Excitonic optical properties of wurtzite ZnS quantum dots under pressure Zaiping Zeng, Christos S. Garoufalis, Sotirios Baskoutas, and Gabriel Bester Citation: The Journal of Chemical Physics 142, 114305 (2015); doi: 10.1063/1.4914473 View online: http://dx.doi.org/10.1063/1.4914473 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Near resonant and nonresonant third-order optical nonlinearities of colloidal InP/ZnS quantum dots Appl. Phys. Lett. 102, 021917 (2013); 10.1063/1.4776702 Electroluminescence of green CdSe/ZnS quantum dots enhanced by harvesting excitons from phosphorescent molecules Appl. Phys. Lett. 97, 253115 (2010); 10.1063/1.3530450 Direct observation of three-photon resonance in water-soluble ZnS quantum dots Appl. Phys. Lett. 92, 131114 (2008); 10.1063/1.2906904 Self-assembled ZnO quantum dots with tunable optical properties Appl. Phys. Lett. 89, 023122 (2006); 10.1063/1.2221892 Photophysical properties of ZnS nanoclusters J. Appl. Phys. 88, 6260 (2000); 10.1063/1.1321027
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.129.120.3 On: Mon, 25 May 2015 23:02:53
THE JOURNAL OF CHEMICAL PHYSICS 142, 114305 (2015)
Excitonic optical properties of wurtzite ZnS quantum dots under pressure Zaiping Zeng,1 Christos S. Garoufalis,1 Sotirios Baskoutas,1,a) and Gabriel Bester2,b)
1 2
Materials Science Department, University of Patras, 26504 Patras, Greece Institut für Physikalische Chemie, Universität Hamburg, 20146 Hamburg, Germany
(Received 15 December 2014; accepted 27 February 2015; published online 17 March 2015) By means of atomistic empirical pseudopotentials combined with a configuration interaction approach, we have studied the optical properties of wurtzite ZnS quantum dots in the presence of strong quantum confinement effects as a function of pressure. We find the pressure coefficients of quantum dots to be highly size-dependent and reduced by as much as 23% in comparison to the bulk value of 63 meV/GPa obtained from density functional theory calculations. The many-body excitonic effects on the quantum dot pressure coefficients are found to be marginal. The absolute gap deformation potential of quantum dots originates mainly from the energy change of the lowest unoccupied molecular orbital state. Finally, we find that the exciton spin-splitting increases nearly linearly as a function of applied pressure. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4914473] Zinc sulfide (ZnS), as one of the most important members in the family of group II-VI wide-gap semiconductors, has been extensively investigated because of its potential applications in flat-panel display, light-emitting diodes (LEDs), infrared windows, electroluminescence, sensors, lasers, and photocatalysis due to its diverse range of possible structures and morphologies, and superior chemical and thermal stabilities (see Ref. 1 and references therein). In bulk form, ZnS usually stabilizes into a cubic zinc-blende (ZB) structure (B3 phase in Struckturbericht notation) at standard temperature and pressure. It transforms into meta-stable hexagonal wurtzite (WZ) structure (B4 phase in Struckturbericht notation) at temperature above 1020 ◦C under ambient pressure.2 The WZ phase is a high temperature phase, but shows a stronger luminescence than the ZB phase, which makes it more favorable in many applications.3 Furthermore, in comparison to the ZB phase, ZnS with the WZ phase presents a higher ionization transition rate, and therefore, has a higher optical gain.4 High pressure can change the structure and morphology of a material and further modify its optical properties. The existing experimental and theoretical studies5,6 on the highpressure behaviour of WZ ZnS are focused on the phase transformations. It is well established that wurtzite ZnS experiences a phase transition into a rocksalt (RS) phase around 12 GPa.5 When the physical dimensions of the bulk WZ ZnS are reduced to be comparable to the Bohr excitonic radius, ZnS quantum dots (QDs) with significant confinement are produced. This type of nanostructure can be fabricated at low temperature (∼400 ◦C [Ref. 3], ∼280 ◦C [Ref. 7], ∼150 ◦C [Ref. 8], ∼110 ◦C [Ref. 9]) and even at room temperature,10 i.e., significantly lower than for bulk material (>1020 ◦C as mentioned before), thus reducing the cost of processing. Furthermore, WZ ZnS QDs exhibit a pressure dependence which is entirely different from the bulk case. As reported by Qadri et al.,3 WZ ZnS QDs with an average size of 25.3 nm a)Electronic mail: [email protected] b)Electronic mail: [email protected]
