Electronic band structure of TiN/MgO-4 × 4 and 5 × 5 nanostructures

We have investigated TiN(001)/MgO(001)-1 × 1, 2 × 1, 2 × 2 and 3 × 3 superlattices previously^1–3) in order to support an experiment⁴⁾ of a TiN thin film on a MgO substrate using molecular beam epitaxy. MgO is a suitable substrate for the growth of the TiN film due to the small lattice mismatch of 1.4%.^1,2) The (001) surfaces calculated in this study are non-polarized and each ideal (001) surface layer consists of an equal number of cation and anion atoms. We present TiN(001) dot/MgO(001) as TiN dot/MgO hereafter. The electronic states of the rectangular TiN dot/MgO-2 × 2 and 3 × 3 superlattices are semiconducting.³⁾ Other TiN/MgO superlattices correspond to metallicity.^1–3) Experimental and theoretical studies of TiN dot/MgO or TiN film/MgO superlattices (interfaces)^3,5,6) have been undertaken. The nanostructured materials are focused on enhancing the thermoelectric properties.^7–10) Therefore, we have calculated the thermoelectric properties (Seebeck coefficients, thermal conductance, and figure of merit [ZT]) of TiN/MgO-1 × 1 and 2 × 2 superlattices using the non-equilibrium Green's function (NEGF) method.^11,12)

We consider larger MgO substrates in this study. The TiN dot structures on MgO-4 × 4 and 5 × 5 substrates are considered, as shown in Fig. 1. In this figure, the shaded stick and rectangle indicate schematically the rectangular and rectangular parallelepiped TiN dots, respectively. These two dot structures are considered in this study.

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The main purpose of this study is to obtain structurally relaxed TiN dot/MgO-4 × 4 and 5 × 5 superlattices. Another important aim is to obtain their electronic properties as nanostructured TiN dot/MgO superlattices. A nonmetallic state is more suitable to enhance thermoelectric properties than a metallic state. Although the ZT value of the TiN/MgO-1 × 1 interface¹¹⁾ is quite small at approximately 0.002 due to its metallicity, that of the TiN dot/MgO-2 × 2 superlattice¹²⁾ is approximately 0.15 due to its semiconducting. It is expected that small TiN dot structures on the large MgO substrates such as 4 × 4 and 5 × 5 will be nonmetallic and have larger bandgap values.

The calculated results (electronic properties) of the TiN dot/MgO-4 × 4 and 5 × 5 superlattices were published.¹³⁾ It has been found that these nanostructured TiN dot/MgO superlattices have band gaps in their electronic states¹³⁾ because the wave function of the TiN dot cannot overlap those of the neighboring TiN dots on the large MgO substrate. The more detailed electronic properties (density of states) and structurally relaxed lattice properties of the TiN dot/MgO-4 × 4 and 5 × 5 superlattices have been investigated in this study. In addition, we consider a generalized density-functional theory (GDFT)^14–17) in order to correct the bandgap values.

The present calculation is based on local density approximation (LDA) in DFT^18,19) with the Wigner²⁰⁾ formula for the exchange correlation. The optimized pseudopotentials by Troullier and Martins^21,22) are used for Ti, Mg, N and O, and their nonlocal parts of the pseudopotentials are transformed to the Kleinman–Bylander separable forms²³⁾ without ghost bands. A partial core correction²⁴⁾ is considered for the Ti and Mg pseudopotentials. The wave function is expanded in plane waves and the cutoff energy is 49 Ry for 4 × 4 and 36 Ry for 5 × 5. The mesh sizes of the sampling k-points in the whole Brillouin zone are 2 × 2 × 1 (=4) for the structural relaxation and 4 × 4 × 1 (=16) for obtaining the detailed electronic properties. This means that the electronic properties (bandgap value, density of states, etc) of the TiN dot/MgO-4 × 4 and 5 × 5 superlattices, which were structurally relaxed in the 2 × 2 × 1 k-point mesh, were obtained in the 4 × 4 × 1 mesh. The cutoff energy and mesh size of the sampling k-points for each system are tabulated in Table I.

