Comparison of ground states of CdS supercells made by 8 cubic unit cells by DFT

 

Nishant T. Tayade*, Purushottam R. Arjunwadkar

Dept. of Physics, Govt. Institute of Science Nagpur, R.T. Road Civil Line, Nagpur-440001 (M.S.) India

*Corresponding Author E-mail: nishanttayade@rediffmail.com

The supercells made by eight unit cells of cubic CdS (supercell-8) and a single unit cell were comparatively analyzed for the ground state using density functional theory (DFT) and optimized its structure for the total energies and other parameters. Their density of states was studied comparatively. The slight variations in DOS and from it, its bandgaps were discussed. Supercell-8 and Supercell-1 are also compared using theoretical calculation to predict significant difference.

 

 

 

INTRODUCTION:

The CdS is a direct band gap semiconductor and found in cubic and hexagonal form [1]. It has wide energy band gap (E_gd) 2.4eV (in bulk) [2][3] and 5.4eV Fermi energy (E_f) at room temperature. Band gap can vary according to dependency on other parameters. This cubic CdS compound has many applications due to its good semiconducting stability and opto-electronics properties [4] [5] such as photocell [6], solar cell [7], display and LED [8,9] etc. The nanoparticles, nano-rods or thin film base materials are main interests of research for such applications [4,10]. A relax and an unrelax structure of the cubic and hexagonal structure has been compared and shown that the cubic has more deformation and to preserved it reconstructed to hexagonal. Their DOS Eg found 1.81eV (for PAW and GGA) [11].

 

The present study is restricted to cubic phase of CdS. At a very small scale of size, a periodic single geometrical structure (supercell) made by few (say p) cubic unit cells is known as supercell-p in a present work. In this paper p=8 and hence such structures (supercells) are known as spercell-8 hereafter. Let us suppose the (n×m×r) scaling is showing that the n, m and r numbers of unit cells are present in the x, y and z directions of the supercell-8 respectively. Then three basic higher symmetry geometric structures for supercell-8 can be written on the selection of the n, m, r as (2×2×2), (4×2×1) and (8×1×1). A supercell with (2×2×2) has cubic in shape and 24 single-unit-cell-surfaces; therefore it is also identified in this paper by 8₂₄ or C-supercell. Supercell-8 with (4×2×1) has rectangular slab shape and 28 single-unit-cell-surfaces; therefore it is also identified in this paper by 8₂₈ or R-supercell. Supercell -8 with (8×1×1) has bar shape and 34 single-unit-cell-surfaces; therefore it is also identified in this paper by 8₃₄ or B-supercell. The study of these supercells at its most relaxed ground state has been aim to optimized and investigated in a present paper along with supercell-1. The supercell-1 i.e. a cubic single unit cell of (1×1×1) with 6 faces is identified as 1₆. It is also called unit cell in present paper.

EXPERIMENTAL:

All supercell-8 contains 64 native (32 CdS and 32 S). All supercells were built in a GUI. The calculations for the 64 natives required atleast 0.26MHz speed processor with 4GB RAM computing machine. By considering various combination of K-points selection the speed surely need more than above. Therefore Intel i5 1.80GHz with 4GB RAM machine was used to run codes. Each calculation took approximately 5 to 6 hours for each supercell-8 which had taken more than 24 hours (continuous) for each supercell-8. All samples took 6-7 days for running the codes in both GUI and in terminal modes for verification. A cubic lattice parameter of the CdS unit cell was taken 5.818Å (kept constant during calculation)) from one of the nanosize standard CdS sample from our lab. It was taken as the fundamental building block (unit) for all supercells. 

 

Computational Methods:

Quantum espresso package [12] and BURAI GUI have been used for the study. Generated codes were run on terminal as well as on GUI support. The PWSCF tool was executed first for DFT calculation. A Davidson digonalization overlapped for each iteration for minimization of the total energy which was estimated by Harris-Foulkes using plain wave self consistency calculation. The ultrafast pseudo-potential with the generalized gradient approximation (GGA)- Perdew–Burke–Ernzerho (PBE) exchange correlation was used and taken from the quantum espresso site [13]. The grids were kept by selecting K-value in automatic mode previously and then change according to need of resolution. DOS tool is used for the calculation of density of states.

 

RESULTS:

The super cells are illustrated in a Fig.1 along with unit cells U-supercell. The plane wave self consistency field calculation’s outcome is given in Table1. Optimization of geometry by plane wave predicts same parameters corresponding to ground state of the concerned supercells are given in Table2. In Supercell-8, surface area increases from C-supercell to B-supercell geometry cell which affects surface to volume ration increases in the order of C-supercell to B-supercell. However the minimum total energy for convergence and the force per atom for the C-supercell is larger than B-supercell and R-supercell. Total stress was found as an exact function of surface to volume ratio.

 

 

Figure 1: Geometrical construction of the Supercells-8 and unit cell with their scaling and names

 

The fermi Energies obtained for scf and optimization are indicated in Table 1 and Table 2. These are also obtained for non self consistency field as 6.3084 for 1₆, 6.4447eV for 8₂₄, 6.4281eV for 8₂₈ and 6.4617eV for 8₃₄. These three tests showed that the fermi energy of B-supercell is largest and of R-supercell is smallest and the cubic supercell-8 has Fermi level in between Slab and bar geometry.

