Tunable band structure and effective mass of disordered chalcopyrite
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Tunable band structure and effective mass of disorderedchalcopyrite
Wang Ze-Lian and Zhao Yong-Hong ∗ College of Physics and Electronic Engineering, Institute of Solid State Physics,Sichuan Normal University, Chengdu 610068, P. R. China
Xie Wen-Hui
Department of Physics, East China Normal University, Shanghai 200062, P. R. China
Abstract
The band structure and effective mass of disordered chalcopyrite photovoltaic materialsCu − x Ag x Ga X ( X = S, Se) are investigated by density functional theory. Special quasiran-dom structures are used to mimic local atomic disorders at Cu/Ag sites. A local density pluscorrection method is adopted to obtain correct semiconductor band gaps for all compounds. Thebandgap anomaly can be seen for both sulfides and selenides, where the gap values of Ag com-pounds are larger than those of Cu compounds. Band gaps can be modulated from 1.63 to 1.78eV for Cu − x Ag x Ga Se , and from 2.33 to 2.64 eV for Cu − x Ag x Ga S . The band gap minima andmaxima occur at around x = 0 . x = 1, respectively, for both sulfides and selenides. In orderto show the transport properties of Cu − x Ag x Ga X , the effective mass is shown as a function ofdisordered Ag concentration. Finally, detailed band structures are shown to clarify the phononmomentum needed by the fundamental indirect-gap transitions. These results should be helpful indesigning high-efficiency photovoltaic devices, with both better absorption and high mobility, byAg-doping in CuGa X . zinc-blende semiconductors still play an important role, owing to theirhigh efficiency and stability in real environments. For example, Cu ZnSnSe (CZTS) andCu(In,Ga)Se (CIGS), which are both based on chalcopyrite CuGa X ( X = S, Se), havebeen considered as potential candidates for photovoltaics. It has been reported recentlythat the efficiency of CIGS has gone beyond 20%, which is close to that of polycrystallinesilicon. Furthermore, photovoltaic devices related to CZTS have gained attention becausethey solely contain abundant and nontoxic materials.
The spectrum of sunshine arriving at the earth ranges from 0.35 to 2500 nm. Therefore,tunable bandgaps are essential for photovoltaic materials, in order to absorb solar fluxas much as possible. The use of doping impurities is the most commonly used method tomodulate the bandgaps, as well as effective mass (EM), of semiconductors.
Phase stabilityand formation energy of uniform impurities in CZTS have been systematically reviewedand classified using the supercell method.
Owing to high manufacturing costs, practicalphotovoltaic devices cannot be made of crystalline CZTS, and hence atomic disordereddefects and impurities are of high importance in practical devices. Although successful inuniform situations, the supercell method is difficult to apply to highly disordered impurities,which are very common in practical photovoltaic devices. In addition, the energy bandstructure is a key factor for both optical and electronic transport properties of photovoltaicmaterials. However, there is a well-known underestimation of semiconductor bandgaps byfirst-principles calculations. There are several common methods to modify the bandgapunderestimation of density functional theory, such as the Heyd–Scuseria–Ernzerhof (HSE)hybrid functional , quasiparticle approximation with Green’s Function (GW) , and themodified Becke–Johnson (mBJ) method. The HSE and GW calculations are extremelyexpensive in calculation, while the mBJ method can only give a partially corrected bandgapof 1.03 eV for CuGaSe , which is still much smaller than the experimental value of 1.68eV. Therefore, it is desirable to find a first-principles scheme to overcome the well-knownbandgap underestimation for semiconductors, which is also capable of treating disorderedimpurities including spin-orbit coupling (SOC), with moderate computational cost.Although CZTS and CIGS are the most important chalcopyrite photovoltaic materials,Cu − x Ag x Ga X ( X = Se, S) have also drawn much attention, owing to their interesting bandanomalies and grain boundary effects. In this work, we focus on the band structure and2
ABLE I. Full optimized lattice constants (a/c) of disordered Cu − x Ag x Ga X ( X = Se, S) givenby SQS method in unit of ˚A. x Cu − x Ag x GaSe Cu − x Ag x GaS EM of Cu − x Ag x Ga X , modulated by disordered Ag atoms. The special quasirandom struc-ture (SQS) method has been proved reasonable for disorder effects in all these compoundswith x ranging from 0.0 to 1.0. The initial guesses of internal parameters are given byRef. 18. All electronic structure calculations are based on fully optimized SQS structures,including both lattice constants and internal parameters. The Vienna ab-initio simulationpackage is used for optimization, with the local density approximation (LDA) for exchange-correlation potential and projector augmented wave for pseudopotentials.
