSiAlS_2012 [19] for the S S configurations and ReaxFF_CHONSSi_2012 [20] for the N S configurations. We chose a Langevin thermostat [21] and run for 5000 steps with a time step of 1 fs. We set the initial velocities (temperature) to be the same as the reservoir temperature and started the MD simulations with Maxwell-Boltzmann distribution of the velocities corresponding to the temperature. We performed only one MD simulation for the S S configurations due to the large calculations and we performed ten different MD simulations for the N S configurations and obtained the average transmissions.

Results and discussion
The electronic structures of the S S in different supercells are shown in figures 1(b)-(e). The atomic concentration of S is 1.56% (2 × 2 × 2), 1.04% (2 × 2 × 3), 0.69% (3 × 3 × 2), and 0.46% (3 × 3 × 3). Compared with the band structures of S S calculated by HSE06 [22], the results gotten by the GGA-1/2 method (figures 1(b) and (c)) show similar features. For instance, the total bandgaps (from VB maximum to CB minimum) calculated by GGA-1/2 method are similar to that calculated by HSE06 [22]. Besides, the and (c) show the band structures of the SS configuration in the 2 × 2 × 2 and 3 × 3 × 3 supercells (the dopant concentration is S1:Si63 and S1:Si215), respectively. Panels (d) and (e) show the density of states (DOS) of the SS configuration in the 2 × 2 × 3 and 3 × 3 × 2 supercells (the dopant concentration is S1:Si95 and S1:Si143), respectively. band structures at around the Fermi energy calculated by GGA-1/2 method, such as the overlapping degree or bandgap between IB and CB, are also similar to that calculated by HSE06 [22]. The results demonstrate the accuracy and effectiveness of the method for these systems. It is known that the traditional GGA method underestimates the bandgaps of semiconductors. The GGA-1/2 method used in this work is a semi-empirical method and it can correct the self-interaction error in local and semi-local exchange-correlation functional for extended systems, which has been found to substantially improve the bandgaps for many semiconductors and insulator systems [15]. Although the GGA-1/2 can compete with the HSE06 for bandgaps of the S-hyperdoped silicon system, it may not describe the orbital contributions correctly for the system. From Zhao and Yang's report, the impurity band of the S-hyperdoped silicon is predominantly composed of the 3s states of S and the 3p states of Si [23]. The precision and limitation of the method for calculating other doped systems still needs further investigation. The results indicate that the IB formed at a low concentration of S (S 1 :Si 243 ) is completely isolated from the CB and VB (the energy difference between the IB maximum and the CB minimum is about 0.12 eV), which is different from the PBE result that the IB and the CB nearly merge at that concentration of S [22]. With increase of content of S, the IB is broadened and gets closer to the edge of the CB. It overlaps with the CB at the concentration of S 1 :Si 95 (for the S 1 :Si 143 , there is still a tiny bandgap). According to previous researches of the chalcogen-hyperdoped silicon, the mergence between the IB and CB might be the critical condition of the I-M transition [10,11]. Based on the assumption, the critical concentration of the substitutional S for the I-M transition should be between S 1 :Si 143 (3.5 × 10 20 cm −3 ) and S 1 :Si 95 (5.26 × 10 20 cm −3 ), which is basically consistent with the experimental results [9]. Nevertheless, due to the great difference between the actual situation and the theoretical simulation, e.g. the actual electronic structure of the material should more complex due to the complicated states of dopant, it is still too early to give an accurate conclusion.
To determine the transition condition and its mechanisms, we first simulated the temperature-dependent conductivity for the S-hyperdoped silicon based on the MD-Landauer method. From the Landauer formula, as shown in equation (1), the conductance is tightly related to the electronic transmission function T L (E, T). f (E, µ, T) is the Fermi-Dirac distribution function.
The transmission function was calculated by standard Green's function method after the MD simulation: where G r/a [E, x(T), L] is the retarded/advanced Green's function, x(T) is the set of displacement coordinates of atoms, L is length of MD region, and Γ L/R (E) is the width in the left/right electrode. The calculation method and parameters used for the transmission function calculations were kept consistent with that for the electronic structure calculations, which could make the transmission spectra match well with the electronic structure of the material. Therefore, there is a better corresponding relation between the electronic structures and the conductivity. The electronic transmission functions of the configurations at different temperature and lengths of MD region are shown in figure 2. At the highest concentration of S (S 1 :Si 63 ), there is a broad transmission band below the Fermi energy, which is consistent with the band structure of the configuration. In addition, the transmission coefficient is still larger at the Fermi energy, which is due to the higher mergence between the IB and the CB at the concentration of S. For the S 1 :Si 63 , the transmission is insensitive to the temperature, but it decreases with the increase of the length of MD region. As the concentration of S decreases, such as the S 1 :Si 95 , it can be found that the transmission remains unchanged when the temperature is below 30 K, but it decreases at the temperature of 40 K. In addition, the transmission at around the Fermi energy is much lower than that of S 1 :Si 63 , because the IB of S 1 :Si 95 is narrower and nearly separated from the CB. When further reduce the concentration of S (S 1 :Si 143 ), the IB is completely separated from the CB and the transmission at around the Fermi energy nearly disappears.
