Electronic and Optical Structure of Wurtzite CuInS2 - The Journal of

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Article 2

Electronic and Optical Structure of Wurtzite CuInS

Stanko Tomic, Leonardo Bernasconi, Barry G. Searle, and Nicholas M. Harrison J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 11 Jun 2014 Downloaded from http://pubs.acs.org on June 16, 2014

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The Journal of Physical Chemistry

Electronic and Optical Structure of Wurtzite CuInS2 Stanko Tomić∗ Joule Physics Laboratory, University of Salford, Manchester M5 4WT, United Kingdom Leonardo Bernasconi STFC Rutherford Appleton Laboratory, Harwell Oxford, Didcot OX11 0QX, United Kingdom Barry G. Searle SFTC Daresbury Laboratory, Daresbury, Cheshire WA4 4AD, United Kingdom Nicholas M. Harrison SFTC Daresbury Laboratory, Daresbury, Cheshire WA4 4AD, United Kingdom Department of Chemistry, Imperial College, London SW7 2AZ, United Kingdom (Dated: May 27, 2014)

Abstract We present a theoretical study of the electronic structure of the wurtzite CuInS2 material. To address reliably some material properties of this new phase we use hybrid density functional theory. Among possible wurtzite polymorphs we have determined the most stable phase on the basis of total energy minimisation. The minimum energy structure exhibits a semiconducting ground state with a band gap of ∼ 1.3 eV in excellent agreement with experimental data. We use time dependent density functional theory to compare the optical response of the chalcopyrite and wurtzite phases, and to identify the nature of the optically active transitions in the vicinity of the absorption edge. Our analysis indicates that the wurtzite CuInS2 structure is a suitable material for photovoltaic applications. Keywords: CIS, DFT, TDDFT, band structure, absorption

1

ACS Paragon Plus Environment


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I.

INTRODUCTION

The potential of the I-III-VI2 materials, in the chalcopyrite phase, or I2 -II-IV-VI2 materials in kesterite phase, for applications in photovoltaics is now well established1–6 . The most prominent members of this family, CuInSe2 and CuInx Ga1−x Se2 , when used as absorbing layers in thin-film solar cells regularly exhibit power conversion efficiencies of ∼20%7–9 . One of the main reasons for such a large power conversion efficiency is the relatively large size of the grains (of the order of a few micrometers) into which they crystallise and the relatively benign nature of the scattering properties of the grain boundaries between them10–12 . The large size of the grains implies that their electronic structure is similar to that of the bulk crystal. Moreover, the alignment of the bottom of the conduction band between the absorbing chalcopyrite layer and the surrounding buffer and transparent conducting layers favours quick drift of the electrons in thin-film solar cell devices13,14 . These chalcopyrite materials also exhibit a very large absorption coefficient of 105 (cm−1 ) above the absorption edge as opposed to the majority of the III-V semiconductors with absorption coefficients of 104 (cm−1 ). Although for I-III-VI2 ternaries the chalcopyrite phase is thermodynamically the most favourable, recent advances in growth techniques make it possible to synthesise other crystal phases, i.e. the wurtzite phase. The wurtzite CuInS2 compound has been the subject of recent research efforts because of its potential as a light-absorbing material in printed and flexible PV devices, light-emitting diodes, nonlinear optical devices, and core/shell quantum dot nano structures. It has been observed experimentally that the band gap of wurtzite CuInS2 is near the red edge of the visible spectrum, and that this material has high optical absorption coefficients and substantial photo-stability. Once assembled in a solar cell device via vapour-phase deposition, it has been shown to achieve the power conversion efficiency of ∼7.5%.15 Several experimental groups have been able to synthesise this material recently in various nanocrystalline forms.16–19