Table I.  Supercell sizes of TiN dot/MgO superlattices and dot shapes as "R" and "RP", cutoff energy [Ry] as "cutoff", number of atoms (N_atom) in the supercell, number of k-points (N_k) are tabulated. "R" and "RP" indicate rectangular and rectangular parallelepiped dots, respectively. TiN/MgO-1 × 1 shows the previous results.^1,2) The experimental bandgap value of MgO bulk is 7.67 eV.²⁷⁾ "GDFT" indicates the corrected bandgap values using the GDFT.^14–17) "(*)" indicates that the number of k-points in the rectangular TiN dot/MgO-2 × 2 superlattice is 100 (10 × 10 × 1) for the GDFT.

Superlattice	Cutoff	${N}_{\mathrm{atom}}$	Nk	LDA	GDFT
TiN/MgO-2 × 1 R	144	28	24	—	—
TiN/MgO-2 × 2 R	72.25	52	16	0.17	0.45(*)
TiN/MgO-2 × 2 RP	72.25	56	16	—	—
TiN/MgO-3 × 3 R	36	112	16	0.54	0.90
TiN/MgO-3 × 3 RP	36	116	16	—	—
TiN/MgO-3 × 3 R	49	76	4	0.54	0.96
TiN/MgO-3 × 3 R	49	76	16	0.54	0.96
TiN/MgO-3 × 3 R	49	76	36	0.54	0.73
TiN/MgO-3 × 3 RP	49	80	4(16)	—	—
TiN/MgO-4 × 4 R	49	132	4	0.57	0.77
TiN/MgO-4 × 4 R	49	132	16	0.57	1.01
TiN/MgO-4 × 4 RP	49	136	4	0.28	1.19
TiN/MgO-4 × 4 RP	49	136	16	0.27	1.12
TiN/MgO-5 × 5 R	36	204	4	0.58	0.94
TiN/MgO-5 × 5 R	36	204	16	0.58	0.87
TiN/MgO-5 × 5 RP	36	208	4	0.19	—
TiN/MgO-5 × 5 RP	36	208	16	0.19	—
TiN/MgO-1 × 1	144	16	16	—	—
MgO (Bulk)	144	2	89	4.60	8.10

A TiN dot/MgO superlattice with a vacuum region is periodically repeated in the supercell, as shown in Figs. 2 and 3. As for a repeated slab size, we used the results of TiN/MgO-1 × 1.^1,2) Two supercell sizes of TiN dot/MgO superlattices were treated in this study. One is modeled by a 4 × 4 supercell containing 132 and 136 atoms. Another is modeled by a 5 × 5 supercell containing 204 and 208 atoms. All atoms in the supercell were fully relaxed in which each ideal TiN dot/MgO superlattice is a starting structure. Our criterion for relaxation is that the maximum force acting on each atom should be less than 5.0 × 10⁻⁴ Ry/Bohr.

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We describe the TiN dot structure on the MgO substrate in detail. One is one Ti and one N atom per layer as a rectangular shape and the other is two Ti and two N atoms per layer as a rectangular parallelepiped shape, as shown in Fig. 1, schematically. Thus, the total number of atoms in the rectangular and rectangular parallelepiped TiN dots are 4 and 8, respectively. The nanostructured TiN dot and the MgO-4 × 4 and 5 × 5 substrates are represented by the two-layer dot as (TiN)₂ and the four-layer slab as (MgO)₄, respectively, and a vacuum region corresponding to a six-layer slab is inserted in between the slabs, as shown in Figs. 2 and 3.

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The equilibrium lattice constants and bulk modulus of the rock salt structures of TiN and MgO in theory (experiment) are 4.25 Å, 282 GPa (4.238 Å,²⁵⁾ 288 GPa²⁶⁾) and 4.19 Å, 147 GPa (4.216 Å,²⁷⁾ 161.5 GPa²⁸⁾), respectively. The calculated lattice constants agree well with the experimental results within approximately 1% and the lattice mismatch of TiN and MgO is 1.4%.