0021-9606/2015/142(11)/114305/5/$30.00
transform into the ZB phase at a pressure smaller than 0.5 GPa. However, reducing the size of the QD leads to an increase in the compressibility and the WZ to ZB transformation occurs at 11.5 GPa for QDs with an average size of 6 nm [Ref. 11]. Further, high-pressure experiments have also been performed to study the phase transformations of WZ ZnS nanobelts12 and very recently nanorods.13 Although valuable insights have been gained about the pressure tolerance for ZnS nanostructures in the WZ phase, information concerning the pressure-dependent optical properties of WZ ZnS QDs remains missing, both theoretically and experimentally. The pressure coefficient a p for an interband transition (e.g., Γ9v → Γ7c for direct gap WZ ZnS) is defined as14 ap =
dEg , dp
(1)
where Eg is the band gap and p is the applied pressure. The deformation potentials ai read14 dEi , (2) dlnV where i = c, v, and g indicates the conduction band (c), valence band (v), gap (g) deformation potentials, respectively, Ei is the corresponding energy, and V is the volume of the crystal. These quantities are very important physical properties which are introduced to describe the electron-phonon interaction and the strain-induced energy level shift in a crystal,15,16 and they are the crucial factors in assessing quantum confinement for holes and separately for electrons in semiconductors and related nanostructures.17–19 Plenty of effort has been made to derive these quantities for materials with diamond and ZB structures14,19 and typical III-V and II-VI compounds with WZ phase20 using first principles calculations. However, such information is missing for bulk WZ ZnS.6 Moreover, it is still an open question how the deformation potential(s) and the pressure coefficient(s) behave when the physical dimensions of the WZ ZnS structure reaches the nanoscale. In the present work, we perform density functional theory (DFT) calculations on bulk WZ ZnS. This delivers the value of the
142, 114305-1
ai =
© 2015 AIP Publishing LLC
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.129.120.3 On: Mon, 25 May 2015 23:02:53
114305-2
Zeng et al.
J. Chem. Phys. 142, 114305 (2015)
pressure coefficient, which is crucial for the construction of the pressure-dependent empirical pseudopotentials for WZ ZnS. The calculations for WZ ZnS QDs are performed using the atomistic empirical pseudopotential method (EPM),21–24 followed by a configuration interaction (CI) approach accounting for the excitonic effects. Deformation potentials and pressure coefficients for bulk wurtzite ZnS. Only few reports with regard to the study of the pressure behaviour of bulk WZ ZnS are available in the literature and are mainly based on DFT within the local density approximation (LDA) or generalized gradient approximation (GGA) (see, for example, Refs. 6 and 25). However, the band gaps of bulk WZ ZnS obtained from these calculations (∼1.89 eV by DFT/LDA and ∼2.12 eV by DFT/GGA [Ref. 25]) are often significantly underestimated in comparison to the well-known experimental value (∼3.75 eV [see Ref. 24 and references therein]). For this reason, we perform DFT calculations based on the hybrid nonlocal exchange-correlation functional of Becke and Lee, Yang and Parr (B3LYP) (Ref. 26). All calculations were performed with the Crystal06 code27 using the basis sets of Refs. 28 and 29 for the Zn and S atoms, respectively. Starting from the relaxed structure, we compress the cell in order to achieve the desired pressure and continue with a fixed volume relaxation. In all calculations, a 2 × 2 × 2 supercell and a 8 × 8 × 8 MonkhorstPack grid are adopted while the convergence threshold for RMS gradient of the structural relaxation was set to 3 × 10−4 atomic unit. We note that the pressure values throughout the work are determined by using the Murnaghan equation of state,30,31 ′