In this paper we examine theoretically the structure, electronic and optical properties of wurtzite CuInS2 . We use hybrid exchange density-functional theory (DFT) in the B3LYP approximation, to study the relative stability of possible competing wurtzite structures. The B3LYP approximation has been shown to yield highly accurate estimates of, not only band gaps, but also the whole electronic dispersion throughout the Brillouin zone, for semi2


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conductor materials in the zinc-blend, wurtzite, and chalcopyrite structures.20–23 For the most stable wurtzite CuInS2 polymorph, we also examine the optically induced creation of a bound exciton, and we estimate its binding energy using hybrid exchange time-dependent density-functional theory (TDDFT). We have recently shown that this approach is capable of describing to high accuracy the optical response of weakly bound excitonic systems.24,25 We have found that only the most stable wurtzite CuInS2 polymorph has a semiconducting ground state, with an optical gap in excellent agreement with the observed absorption spectrum.16,18 Our paper is organised as follows. After the Introduction, Sec. I, we outline the theoretical details in (Sec. II). In Sec. III we present and discuss our results and draw conclusions in the Sec. IV.

II.

THEORETICAL MODEL

The theoretical results presented in this paper are based on DFT calculations performed using the all-electron Gaussian basis set CRYSTAL code26,27 . The hybrid B3LYP exchangecorrelation functional28,29 is used in all calculations. This functional has been shown to provide a reliable description of geometry, electronic structure and energetics of wide ranges of condensed matter systems. In particular, B3LYP provides accurate estimates of the band20,21 and optical gaps24,25 of semiconductors and various classes of insulating materials. Brillouin zone integrations are carried out using an 8 × 8 × 8 Monkhorst-Pack mesh, and a denser 16 × 16 × 16 Gilat net was used in the evaluation of the Fermi energy. Polarised triple-valence Gaussian basis sets are used throughout. In the case of In, a pseudopotential is used to describe the core electrons.30 These basis sets and the pseudopotential have been validated in previous studies.22,23 Static response properties are estimated from the converged set of Kohn-Sham orbitals, ψkn (r) = eik·r ukn (r) and their eigenenergies ǫkn . The intensity of the light transmitted at an energy ~ω, I(~ω), is assumed to follow the Beer-Lambert law I(~ω) = I0 (~ω)e−A(~ω)x ,

(1)

where I0 is the intensity of the incident light, A(~ω) is the attenuation coefficient and x is the optical path length in the material. The probability of absorption can then be expressed 3



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in terms of the absorption cross section (ACS) σ(~ω) = A(~ω)/N,

(2)

where N is the atom number density. The ACS is then given by Nvir ,Nocc. V 13 X X σ(~ω) = |dkn→km |2 δ~ω−ǫkm +ǫkn , wk Na k∈BZ m,n

(3)

where V is the unit cell volume, Na is the number of atoms per unit cell, the sum over k-points is restricted to the first Brillouin Zone (BZ), wk are Monkhorst-Pack integration weights, and Nvir and Nocc are the numbers of unoccupied and occupied bands respectively. The vertical transition probabilities are given by Z dkn→km = i dr u∗km (r)∇k ukn (r).

(4)

The calculation of the matrix elements of the ∇k operator is carried out analytically using the method of Ref. 31. The explicit dependence of the transition probabilities on the Cartesian direction ν = x, y, z of light incidence are given by integration over the volume of a single cell: dνkn→km

=i

Z

dr u∗km (r)∇kν ukn (r),

(5)

where ∇kν = ∂/∂kν . In the limit of an infinite basis set, the relation ˆr = i∇k holds.32 The lowest many-body electronic transitions, and, in particular the optical gap of each system, are determined by solving linear-response time-dependent density-functional theory33–35 (TD-DFT) equations with B3LYP eigenvalues as zeroth order states, and a non-local B3LYP exchange-correlation kernel. Optical transition energies are then computed from the poles of the TD-DFT dynamical polarisability tensor α(~ω). We have followed a procedure similar to the one described in Refs.24,25 , in which the poles are determined to arbitrary accuracy from a scan of the energy profile of tr[α(~ω)]. In this method, the lowest pole corresponds to the optical gap Eo of the system under study. Since for weakly bound excitonic systems the B3LYP band gap, Eg , provides an accurate estimate of the experimental band gap,20,21 and the TD-B3LYP optical gap is found to be very close to experiment, we identify the exciton binding energy with the quantity Eb = Eo − Eg . 4