The kineticenergy cutoff for a plane wave basis is set to 800 eV and a k -mesh of 4 × × X andAgGa X , as listed in TABLE I. It can be seen that sulfides have smaller lattice constantsthan selenides. For both sulfides and selenides, a increases with increasing Ag concentration,while there is a maximum value of c at approximately x = 0 .
75 and 0.5 for the sulfides andselenides, respectively. This indicates that there is a compression strain along the z -directionfor all compounds when the Ag concentration is larger than the threshold value.The electronic structure calculations are based on the linear combination of atomic orbital(LCAO) method as implemented in Nanodcal package. A set of optimized double- ζ polarization (DZP) LCAO basis is used for all atoms, together with a real space grid energycutoff of 300 Ry and a k -grid of 4 × × − .In this work, we use a LDA plus correction method (LDA+C) to modify the bandgapsof Cu − x Ag x Ga X . It is shown that experimental gap values can be achieved by choosingappropriate orbit-dependent correction parameters for each element, as given in TABLE3 ABLE II. Orbit-dependent correction values in LDA+C calculation for all elements inCu − x Ag x Ga X . Element s p dCu 0.00 0.00 4.73Ag 0.00 0.00 5.46Ga 3.00 0.00 0.00Se 3.00 0.00 0.00S 3.90 0.28 9.40 II. The calculated bandgaps for Cu − x Ag x GaSe and Cu − x Ag x GaS for x = 0.0, 0.25, 0.5,0.75, and 1.0 are shown in FIG. 1. Spin-orbit interaction is important for the electronicstructure of semiconductors, which is included in all calculations. The solid red diamondsshow the results including SOC, while the hollow blue diamonds show those without SOC. Itcan be seen that there is a gap minimum for Ag concentration near 0.5 for both sulfides andselenides. The results show that SOC interaction can decrease the bandgaps significantlyfor selenides, but only modestly for sulfides. The calculated bandgaps are consistent withexperimental and semiclassical values.EM is important for the electronic transport properties in photovoltaic materials.Anisotropic EM is defined as follows, m ∗ i = ~ d E/dk i , ( i = x, y, z ) . (1)The calculated EM of Cu − x Ag x GaSe is shown in FIG. 2. These results indicate thatEM behaves similarly in the k x - and k y -directions, where the hole EM increases with Agconcentration and the electron EM decreases with it. Moreover, SOC can decrease the holeEM to approximately 0.07 m , but keep the electron EM unchanged. Owing to the factthat the optimized lattice constants of SQSs deviate significantly from the cubic structureas given in TABLE I, EM in the k z -direction differs significantly from those in the k x - and k y -directions. The hole EM in the k z -direction remains nearly constant for Ag concentrationranging from 0.0 to 0.75, and then decreases slightly until x = 1 .
0. However, the electronEM decreases remarkably with increasing Ag concentration. For Cu − x Ag x GaS , there isa maximum at x = 0 .
25 for electron EM and SOC has little effect on EM in the k x - and4 IG. 1. Bandgaps as a function of Ag concentration, (a) for Cu − x Ag x GaSe and (b) forCu − x Ag x GaS . The blue hollow diamonds show that without SOC, while the red solid diamondsshow that with SOC included. k y -directions. EM in the k z -direction behaves similar to that of Cu − x Ag x GaSe .There are several type of EMs for different purposes, such as EM for density of statesand conductivity, among others. For photovoltaic applications, EM for conductivity is ofmost importance, which is related to photocurrent and defined as follows, m ∗ cond = 3[ 1 m ∗ x + 1 m ∗ y + 1 m ∗ z ] − . (2)The calculated m ∗ cond as a function of Ag concentration is shown in FIG. 3. FIG. 3 (a) shows m ∗ cond of Cu − x Ag x GaSe . The hole EM increases slowly from 0.42 m and reaches a max-imum of 0.46 m at approximately x = 0 .