P = (B/B′)[(V0/V)B − 1],
(3)
where B = 75.8 GPa and B = 4.41 (Ref. 32) are the experimental bulk modulus and its first pressure derivative, respectively. V0 and V are the volumes of the supercell in the absence of pressure and under applied pressure, respectively. We perform the calculations on bulk ZnS within a pressure range of 0–10 GPa with a step of 2 GPa. At zero pressure, our DFT/B3LYP results give a band gap 3.62 eV for bulk WZ ZnS, which is in a good agreement with the experimental value of 3.75 eV (see Ref. 24 and references therein) and significantly better than other DFT results.25 Under pressure, both the conduction band minimum (CBM) and valence band maximum (VBM) shift to higher energy, but the shift of the CBM is dominant. The gap deformation potential dE ag = d ln gV = −4.67 eV, which is in a good agreement with the experimental value for bulk ZB ZnS (∼ − 5.0 eV [see Ref. 33]). The obtained gap values as a function of pressure are fitted according to the following quadratic function: ′
Eg = E0g + bp + cp2,
(4)
which gives a linear and quadratic pressure coefficients b = 63.2 meV/GPa and c = −1.48 meV/GPa2, respectively. These two values are in a very good agreement with the DFT/LDA results (b = 65.3 meV/GPa and c = −1.0 meV/ GPa2, see Ref. 6) and also with the experimental values of bulk ZB ZnS (b = 63.5 meV/GPa and c = −1.31 meV/GPa2, see Ref. 33). These are expected from the fact that the nearest-neighbor tetrahedral environment is very similar in
both ZB and WZ structures.14 Based on DFT/B3LYP, similar calculations have been done for WZ ZnO. We obtain a pressure coefficient a p = 20 meV/GPa, which is close to the experimental value 24.7 meV/GPa.34 Deformation potentials and pressure coefficients of wurtzite ZnS quantum dots. After knowing detailed information about the bulk WZ ZnS, we turn our attentions to the WZ ZnS QDs. We have studied QDs with three different sizes with diameter D = 3.5 nm, 4 nm, and 4.5 nm, respectively. QDs with these sizes can be experimentally fabricated by using the synthesis methods proposed in Refs. 3, 7, and 8. Our smallest QD (e.g., with D = 3.5 nm) has 1148 atoms without the surface passivation, thus being far beyond the capability of modern ab initio calculations. The QDs are cut from the bulk material with approximately spherical shape by using the experimental structure parameters (see Ref. 24), taking into account the atomistic nature of the structure. The dangling bonds at the surface of the QDs are passivated by high-band-gap artificial material, as successfully practiced previously.22–24,35 The pressure is created by suitably compressing the lattice constants in both the in-plane and outof-plane directions. The geometry optimization is performed by using Keating’s valence force field model for non-ideal WZ materials36 in order to minimize the total elastic energy, taking into account the bond-stretching, bond-bending, and bondangle coupling interactions. The two bond stretching constants (α and α ′) and two bond bending constants ( β and β ′) are taken as α = 38.98 N/m, α ′ = 42.38 N/m, β = 64.99 N/m, β ′ = 69.68 N/m, respectively. It gives the elastic constants, C11 = 124.05 GPa, C12 = 57.41 GPa, C13 = 50.03 GPa, C33 = 138.55 GPa, C44 = 31.85 GPa, C66 = 33.32 GPa, and bulk modulus B = 77.9 GPa, which reproduce almost exactly the corresponding experimental values,32,37 C11 = 124.2 GPa, C12 = 60.1 GPa, C13 = 45.5 GPa, C33 = 140.0 GPa, C44 = 28.6 GPa, C66 = 32.0 GPa, and B = 75.8 GPa, respectively. The relaxed structure is then taken as the input geometry. The crystal potential of the system is generated as a superposition of the atomic empirical pseudopotentials which are strain-dependent and given as follows:17 vα (r, ε) = vαeq (r, P = 0)[1 + γα Tr(ε)],