(6)

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Although this identification is questionable from a rigourous point of view, we regard it as a satisfactory working approximation for the physical binding energy, at least in the limit of a purely Wannier-Mott exciton model. We emphasise the considerable accuracy of our TD-DFT (B3LYP) calculations in the estimation of both exciton absorption and exciton binding energies. These results extend our earlier findings24,25 concerning the reliability and accuracy of hybrid TD-DFT in the description of weakly bound Wannier-Mott excitons in small band gap materials. For this class of materials, we do indeed reproduce exciton properties at levels at least comparable to two-particle Green’s function theory (GW+BSE, see e.g.34 ) results. In the hybrid TD-DFT approach used in this work, the optical excitation energies ω, corresponding to the poles of the coupled polarisability,24 can formally be derived from the solution of a generalised anti-Hermitian eigenvalue equation35–37 (A − B)(A + B)|X + Y i = ω 2 |X + Y i,

(7)

where X and Y are left and right eigenvectors. The matrices A and B are given by37 k k ,kb kj

a i Aai,bj

= δij δab δki kj δka kb (εka a − εki i ) + (aka iki |jkj bkb )−cHF (aka bkb |jkj iki )

(8)

+ (aka iki |fxc |jkj bkb ) k k ,kb kj

a i Bai,bj

= (aka iki |bkb jkj )−cHF (aka jkj |bkb iki ) + (aka iki |fxc |bkb jkj ).

where εks s are unperturbed B3LYP energies, with s = i, j, ... and s = a, b, ... for occupied and unoccupied states respectively at the point ks of the first Brillouin Zone, and (pkp qkq |rkr sks ) are two-electron integrals in Mulliken notation. cHF is the fraction of Hartree-Fock exchange used in the exchange-correlation functional (cHF = 0.2 in B3LYP) and the exchange-correlation kernel fxc , which is used here in its adiabatic approximation,38 is given by the functional derivative of the exchange-correlation potential with respect to the density response. The appearance of excitonic features in the spectrum can be ascribed to the inclusion of the non-local Hartree-Fock response contributions in A and B, corresponding to the terms (aka bkb |jkj iki ) and (aka jkj |bkb iki ) in Eqns. 8. In particular, we observe that, for a periodic insulator, the inclusion of exact (Hartree-Fock) exchange in the TD-DFT response equations results in an explicit dependence of the exchange-correlation energy functional on the macroscopic polarisation (or, equivalently, in an ultra-non-local dependence on the surface electron density), which in turn brings about a O(1/q 2 ) divergence 5



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in the exchange-correlation kernel for small wavevectors (q → 0).39–41 This is an essential condition for a response kernel to account for the formation of bound excitons.

III.