As a result of the previous calculations,³⁾ it has been found that the outward displacement of N and inward displacement of Ti on the first layer (=top layer) of the TiN dot/MgO-2 × 1 and 2 × 2 superlattices characterize the stable surface configuration listed in Table II. In this study, "inward" and "outward" are defined as the direction towards the vacuum region along the c-axis, as shown in Fig. 1. From Table II, the inward and outward displacements of the Ti and N atoms are relatively small within 0.01 with the exception of the 2 × 1 case and rectangular TiN dot/MgO-2 × 2 superlattice. The trend of the TiN dot/MgO-2 × 1 and 2 × 2 superlattices is consistent with rumpled transition metal carbide and nitride surfaces.^29,30) In contrast, the inward displacement of N and outward displacement of Ti on the first layer of the TiN dot/MgO-3 × 3³⁾ and 4 × 4 superlattices and the rectangular TiN dot/MgO-5 × 5 superlattice characterize the stable surface configuration listed in Table II. The height difference between the inward displacement of N and outward displacement of Ti on the first layer in the rectangular parallelepiped TiN dot/MgO-4 × 4 superlattice is quite small at 0.001 Å, as shown in Table II and Fig. 2 (right panel). The outward displacement of N and inward displacement of Ti on the first layer of the rectangular parallelepiped TiN dot/MgO-5 × 5 superlattice characterize the stable surface configuration and its height difference is 0.08 Å. It is likely that the TiN dots on the MgO-4 × 4 and 5 × 5 substrates are more isolated than the 2 × 1 and 2 × 2 cases and this leads to the disappearance of the feature of the TiN surface³⁰⁾, but not for the rectangular parallelepiped TiN dot/MgO-5 × 5 superlattice.

The side views of the relaxed structures of TiN dot/MgO-4 × 4 and 5 × 5 superlattices are depicted in Figs. 2 and 3, respectively. As mentioned before, the trend of inward and outward displacements for Ti and N in the rectangular TiN dot/MgO-4 × 4 and 5 × 5 superlattices is the same as shown in Figs. 2 and 3 (left panels). The first layer of the rectangular parallelepiped TiN dot/MgO-4 × 4 superlattice is almost flat, as shown in Fig. 2 (right panel). In contrast, the trend of inward and outward displacements of the rectangular parallelepiped TiN dot/MgO-5 × 5 superlattice is the same as the 2 × 1 and 2 × 2 cases.³⁾

It is established through calculations that the rectangular TiN dot/MgO-4 × 4 and 5 × 5 superlattices correspond to semiconductor. The rectangular parallelepiped TiN dot/MgO-4 × 4 and 5 × 5 superlattices also correspond to semiconductor, although those of the 2 × 2 and 3 × 3 superlattices correspond to metallicity.³⁾ It is expected that the bandgap values of the TiN dot/MgO-4 × 4 and 5 × 5 superlattices will be larger than those of the TiN dot/MgO-2 × 2 and 3 × 3 superlattices because the band gap of MgO (bulk) is quite large at 4.60 eV (7.67 eV) in LDA (experiment²⁷⁾). The bandgap values in the TiN dot/MgO-4 × 4 and 5 × 5 superlattices are tabulated in Table I. The previous results³⁾ of the TiN dot/MgO-2 × 2 and 3 × 3 superlattices are also tabulated in Table I enabling us to compare them with the present results. The bandgap values of the rectangular TiN dot/MgO superlattices are larger than those of the rectangular parallelepiped TiN dot/MgO superlattices. As for the rectangular dot, the bandgap value increases with increasing the size of the MgO substrate, although the difference between 3 × 3 and 5 × 5 is quite small at approximately 0.01 eV. In contrast, the bandgap value of the rectangular parallelepiped TiN dot/MgO-5 × 5 is smaller than that of the rectangular parallelepiped TiN dot/MgO-4 × 4 from Table I.