(5)
vαeq (r, P
where α is an atomic index, = 0) is the atomic empirical pseudopotential at zero pressure, which has been chosen as exponential type and well-tested. The parameters are V −V given in Ref. 24. Tr(ε) = V 0 is the local hydrostatic strain, 0 and γα is a fitting parameter. In our calculations, γα is taken as 4.11 for α = Zn and 0 for α = S, respectively, which reproduce exactly the pressure coefficient obtained from our DFT/B3LYP results (e.g., 63.2 meV/GPa for the bulk material). The conduction band and valence band deformation potentials are ac = −4.86 eV and a v = −0.13 eV, respectively. The resultant gap deformation ag = −4.73 eV, which agrees very well with the previous mentioned DFT results (e.g., ag = −4.67 eV) and the corresponding experimental value for bulk ZB ZnS (e.g., ag = −5.0 eV). With the crystal potential specified, the single-particle electronic states and energies are calculated by the plane-wave atomistic EPM.21–24,38 The many-body excitonic properties are calculated via CI.39 The excitonic wave functions are expanded in terms of single-substitution
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.129.120.3 On: Mon, 25 May 2015 23:02:53
114305-3
Zeng et al.
Slater determinants constructed from the single-particle wave functions of electrons and holes. The corresponding manybody Hamiltonian is solved in the CI scheme. The Coulomb and exchange integrals are screened by the position dependent and size-dependent screening function proposed by Resta,40 which gives a physically smooth transition from short range (unscreened) to long range (screened).39 The values of the parameters used in Resta model for WZ ZnS are given in Ref. 24. It should be noted that the dielectric screening is assumed to be pressure independent, and four states from the conduction band and ten states from the valence band (without accounting for the spin degree of freedom) are included in the CI calculations. We have performed calculations for pressure smaller than or equal to 2 GPa, which is far below the experimentally obtained WZ to ZB phase transformation point (∼11.5 GPa) for similarly sized QD (∼6 nm) (Ref. 11). The calculations are performed at single-particle level (without Coulomb, exchange and correlation effects) and at CI level (with Coulomb, exchange and correlation effects), and the results are depicted in Figs. 1(a) and 1(b), respectively. We find that the gaps of the QDs increase nearly linearly with increasing
J. Chem. Phys. 142, 114305 (2015)
the applied pressure for all the QD sizes. A quadratic fit (according to Eq. (4)) of the calculated single-particle gaps gives the pressure coefficient a p = 49.16 meV/GPa, 51.37 meV/GPa, and 53.03 meV/GPa for D = 3.5 nm, 4 nm, and 4.5 nm, respectively. In comparison to the bulk sample (a p = 63.2 meV/GPa), the pressure coefficients for QDs are significantly lowered, by as much as 22% for our smallest QD. Both the conduction band (e.g., ac ) and valence band (e.g., a v ) deformation potentials for the QDs, as shown in Table I, are significantly smaller than the bulk values. In comparison to a v , ac is much larger in absolute value and it has a dominant contribution to the gap deformation potential ag . Furthermore, Table I shows that QD deformation potentials are highly size-dependent and they are decreased (in absolute value) with enhancing the quantum confinement. For our smallest QD, the gap deformation potential is reduced by as much as 22%. Yet, the signs of all the deformation potentials remain the same as those in the bulk sample. To understand the physical origin of the size-dependence of deformation potentials, we follow the model proposed by Li and Wang41 and analyze the QD deformation potentials by using the bulk deformation potentials at different k-points. For simplicity but without loss of generality, we will take ac of our smallest QD as an example in the following discussion. First, we project the QD wave function of the