Based on the transmission functions, the temperature dependent resistance and conductivity are obtained and the results are shown in figure 3. For the configurations of S 1 :Si 63 and S 1 :Si 95 , it can be observed that the resistance increases linearly with the length of MD region (figures 3(a) and (b)). However, the resistance increases exponentially with the length for the configuration of S 1 :Si 143 , as shown in figure 3(c). It does not mean that the resistance of S 1 :Si 143 is not ohmic. We presume that the exponentially increasing resistance should be due to the electron localization at the lengths of MD region, which have been pointed out in the relevant calculations [24,25]. Even so, it does not affect our conclusions in this work, because our primary concern is the changes of the resistance with temperature. According the results, the resistivity (ρ) of S 1 :Si 63 and S 1 :Si 95 can be calculated and their conductivities (σ = 1/ρ) are shown in figure 3(d). We can observe that the conductivity of S 1 :Si 63 is temperature insensitive at this low temperature range (0-40 K). For configurations with lower S content (S 1 :Si 95 ), the conductivity also remains basically invariable when the temperature is increased from 2 K to 30 K. The conductivity of S 1 :Si 143 cannot be determined due to the deficiency of the calculation, but it can be observed obviously that the resistance drops sharply as the temperature increase. Therefore, we can safely conclude that the conductivity of S 1 :Si 143 will increase rapidly with the temperature increases. The calculational results are basically consistent with experiment results of the S-hyperdoped silicon [9]. From the definition of metal and insulator, configurations of S 1 :Si 63 and S 1 :Si 95 Figure 3. (a)-(c) Resistance vs length of the MD region for the configurations of S1:Si63, S1:Si95, and S1:Si143, respectively. The resistance is obtained by calculating the Landauer formula. (d) Calculational temperature-dependent conductivity of the S1:Si63 and S1:Si95 configurations. The conductivity is gotten from the resistivity which is calculated from the linear relationship between the resistance and the length of MD region. should be in metallic state, while the configuration of S 1 :Si 143 is in insulator state. From the details of the temperature-dependent conductivity in table 1, we can find some slight variations of conductivity with temperature for the S 1 :Si 63 and S 1 :Si 95 . The conductivity increases slightly as the temperature increases from 2 K to 20 K, and it decreases when the temperature further increases. Especially for the S 1 :Si 95 , the conductivity decreases obviously at 40 K. The slight increase of the conductivity with the rise of temperature at lower temperature range is also in accordance with the experiment results [9]. The decrease of the conductivity at higher temperature should be due to the affect of the lattice vibration, which account for the majority gradually with the increase of temperature. The results indicate that the material is a little different from a typical metal and it exhibits some semiconductor properties even at high concentration of S. It should be necessary for the applications, because a typical metal would not be used for semiconductor devices. For the S 1 :Si 143 configuration, because of the separation the IB and the CB, the resistance is much larger than that of S 1 :Si 63 and S 1 :Si 95 . In addition, due to the tiny gap between the IB and the CB, the thermal activation of electron cannot be ignored and it is greatly affected by the temperature. It should be the reason that the resistance of the S 1 :Si 143 decrease dramatically with the increase of temperature. According to the results, combined with the electronic structures, it can be concluded that the condition of the I-M transition for the S-hyperdoped silicon is the mergence between the IB and CB. The metallicity would get stronger and close to a typical metal as the mergence increases. The conclusion can be also applicable to the Se-and Te-hyperdoped silicon, which shows similar electronic structures as the S-hyperdoped one [10,11,22]. However, the I-M transition condition of the chalcogen-hyperdoped silicon will have an adverse effect on its applications as an IBSC, because the mergence between the IB and the CB would make it deviate from a standard IB material. In the following, we took the N S as research object to investigate the I-M transition of a standard IB material and the results are shown in figure 4. Figures 4(a)-(c) show the DOS of the configurations with different concentration of N. It can be found that the characteristic of the IB calculated by GGA-1/2 and Slater-Koster exhibits little difference. Although the bandgaps calculated by the Slater-Koster method are overestimated for these configurations, it does not influence the conclusions because the resistance is mainly determined by the electron state around the Fermi energy. The IB is partially-filled with electrons for the three configurations and it gets narrow with the decrease of the concentration of N. The electronic structure of the N S is distinct from that of the chalcogen-hyperdoped silicon and it is closer to that of a standard IB model. The transmission functions of these configurations were calculated by the Slater-Koster method, which is far more efficient than the GGA-1/2 method. To get more accurate results, we performed ten different MD simulations