RESULTS AND DISCUSSION

Chalcopyrite (CP) CuInS2 adopts the I ¯42d space group. Each S anion is tetrahedrally coordinated to two Cu cations and two In cations, as shown in Fig. 1(a). The S anion adopts an equilibrium position that is closer to Cu than In. The anion displacement, u, measures the extent of the unequal bond lengths in the system; it is given by u = 0.25+(RCuS +RInS )/a where RCuS and RInS are the Cu-S and In-S bond lengths, respectively, and a is a lattice parameter in [100] plane. A wurtzite structure of I-III-VI2 compounds can be generated with several structural arrangements, all of them satisfying the charge neutrality condition. We have examined the three characteristic CuInS2 wurtzite structures: (b) the simple wurtzite structure in which the occupation of the cation site alternates from Cu to In along the vertical [0001] stacking direction; this means that along [0001] the first S atom in bonded to 3 Cu and 1 In cations, and the following S atom is bonded to 1 Cu and 3 In cations, Fig. 1(b); (c) the wurtzite structure in which the cation sequence along the [0001] direction is Cu-Cu-In-In, which can be described in a periodic cell with the primitive cell doubled along the [0001] direction; this means that the first S atom is surrounded 3 Cu and 1 In cations, the second S atom in the sequence is bonded to 1 Cu and 3 In cations, the third S is bonded to 1 Cu and 3 In, and the fourth S atom is bonded to 3 Cu and 1 In cations, Fig. 1(c); (d) the wurtzite structure in which the a lattice constant is doubled and cation sites are occupied by Cu or In atoms along the [01¯10] direction and the occupancy alternates from Cu to In along the [10¯10] direction; in this structure all S anions are bonded to 2 Cu and 2 In cations. In Table I we show the energies of all the structures examined. The chalcopyrite structure is found to be the thermodynamically most stable phase. Among the three wurtzite structures considered, the WZ-c is the most stable. At the same time we have also found that the WZ-c is the only wurtzite structure exhibiting a non-vanishing direct energy gap at the Γ point. The predicted energy gap of 1.322 eV for the optimised geometry and 1.24 eV for the experimental structure, are in very good agreement when compared to the experimentally measured value for the band gap in the wurtzite phase of 1.282 eV.18 Those 6


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(a)

(b)

(c)

(d)

FIG. 1: (Colour online) Crystal structure of CuInS2 : (a) in chalcopyrite structure, (b-d) wurtzite structure. For details of different WZ structures see the main text.

gaps are very close to the maximum of the Planck spectral distribution of sunlight. We have also estimated the equilibrium lattice parameters of the WZ-c structure, Table II. We can see an overestimation of the lattice constants by 3%, which is typical of B3LYP, while the cation-anion displacement factor deviates from 3/8 by ∼ 2% which causes additional crystal field splitting that will be discussed below. To explain the appearance of the metallic behaviour in structures WZ-a and WZ-b we consider the possible cation-anion alternation patterns in I-III-VI2 WZ structures. In the cubic structure every S atom is surrounded by 2 Cu and 2 In atoms. Since every atom is tetrahedrally bonded we assume 1/4 of its formal charge is contributed to each neighbour. The sum of the charges surrounding a S is than 1/4+1/4+3/4+3/4 = +2, and the -2 on the S is perfectly compensated by its nearest neighbours. In the perfect wurtzite structure each S has either 3 In and 1 Cu or 1 In and 3 Cu neighbours and the charge is not balanced by the nearest neighbours. This means that charge must be shared beyond the local bonds implying a more delocalised electronic structure and thus a greater tendency to stabilise the metallic state. The sets of 3 atoms that change the local coordination are related by the symmetry axis along the crystalline c axis, so a supercell along this axis can’t change the coordination. In order to produce a S coordination analogous to the cubic, chalcopyrite, structure, and produce a more local electronic structure, we must break this symmetry by forming a su7



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TABLE I: Total energies and energy gaps, on optimised and experimental structure of the CuInS2 , and experimentally measure gaps, in CP and three representative WZ structure. CuInS2

Etot

CP

-96.581807

WZ-a

-96.581219

WZ-b

-96.580084

WZ-c

-96.581688

Eg (eV) [opt.] Eg (eV) [exp.] Eg (eV) 1.439

1.566

1.53

1.322

1.282

1.24

TABLE II: Lattice constants a and c and the anion displacement factor u for CuInS2 in CP and WZ-c structure. In brackets are given experimental values. CuInS2

a (˚ A)

c (˚ A)

u

CP

5.689 (5.561) 11.442 (11.116) 0.237 (0.229)

WZ-c

7.998 (7.812) 6.631 (6.429) 0.367 (0.375)