75, while the electron EM quickly decreases from0.40 m to 0.23 m with Ag concentration increasing from 0.0 to 1.0. m ∗ cond of Cu − x Ag x GaS is shown in FIG. 3 (b). The hole EM increases from 0.76 m and reaches the maximum of0.99 m at approximately x = 0 .
75. The electron EM begins with 0.53 m at x = 0 .
0, thenreaches a maximum of 0.57 m at approximately x = 0 .
25 and a minimum of 0.43 m at5 IG. 2. Anisotropic EM of Cu − x Ag x GaSe , where (a), (b), and (c) show the x -, y - and z -directions,respectively. In each subfigure, the solid and/or hollow triangles show hole and/or electron EM,respectively. The upward and downward triangles show those with and without SOC. x = 1 .
0. Generally speaking, a smaller EM implies a higher carrier mobility. Therefore, theelectron mobility of Cu − x Ag x GaSe can be promoted by increasing Ag concentration, whileaccompanied by some reduction of hole mobility. For Cu − x Ag x GaS , the electron mobilitycan be promoted only when the Ag concentration is larger than 0.25, and accompanied by alarge reduction of hole mobility. On balance, the mobility of selenides is higher than that of6ulfides for both electrons and holes, and should be enhanced by increased Ag concentration. FIG. 3. EM for conductivity of (a) Cu − x Ag x GaSe and (b) Cu − x Ag x GaS , where the upwardand downward solid triangles show hole EM for with and without SOC, and hollow triangles forelectron EM. Optic absorption is one of the most important properties of photovoltaic materials. Theoptical transitions between any bands must obey conservation of both energy and momen-tum. Indirect-gap semiconductors undergo a phonon-assisted transition. However, for fun-damental transition of direct-gap semiconductors, phonons are unnecessary, which typicallyresults in higher absorption. On the other hand, direct bandgap will unfortunately resultin short minority carrier lifetime. The detailed band structure of Cu − x Ag x GaSe is shownin FIG. 4, where ∆ k Ci , ( i = x, y, z ) shows the deviation of the conduction band minimum(CBM) and ∆ k Vi indicates the deviation of the valence band maximum (VBM) from the Γpoint. ∆ q i = ∆ k Ci − ∆ k Vi is the phonon momentum needed by the fundamental indirect-gaptransition. It can be seen from FIG. 4(a) and (b) that SOC causes the top valence band inthe k x -direction of CuGaSe to split about 0.005 Bohr − , but leaves the bottom conduction7 IG. 4. Anisotropic band structure of Cu − x Ag x GaSe , where (a) and (b) for conduction bandsand valence bands in the k x − direction, respectively. The blue lines show bands without SOC,while the red dot line those including SOC. (d) and (e) show band structure in k z − direction.The momentum of phonon needed by fundamental transitions in the k x − and k z − directions arepresented in (c) and (f). band unsplit. FIG. 4(c) shows ∆ q x as a function of Ag concentration, which indicates thata larger phonon momentum is needed for larger Ag concentration compounds. The bandstructure in the k y -direction is absolutely similar to that in the k x -direction. The CBM andVBM of CuGaSe in the k z -direction are different and shown in FIG. 4(d) and (e), whereit can be seen that VBM remains in the Γ point, regardless of SOC or an increase in Agconcentration. When there is no Ag, SOC causes a splitting of CBM, which disappearsquickly with increasing Ag concentration. This means that a reduction of Ag ions can leadto a direct bandgap in the k z -direction of Cu − x Ag x GaSe .In summary, we calculate the electronic band structure and effective mass of disorderedquaternary chalcopyrite Cu − x Ag x Ga X ( X = Se, S) as a function of Ag concentration.Disorder structures are depicted by SQS method and a LDA+C potential is used to correctthe well-known bandgap underestimation of the first-principles calculation. SOC is included8or all compounds. For both sulfides and selenides, the bandgap minimum appears at 50%Ag concentration. Meanwhile, bandgaps of AgGa X are bigger than CuGa X , which isconsistent with the abnormal band structure of the previous study. The anisotropic EMsof Cu − x Ag x Ga X have been calculated, from which the conductivity related to EM canbe derived. The results show that the electron EM of Cu − x Ag x Ga Se decreases withincreasing Ag concentration, and the hole EM remains nearly constant. However, thereare clear maxima at x = 0 .
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