and obtained average transmissions, which are shown in figures 4(d)-(f). We can observe that the transmissions match well with the corresponding DOS results, which indicates the reliability of the calculations. For the configurations with higher concentration of N, such as N 1 :Si 63 , N 1 :Si 95 , we can find that the transmissions increase slightly with the increase of temperature. While for the configuration of N 1 :Si 143 , the transmission increases obviously with the increase of temperature. The local density of state in figure 4(g) shows that the IB is formed by the hybridization of the states of Si and N, and the states of Si account for the majority of the IB. Based on the transmission results, the resistance of these configurations are calculated from the Landauer formula. As shown in figure 4(h), the resistance of N 1 :Si 63 and N 1 :Si 95 is basically stable with the change of temperature, which is similar to the property of a metal. While the resistance of N 1 :Si 143 is much larger than that of N 1 :Si 63 and N 1 :Si 95 and it is sensitive to the temperature, i.e. it drops obviously along with the increase of temperature. The results indicate that the N 1 :Si 143 should still be a semiconductor. Due to the limitation of the calculational method, the calculations for the configurations with lower concentration of dopant, i.e. larger supercells, would be less reliable. Even so, it can be concluded that the I-M transition should be occurred at the concentration between N 1 :Si 143 and N 1 :Si 95 . From the DOS results it is not difficult to perceive the reason for the phenomenon. For the N 1 :Si 143 configuration, the localization of the carriers in the IB is still strong due to the narrow band and so the carrier properties, such as the mobility, are susceptible to the temperature. With the expansion of the IB, the carrier localization is weakened and the sensitivity of the mobility to the temperature is reduced. The details of the resistance are shown in table 2. We can observe that the variation of the resistance with temperature for the metallic state (N 1 :Si 63 and N 1 :Si 95 ) is also a little different from the typical metal, i.e. the resistance reduces slightly with the increase of temperature. Compared with the S-hyperdoped silicon, we can find that the N-hyperdoped silicon (standard IB material) exhibits a markedly different mechanism of the I-M transition. Even so, the metallic states of the two types of materials have similar features, i.e. both of them exhibits some semiconductor properties at low temperature. For the standard IB material, the Mott transition theory seems no longer active. From the Mott theory, the critical carrier concentration of the I-M transition for the group-IV semiconductors can be calculated by equation (3), where a H is the effective Bohr radius of the donor electrons and it is calculated by equation (3) In equation (4), E is the activation energy of the localized states. Therefore, the critical concentration obtained from the Mott theory should be the critical condition of the mergence between the IB and the CB, not the I-M transition condition. The Mott theory could be applicable to the models of which the IB is filled, such as the chalcogen-hyperdoped silicon [9][10][11]. For the standard IB material, which has a partially-filled IB, the I-M transition would be occurred when the localization of the carriers in the IB is weakened to the levels that the mobility is insensitive to the temperature. Therefore, the I-M transition of the standard IB material is a gradual process due to its transition mechanism, which is clearly reflected from figure 4(h). It should be noted that the method used for calculating the resistance in this work do not properly count strong electron correlations, which is closely associated with the Mott transitions in many d-electron systems [28]. For other systems, such as the N-hyperdoped silicon in this work, the strong electron correlations may also associate with the I-M transition, which should need further investigation.

Conclusion
In conclusion, the critical condition and mechanism of the I-M transition for the two types IB materials were comprehensively studied by the temperature-dependent transmission computations. The computational results of the IB material based on the chalcogen-hyperdoped silicon agree well with the experimental results. The results indicate that the I-M transition of the chalcogen-hyperdoped silicon would be occurred when the filled IB is expanded and merged with the CB. For the N S configuration in silicon, which can be regarded as a standard IB material, the sensitivity of the carrier mobility to the temperature is reduced when the partially-filled IB is expanded and the localization of the carriers in the IB is weakened. As a result, the I-M transition would be occurred. For this case, the Mott transition theory would not be applicable and the I-M transition is a gradual process, which is quite different from the chalcogen-hyperdoped silicon. Both of the two types IB materials at the metallic state are not exactly the same as the typical metals and they exhibit some semiconductor properties at low temperature. This work provides a new way for studying the I-M transition in the IB material and gives us a deeper insight into its critical conditions and mechanisms, which is crucial in developing high-efficiency IB photovoltaic materials.

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.