8


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percell in the a/b plane. In order to observe experimentally a symmetric wurtzite structure it would have to be formed as an average structure based on random substitutions of these lower symmetry local clusters. The electronic band structure of all CuInS2 systems examined are shown in Fig. 2. For the chalcopyrite CP-CuInS2 (a) and final WZ-c (d) structures we plot band energies for the optimised (solid lines) and the experimental (dotted lines) lattice parameters. Both structures exhibit a non-zero direct energy gap at the Γ point. By contrast WZ-a shows a semi-metallic behaviour with the top of the valence band and the bottom of the conduction band degenerate only at the Γ point. WZ-b shows metallic features. In Table III we list the most important band energies at the Γ point, their degeneracies, and relevant energy gaps with associated polarisation dependent optical matrix elements, Pifν = −i

~ hi|ˆ eν · p|f i, m0

(9)

where ν ∈ (x, y, z) , and eˆν is the unit polarisation vector along the Cartesian coordinate ν.

Eq. 9 is related to Eq. 5 by Pifν = (Ef − Ei ) · dνki→kf , and is given in units of eV˚ A. From the band dispersion shown in Fig. 2, we also estimate effective electrom masses, m∗−1 = ~−2 [∂ 2 E(k)/∂k 2 ], at the bottom of the CB and for the first two top VBs, for both CP and WZ-c CuInS2 structures. Results are listed in Table IV.

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FIG. 2: Electronic band structure of optimised CuInS2 (solid lines) obtained by using DFT n B3LYP approximation around the Γ point: (a) chalcopyrite structure along Z–Γ–X path, (b-d) WZ structures along K–Γ–M path. For details of different WZ structures see the main text. Dotted line in (a) and (d) represents the band structure of the experimental lattice.

9



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TABLE III: Electron eigen energies at Γ point of the BZ at the bottom of the conduction band (Γ1 ) and for the top three bands in the valence band. Symbol × is used to mark the degree of the band degeneracy. The top of the valence band in all materials is set to 0 eV. Relevant energy gaps, Eg , and associated polarisation dependent optical dipole matrix elements, Pν , are also listed for the reference GaAs in zinc-blend structure, CuInS2 both in chalcopyrite and WZ-c structure. State Energy (eV) Eg (eV) Px (eV˚ A) Py (eV˚ A) Pz (eV˚ A) Eb (meV) GaAs

Γ1

1.44

Γ15v 0.00×3

1.44

9.68

9.68

9.68

CP-CuInS2 Γ1

1.44

Γ5

0.00

1.44

0.00

0.00

4.90

18

Γ4

-0.063×2

1.50

5.38

5.38

0.00

27

WZ-CuInS2 Γ1

1.320

′

0.000

1.32

3.78

1.56

0.00

10

Γ6

′′

-0.088

1.41

1.96

4.44

0.00

7

Γ1

-0.130

1.45

0.00

0.00

5.21

11

Γ6

TABLE IV: Electron effective masses at the Γ point along Γ → X and along Γ →M direction for CP and WZ-c structures of CuInS2 respectively, on optimised (experimental) lattices obtained by B3LYP. All masses are given in the units of free electron mass m0 . CP-CuInS2

WZ-CuInS2

m∗e1 (m0 ) 0.153 (0.156) 0.173 (0.170) m∗h1 (m0 ) 0.958 (0.800) 2.181 (2.746) m∗h2 (m0 ) 2.507 (2.061) 0.203 (0.209)