We consider the GDFT^14–17) in order to correct the bandgap value. The bandgap value is simply corrected by using GDFT. It is advantageous to obtain the bandgap value in a large system such as the 4 × 4 and 5 × 5 cases. The bandgap value of bulk MgO in LDA is corrected to 8.10 eV in the GDFT calculation. This value is in better agreement with the experimental value (7.67 eV²⁷⁾). The corrected bandgap values of the TiN dot/MgO superlattices are also tabulated in Table I. Most of the 3 × 3–5 × 5 cases are approximately 1.0 eV and this is twice as large as those in LDA. The electronic states of the rectangular parallelepiped TiN dot/MgO-2 × 2 and 3 × 3 superlattices are metallic in GDFT. The band-gap values of the rectangular parallelepiped TiN dot/MgO-4 × 4 is maximum with 1.12 eV for 16 k-points (1.19 eV for 4 k-points) although that of the rectangular TiN dot/MgO-5 × 5 is maximum in LDA. It is impossible to compare the corrected bandgap values with the experimental values, since there is no experimental observation of the bandgap values for nanostructured TiN dot/MgO superlattices.

The band gap in the rectangular TiN dot/MgO-4 × 4 (5 × 5) superlattice appears and its value is approximately 0.57 eV (0.58 eV), although it could be underestimated the bandgap value due to inaccuracy of the DFT-LDA calculation. The bandgap value in LDA is insensitive to computational conditions (number of k-points, cutoff energy) from Table I and this implies the DFT-LDA calculation is sufficiently converged. The corrected bandgap values of the rectangular TiN dot/MgO-4 × 4 and 5 × 5 superlattices using GDFT are 1.01 and 0.87 eV (16 k-points), respectively. The bandgap values of the rectangular parallelepiped TiN dot/MgO-4 × 4 and 5 × 5 superlattices are 0.27 and 0.19 eV in the DFT-LDA calculation, respectively. The corrected value of the rectangular parallelepiped TiN dot/MgO-4 × 4 superlattice is 1.12 eV (16 k-points). We cannot obtain the band gap of the rectangular parallelepiped TiN dot/MgO-5 × 5 superlattice and its electronic state is metallic in GDFT, although that in LDA corresponds to semiconductor. It is necessary to investigate whether the result in GDFT is correct or not, because the GDFT calculation is sensitive to calculational convergence and conditions (number of k-points, cutoff energy, etc).

It is obvious that the electronic state of the TiN dot/MgO superlattice depends on its dot shape and the size of the MgO substrate. The bandgap value of the TiN dot/MgO superlattice could be controllable to vary its TiN dot shape and MgO substrate size.

We have calculated the density of states (DOS) of relaxed TiN dot/MgO superlattices. The detailed DOS were obtained under condition of 49 Ry (4 × 4), 36 Ry (5 × 5) and 16 k-points (4 × 4 × 1). It is sufficient to represent electronic properties (DOS) in this mesh. The DOS of the rectangular and rectangular parallelepiped TiN dot/MgO-4 × 4 and 5 × 5 superlattices are shown in Figs. 4–7, respectively, and they all correspond to semiconductor. More detailed (expanded) DOS around the band gap are inset in each figure. These DOS curves are obtained in the DFT-LDA calculation and not corrected by GDFT.