 

 

 

Table 1: Parameter from scf calculations

Supercells With identifier	Total Energy (Ry)	Total Force (Ry/Bohr)	Total Stress (kbar)		Fermi Energy (eV)
			AA	AB
U-supercell, 16	-541.28579160	0.077127	34.89	8.92	6.3043
C-supercell, 824	-4330.28633451	0.217804	34.78	8.92	6.3050
R-supercell, 828	-4330.34510075	0.218130	33.89	8.97	6.3684
B-supercell, 834	-4330.34714219	0.217905	33.79	8.99	6.2912

AA -> XX,YY, ZZ; AB -> XY, YZ, XZ

 

Table 2: Parameters from BFGS geometric optimization

Supercells identifier	Surface to Volume Ratio	Total Energy (Ry)	Total Force (Ry/Bohr)	Total Stress (kbar)		Fermi Energy (eV)
				AA	AB
16	1.03	-541.2974517208	0.000697	32.71	0.12	6.3027
824	0.51	-4330.3796094019	0.002342	32.72	0.10	6.3027
828	0.6	-4330.4380487730	0.002327	31.61	0.10	6.3669
834	0.73	-4330.4400982761	0.002341	31.55	0.10	6.2893

 AA -> XX,YY, ZZ; AB -> XY, YZ, XZ

The contributions of density of states of Cd and S are illustrated in Fig.2 along with total DOS of CdS in all supercells. It showed that the minima of conduction band are formed by a mixture of the Cd’s 5s-orbital and the S’s 3p-orbital and the maxima of the valence band are formed from the 3p-orbital of S atoms. The nature of the PDOS of supercell-8 is different.

 

 

Figure 2: PDOS of an electrons of Cd, S and CdS in supercells 1, 8₂₄, 8₂₈ and 8₃₄ . (Colour code: Red-Cd, Blue-S, Black-combine CdS)

DISCUSSION:

More atoms correspond to higher Density of states. It has been proved in comparing DOS of 16 and all supercell-8 shown in Figure3. Convolutions of states and broadening of bands occurring in supercell-8 with more than four times of a DOS of supercell-1 (i.e. single cell are being observed in calculation). For a lowest valence states, in Fig.3(a), the DOS of R-supercell and B-supercell are same and formed single peak band (of s-orbital states of S ) instead convoluted two of C-supercell. In a Fig.3(b) all supercell-8 have shown nearly same pattern of the band formed by d-orbital states of Cd. States around Fermi energy (Fig.3(c) and (d)) are small variations for all supercell-8. From the cubic to bar or to slab geometry, some peaks are tending to flat. C-supercell has observed fewer states per eV compare to the R- and B- supercell-8s. The conduction states has slightly affected for the same number of unit cell used to turn its cubic to bar or to slab structure.    However, the DOS of R-supercell and B-supercell have slight difference and have different fermi energy (E_f), force and stress and densities (table1).

 

 

Figure 3: Comparison of the total density of states of supercells with closer observation of the variation in DOS in supercells-8.

A band gap of CdS unit cell (i.e 1₆) was obtained from DOS and found 2.11eV and it is narrowing in the supercell-8 (see figure4). This is reasonable and agreed with found in literature for CdS[2][3][4][7][11][14]. It is outcome of difference of Cd’s s-states of uppermost valence band and S’s 3p-states in lowest conduction band. It is found at Γ-points (the center of Brillion zone) for 1₆ using band tool. From DOS, the order of band gap is found as 1₆>>8₃₄>8₂₈>8₂₄. This indicates supercells have less band gap compared to a single unit cell of cubic CdS. The maximum surface structures have maximum band gap energy in superlattice-8. These suggest the Bandgap must be directly proportional to the surface to volume ratio at constant p of supercell.

 

All these Eg were obtained by general practice. However, we notice that, when the band gap is calculated by taking the first peak (as uppermost state) of top valence band and first peak (as lowest state) of bottom conduction band from the E_f has been taken in account then it is found 2.6eV for 11 which has matched with an experimental data. Although, the order of band gap remain unchanged.

 

Figure 4: Energy Band gap from DOS

 

In added to P. P. Favero et. al. finding [11], our result showed the BFGS geometric optimization and self consistency field calculation for 8₂₄ (cubic) has lowest Hamiltonian than 8₂₈ (slab/film like) and 834 (bar/rod like). Thus, comparatively cubic needs to work less for preservation of lattice against deformation.

 

If 8₂₄, 8₂₈ and 8₃₄ are consider as a restricted number of unit cell’s supercell representative of dot, film and rod, then variation of the properties can be inferred from these supercells and can be extended to big structure in nanosize. Supercell 8₂₄ and 1₆, concerned to (2×2×2) and (1×1×1) respectively (from Fig.1, Fig.2, Fig.3 and Table1), shown the decrement in band gap and slightly increment in fermi energy on the increment of dimensions of the cubic structure. (This study undergo only with eight unit cell made supercells.)

 

CONCLUSION:

According to the optimization supercell-8 at its ground state, the properties are found dependent on its (geometric) structure. Density of state also agreed with this. The supercell with more unit cells was found with more Hamiltonian than same geometry and the unidirectional supercell has with slightly more bandgap. This study has clearly signified the fact that there is a slight difference in different structures made by same number of unit cells.

 

ACKNOWLEDGEMENT:

Authors are acknowledging Asist. Prof. Nitin Bijewar, UDP, University of Mumbai for meaningful discussion and his guidelines on Quantum espresso tool for DFT. We also thankful the Quantum Espresso and BURAI team and acknowledge them here.

 

CONFLICT OF INTEREST:

There is no conflict of interest.

 

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Received on 17.11.2017         Modified on 02.12.2017

Accepted on 20.12.2017         © AJRC All right reserved