10


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To understand the optical activity of CP and WZ CuInS2 , as a reference material we chose zinc-blende (ZB) GaAs. This material has a direct energy gap at Γ of 1.44 eV, which is very close to CP-CuInS2 , Table III. GaAs has an isotropic band structure near the Γ point, and a triply degenerate valence band maximum, in the absence of spin-orbit interactions. The optical matrix elements along all three polarisations are therefore identical. The CP structure has a tetragonal lattice, and the top of the valence band is split into Γ5 and Γ4 symmetries, which are doubly degenerate. We have verified that the in-plane components are identical, PΓx1 ,Γ4 = PΓy1 ,Γ4 , while its z component is dark, and only PΓz1 ,Γ5 is bright. Although the first few optical matrix elements are smaller in CP-CuInS2 than in zinc blend ZB GaAs, (used here as a reference material), owing to a much higher density of states within the valence band (in the region of ∼-2 eV from the valence band maximum),

the absorption of CP-CuInS2 is much higher than the one of ZB GaAs, ∼ 105 cm−1 versus ∼ 104 cm−1 . In WZ-c CuInS2 , in contrast to conventional wurtzite structures, we observe

an additional crystal field splitting, originating from the inequivalence of Cu and In, at the cation sites. This removes the degeneracy of the valence band top at the Γ6 point. We label ′

′′

these different points in the valence band Γ6 and Γ6 . From the computed band structure, we estimate this additional crystal field splitting to be ∆cf = 88 meV. Another striking feature of the WZ-c structure, as compared to a perfect tetragonal or perfect wurtzite lattice, is that the optical activity of the in-plane (x and y) components are not identical, i.e. PΓx ,Γ′ + PΓx ,Γ′′ 6= PΓy 1

6

1

6

′ 1 ,Γ6

+ PΓy

′′ 1 ,Γ6

. We estimate that the degree of optical anisotropy is

|P y 2 −P x 2 |/|P y 2 +P x 2 | = 4.4%. As expected, Γ1c → Γ1v transitions are only optically active

in the z− direction as in conventional WZ materials. In Figs. 3 and 4 we plot the absorption attenuation coefficients A(~ω) computed at the B3LYP level for CP-CuInS2 and WZ-c CuInS2 in their optimised and in experimental lattices. To reach the satisfactory accuracy the dipole matrix elements are sampled inside the first BZ on 24×24×24 Monkhorst-Pack mesh in the case of CP-CuInS2 , and 14×14×14 Monkhorst-Pack mesh in the case of WZ-c CuInS2 . In all absorption spectra the energy conservation criteria, δx function in Eq. 3, is √ √ relaxed by the Gaussian line function, exp[−(x/ 2∆)2 ]/( 2π∆), broaden for ∆ = 25 meV. From the absorption attenuation spectra presented in Fig. 3, we can see that both structures exhibit almost identical A(~ω) over wide energy ranges (0-10 eV). A small discrepancy was only observed for the CP structure for energies lager than 6 eV, corresponding to a slight change in the CB density of states for the optimised structure relative to the exper11



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imental one. By contrast, for the WZ-c structure the absorption attenuation spectra for both optimised and experimental structures, show a striking similarity in the whole range of energies considered. For optoelectronic applications, the most important region of the absorption attenuation spectra, under normal doping concentrations and photon excitation energies, lay in the vicinity of the absorption edge, i.e. just few hundreds of meV above the absorption edge. Consequently we show in Fig. 4 a detail view of this energy region for all structures considered. The chalcopyrite structure reaches an absorption maximum of ∼ 105 cm−1 soon after the absorption edge, as already reported for many Cu containing chalcopyrite structures. However the CuInS2 in WZ-c structure also shows a significant absorption coefficient, of the order of ∼ 8×104 cm−1 . This value is 7–8 times higher than