There are sharp DOS curves and peaks around the band gap from the DOS figures, as shown in Figs. 4–7 (see each inset). These sharp DOS curves and peaks may enhance thermoelectric properties.³¹⁾ From Figs. 4 and 6 (see insets), there are two peaks around the band gap. One of the peaks is completely occupied and the other is unoccupied in the rectangular TiN dot/MgO-4 × 4 and 5 × 5 superlattices. The corresponding two peaks in the rectangular parallelepiped TiN dot/MgO-4 × 4 and 5 × 5 superlattices are occupied completely, as shown in Figs. 5 and 7 (see insets). In all cases, the two peaks are dominated by Ti states, as shown in the insets of Figs. 4–7. These states mainly consist of 3d-states, as shown in Fig. 8. Figure 8 is the projected DOS³²⁾ (=partial DOS) of Ti (s, p and d) states in the rectangular TiN dot/MgO-4 × 4 superlattice. The 3d-states (orange curve in Fig. 8) are widely spread around −4 ~ 3 eV. In particular, unoccupied DOS above the Fermi level (0 ~ 2 eV) predominantly consists of the 3d-states, as shown in the inset of Fig. 8. This trend is invariant in all TiN dot/MgO-4 × 4 and 5 × 5 superlattices. The location of s- and p-states for N, O and Mg is roughly indicated in Figs. 4–7, respectively.

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Although the DOS of the calculated TiN dot/MgO superlattices are similar to each other on the whole, due to the structurally same layer numbers as (TiN)₂/(MgO)₄, more detailed shapes of DOS around the band gap are different to each other in the rectangular parallelepiped TiN dot cases, as shown in Figs. 5 and 7 (see insets). It is likely that this difference is caused by the structural difference of the rectangular parallelepiped TiN dots on the MgO-4 × 4 and 5 × 5 substrates, as shown in Figs. 2 and 3 (right panels). There are two peaks around −14 ~ −13 eV in the rectangular TiN dots/MgO-4 × 4 and 5 × 5 superlattices, as shown in Figs. 4 and 6 and these peaks consist of s-states of N. Similarly, there are four peaks (s-states of N) around −15 eV in the rectangular parallelepiped TiN dots/MgO-4 × 4 and 5 × 5 superlattices, as shown in Figs. 5 and 7. The upper two peaks are quite close to each other in Fig. 5.

We have calculated the internal lattice and electronic properties of rectangular and rectangular parallelepiped TiN dot/MgO-4 × 4 and 5 × 5 superlattices using the total-energy pseudopotential method. Their internal lattice parameters were fully relaxed. The electronic properties of the various TiN dot/MgO superlattices (2 × 1 and 2 × 2-5 × 5) have been definitively established in the present and previous³⁾ studies. The electronic states of all rectangular TiN dot/MgO superlattices (2 × 2–5 × 5) are semiconducting with the exception of the 2 × 1 case. The rectangular parallelepiped TiN dot/MgO-4 × 4 and 5 × 5 superlattices also correspond to semiconductor. The maximum bandgap value is 0.58 eV (rectangular, 5 × 5) in LDA. The corrected bandgap values were obtained by using the GDFT calculation. It is necessary to compare the corrected bandgap values with the results using other accurate methods (GW, exact exchange, hybrid functionals, etc) in order to evaluate the validity of GDFT. The electronic state of each TiN dot/MgO superlattice depends on the shape of the TiN dot and the size of the MgO substrate. Therefore, the TiN dot shape and MgO substrate size are important to control the bandgap value. In addition, it is expected that the growth and observation of TiN dot/MgO superlattices in experiment will be realized.

Our next task is to investigate in detail various types of TiN dot or various defect (impurity, vacancy, etc)-doped MgO substrate and TiN dot in order to lower its thermal conductivity and tune the Fermi level. Furthermore, it is necessary to consider thermoelectric properties (thermal conductivity, Seebeck coefficient, figure of merit, etc) of the large TiN dot/MgO superlattices using NEGF^11,12) in the future.

We would like to extend our thanks to Dr. T. Mori of NIMS for continual support. This study is supported by JST CREST Grant Number JPMJCR15Q6, Japan. The numerical calculations were performed using Z420, Z440, Z620 and Z640 workstations (HP), and the numerical materials simulator [Altix, SGI] in NIMS. Lattice structures were depicted by using "Materials Studio" (BIOVIA).