LUMO state into the bulk Bloch functions {unk eik r } of band index n and supercell allowed reciprocal vector k within the first Brillouin zone. After summing over the bands, one can get the k-spectral ikr 2 weight Wc (k), where Wc (k) = ∞ ⟩| . We n=1 |⟨ψLUMO(r)|unke find that the QD LUMO state originates mainly from the lowest bulk conduction band. The k-spectral weight Wc (k) at Γ point only contributes around 5% of the total weight and the vast majority of the weight comes from the off-Γ k-points. This remains true when the pressure is applied, although the weight at Γ point is slightly increased. Consequently, one can approximately write the eigenenergy of the QD LUMO state as the following:41 E LU MO = Wc (k)E(k), (6) k
where E(k) is the energy of the lowest bulk conduction band at k point. Using the above equation, the QD conduction band TABLE I. Conduction band (a c ), valance band (a v ), electronic gap (a g ), and excitonic gap (a gX ) deformation potentials (in unit of eV) for bulk WZ ZnS and its quantum dots obtained from the EPM calculations. The superscript A, B, C, and X indicates the bulk A-band, B-band, C-band, and exciton, respectively. Quantum dots Diameter (nm) 3.5 4.0 4.5
ac
av
ag
a gX
−3.72 −3.89 −4.02
−0.038 −0.046 −0.057
−3.68 −3.84 −3.96
−3.63 −3.79 −3.92
Bulk FIG. 1. Single-particle gap (a) and optical band gap (b) as a function of the applied pressure for three WZ ZnS quantum dots with sizes 3.5 nm, 4 nm, and 4.5 nm in diameter. Each line represents a quadratic fit (according to Eq. (4)) to the calculated gap values.
ac −4.86
a vA
a vB
aC v
ag
−0.13
−0.13
−0.37
−4.73
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.129.120.3 On: Mon, 25 May 2015 23:02:53
114305-4
Zeng et al.
J. Chem. Phys. 142, 114305 (2015) dE
deformation potential (e.g., ac = d LUMO ln V ) can be calculated as41 dWc (k) dE(k) ac = [ E(k) + Wc (k) ], (7) d ln V d ln V k where V is the volume of the QD. It can be seen from Eq. (7) that ac has two terms and the latter one is dominant. Comparing to the bulk sample, the absolute value of this term (e.g., Wc (k) dE(k) d ln V ) decreases in the QD since the bulk deformation potential (e.g., dE(k) d ln V ) at off-Γ points is smaller than the value at Γ-point (see Fig. 2). This in turn explains why the deformation potentials of the WZ ZnS QDs are smaller than the bulk values. The weight of the off-Γ points increases with enhancing the quantum confinement. This explains why the deformation potentials decrease with decreasing the QD size. Comparing to the single-particle gaps, the optical band gaps are considerably decreased for all the pressures, especially for smaller QDs, as expected (see Fig. 1). The pressure coefficients at CI level appear to be slightly smaller than their single-particle counterparts and they are 48.47 meV/GPa, 50.73 meV/GPa, and 52.43 meV/GPa for D = 3.5 nm, 4 nm, and 4.5 nm, respectively. It turns out that considering the excitonic effects only leads to a further slight decrease in the pressure coefficient, even for our smallest QD (only ∼1.4%). Similar behaviour has also been found for the gap deformation potential (see Table I). In other words, the excitonic effects seem to have only marginal influence on the pressure coefficient and the gap deformation potential, even for WZ ZnS QDs with sizes comparable to the Bohr radius. Finally, except the valence band deformation potential for bulk A-band, we also present in Table I the corresponding ones for bulk B-band and C-band for possible comparisons with future experiments. Pressure-dependent exciton spin-splitting (SS) and fine structure splitting (FSS) of wurtzite ZnS quantum dots. For all the QDs studied herein within the pressure range 0–2 GPa,
FIG. 2. Conduction band deformation potential in absolute value |a c | of bulk WZ ZnS in the k-space obtained by EPM. The wave vector is in unit of 2π a in the direction of Γ (0,0,0) to M (0,0.5,0), and it is in unit of 2π in the direction c of Γ (0,0,0) to A (0,0,0.5), where a and c are the lattice constants of bulk WZ ZnS.