the majority of III-V semiconductors,42 qualifying this new phase as a potentially good absorbing material for photovoltaic applications. Moreover this structure exhibits such a large A(~ω) at the direct Eg of 1.32 eV at the Γ point, which also is very close to the maximum of the sun’s photon Planck emission spectra maximum. As will be shown below, the optical spectrum of both CP- and WZ-CuInS2 exhibits excitonic features at low absorption energies. Although the appearance of exciton states may alter the shape of the spectral profile near the absorption edge relative to the independent-particle B3LYP estimate, we remark that calculated exciton binding energies are very weak (∼ 10−2 eV) when compared to the independent-particles energy gaps (∼1.3 eV). In this case, the independent-particle B3LYP absorption spectrum therefore constitutes a very reasonable, and, as we have shown, considerably accurate approximation to the real spectrum of these materials. The optical response of CP-CuInS2 and WZ-c is computed using the TD-B3LYP DFT, to characterise the presence of possible excitonic features in the absorption spectrum. In Fig. 5 we show the diagonal components of the many-body polarisability obtained from TD-B3LYP using the self-consistent coupled-perturbed method described in Refs.24,25 . The different components of the polarisability tensor αxx , αyy and αzz represent the response of the electrons to radiation applied along the Cartesian directions ν = x, y and z respectively. The frequencies at which ανν shows a discontinuity (pole) correspond to optically allowed electronic transition energies. In CP-CuInS2 , Fig. 5(a) the lowest pole, corresponding to the optical gap of the system, occurs for radiation incident along the z direction at an energy Eo = 1.422 eV. A second pole for radiation incident in the xy plane is observed at 1.476 eV. The optical response of the CP-CuInS2 thus mirrors the in-plane anisotropy of 12


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6

1.0x10

(a)

opt exp

(b)

5

Absorption (1/cm)

8.0x10

5

6.0x10

5

4.0x10

5

2.0x10

WZ

CP 0.0 0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Energy (eV)

Energy (eV)

FIG. 3: B3LYP absorption spectra of CP-CuInS2 (a) and WZ-c (b). Absorption is expressed in terms of the Beer-Lambert attenuation coefficients, A(~ω), cf. Eqns. 1 and 2.

6

10

(a) Absorption (1/cm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


opt exp

(b)

5

10

4

10

WZ

CP 3

10

0

1

2

3

0

1

2

3

Energy (eV)

Energy (eV)

FIG. 4: B3LYP absorption spectra around the absorption edge of CP-CuInS2 (a) and WZ-c (b). Absorption is expressed in terms of the Beer-Lambert attenuation coefficients, A(~ω), cf. Eqns. 1 and 2.

this system. From the values of Eo and Eg , Table III, we estimate an exciton binding energy Eo − Eb = 18 meV, in excellent agreement with the experimental value of EX = 20 meV. 43

For the WZ-c sample, Fig. 5(b), we observe an anisotropy along all Cartesian directions,

with the lowest optical transition occurring for radiation incident along the x− direction.

13



The calculated exciton binding energy is in this case predicted to be Eb = 10 meV. We also observe the occurrence of further sharp absorption features at 1.313 eV and 1.401 eV along the x direction, 1.313 eV and 1.401 eV along the y direction, and 1.439 eV along the z direction. Although the lines in the x− and y− spectra appear at identical energies, a small difference between the line position in x− and y− spectra is to be expected due to the in-plane optical anisotropy discussed above. This optical anisotropy could introduce uncertainty in our predicted exciton binding energies listed in Table III, for WZ-c CuInS2 of the order of δ . 1 meV. These absorption peaks at energies higher than the optical gaps should be interpreted as fundamental absorption energies of different excitons, rather than higher-energy absorption peaks in the Rydberg series of the exciton absorbing at the energy of the optical gap.

6.0x10

3

Polarizibility ανν (Bohr )

3

αxx=αyy 3

4.0x10

αzz

3

2.0x10

0.0 0.0

0.1

1.2

1.3

1.4

1.5

1.4

1.5

Energy (eV) 3

6.0x10

αxx

3

4.0x10

3

Polarizibiility ανν (Bohr )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

3

2.0x10

3

6.0x10

αyy

3

4.0x10

3

2.0x10

3

6.0x10

αzz

3

4.0x10

3

2.0x10

0.0 0.0

0.1

1.2

1.3

Energy (eV)

FIG. 5: Energy dependence of the diagonal components of the many-body polarisability tensor computed using TD-B3LYP for CP-CuInS2 (a) and WZ-c (b). Symbols are the calculated values, the lines are guides to the eye.

The picture emerging from our calculations for both CP-CuInS2 and WZ-c is therefore 14


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that of highly anisotropic absorption media, whose spectrum is dominated by intense excitonic features in the visible region, with exciton bindings being smaller in the WZ-c structure. Smaller values of the relative dielectric constants and refractive indexes, Table V, in WZ-c and CP structure of CuInS2 suggest that the difference in the exitonic binding energies is a consequence of the much weaker electron and/or hole localisation in WZ-c, see Fig. 2(a) and (d). This is likely to be related to a higher concentration of states near the valence band top of WZ-c CuInS2 , see Fig. 2(a) and (d). In turn, this may lead to a larger number of one-particle states contributing to the relaxed (many-body) hole wave function in TD-DFT. We remark that the approach used in our work is based on an analytical calculation of the Hartree-Fock exchange integral series for periodic systems,26,27 and it is therefore extremely accurate and efficient. Work is in currently progress to address the applicability of our approach to wide-gap insulators and charge-transfer excitons,44 and a more exhaustive description of the importance of non-local Hartree-Fock exchange in the description of excitons in insulators will be presented elsewhere.

IV.

CONCLUSIONS

We have examined the structural, electronic and optical properties of CuInS2 in the chalcopyrite and wurtzite structures using density-functional theory and time-dependent density-functional theory at the hybrid (B3LYP) level of theory. We have characterised three possible wurtzite CuInS2 lattices, only one of which has a non-vanishing band gap. In this system (WZ-c), the ideal wurtzite structure is perturbed by the presence of two inequivalent cation sites, which results in the removal of the degeneracy near the top of the valence band (Γ6 ) and contributes an additional crystal field splitting of 88 meV. Both CP-CuInS2 and WZ-c are found to absorb strongly in the visible region, with LambertBeer attenuation coefficients ≥ 5 times higher than conventional III-V semiconductors. In both materials, the optical response in the visible region is dominated by intense excitonic features, with binding energies of the order of 20-30 meV in CP-CuInS2 and ∼ 10 meV in WZ-c. The latter system also exhibits a sizeable (4.4%) anisotropy in the optical response in the xy plane, in contrast to a perfect tetragonal (CP-CuInS2 ) or a perfect wurtzite structure.

15



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

TABLE V: Refractive index n(~ω) and diagonal dielectric tensor components ǫ(~ω) at ~ω = 0 and 1.0 eV CP-CuInS2 WZ-CuInS2 n (E = 0)

2.447

2.461

n (E = 1 eV)

2.808

2.829

ǫxx (E = 0)

5.972

5.705

ǫyy (E = 0)

5.972

5.841

ǫzz (E = 0)

5.989

6.056

ǫxx (E = 1 eV)

7.848

7.507

ǫyy (E = 1 eV)

7.848

7.666

ǫzz (E = 1 eV)

7.884

8.002

16


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Acknowledgements

We acknowledge support from STFC Scientific Computing Department, UK. This work was supported by EPSRC, UK, through a Service Level Agreement with STFC Scientific Computing Department. ST also wishes to thank the Royal Society, London, grant ”High Performance Computing in Modelling of Innovative Photo-Voltaic Devices.” The authors wish to thank Class Persson for critical reading of the manuscript.

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∗

Electronic address: [email protected];Phone:+44(0)1612-953847

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21



WZ-c CuInS2 6

6

Wurtzite CuInS2

5

10

opt exp

4 5

10 3 Energy (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

2 1

4

10

0

WZ

-1 3

-2

10 L

(c)

Γ

M

0

1

2

3

Energy (eV)

(d)


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Electronic and Optical Structure of Wurtzite CuInS2 - The Journal of

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