the lowest exciton stems mainly from the HOMO-LUMO transition, where the HOMO state exhibits an S-type envelope function and originating from the bulk A-band (>88%), while the LUMO state also presents an S-type envelope but has a pure parentage from bulk Γ7c band (CBM) (∼93%). This exciton is the so-called A-exciton, which consists of two degenerate “dark” states and two singly degenerate, in-plane polarized “bright” states. The energy difference between the first “dark” and the first “bright” states is defined as the exciton spin-splitting. This splitting is usually caused by the electron-hole exchange interaction.42 Its pressure dependence for various QD sizes is shown in Fig. 3. We find that the exciton spin-splitting is in the range of meV and appears to be a decreasing function of the QD diameter, as expected. Similar behaviour has been experimentally found for WZ CdSe QDs which have the same “dark” exciton mechanism as our WZ ZnS QDs (e.g., spin-forbidden [Ref. 43]). Remarkably, for a specific QD, the exciton spin-splitting increases nearly linearly with applied pressure. For our smallest QD, the increase can be as much as 7% at pressure 2 GPa. This nearly linear pressure-dependence of the exciton spin-splitting is attributed to the enhancement of the electron-hole wave function overlap by the applied pressure. Finally, we fit the pressure-dependent exciton SS by the following quadratic function: SS SS SS 2 ESS P = E0 + b p + c p ,
(8)
SS where ESS 0 is the spin-splitting at zero pressure. b is the linear coefficient (in unit of meV/GPa), which is 0.15, 0.1, and 0.07 for dot sizes, 3.5 nm, 4 nm, and 4.5 nm, respectively. cSS is the quadratic coefficient (in unit of meV/GPa2), which are −0.016, −0.011, and −0.0078 for the previously mentioned three QDs. Another important optical property we study in the present work is the exciton fine structure splitting (FSS), referring to the splitting of the first two “bright” exciton states. The origin of this splitting is attributed to the combined effects of the atomistic symmetry, spin-orbit interaction, and the electronhole exchange interaction.44 The FSS is usually in the range of 10−6 eV (e.g., µeV). We find that the FSS is a decreasing
FIG. 3. Exciton spin-splitting as a function of the applied pressure for three WZ ZnS QDs with diameter D = 3.5 nm, 4 nm, and 4.5 nm. Each solid line represents a quadratic fit to the calculated values of the exciton spin-splitting.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.129.120.3 On: Mon, 25 May 2015 23:02:53
114305-5
Zeng et al.
function of the QD size, irrespectively of the applied pressure. However, only a very marginal increase is observed when the pressure is applied. For example, the FSS is 1.07 µeV for our largest QD at zero pressure and it is increased by only 0.14 µeV at pressure 2 GPa. To summarize, we have studied the pressure-dependent optical properties of WZ ZnS colloidal QDs in the presence of strong quantum confinement effects by using the empirical pseudopotential method and configuration interaction approach. The pressure coefficients of the QDs are substantially lower than their bulk counterpart obtained by performing DFT/B3LYP calculations. For our smallest QD, the pressure coefficient is reduced by as much as ∼23% in comparison to the bulk sample. The excitonic effects are found to have only marginal effects on the pressure coefficients. The gap deformation potential is dominated by the conduction band deformation potential in both the bulk sample and QDs. However, in comparison to the bulk sample, the absolute gap deformation potential is smaller and it is decreased by enhancing the quantum confinement. This is analyzed by using the bulk deformation potential at off-Γ points. The exciton spin-splitting increases nearly linearly with the applied pressure. It is enhanced by as much as 7% at a pressure of 2 GPa for our smallest QD. Concerning the exciton fine structure splitting, only a very slight increase is observed when pressure is applied. This work was supported by the project “Thales” under Contract No. MIS-380252. 1X. Wang, H. Huang, B. Liang, Z. Liu, D. Chen, and G. Shen, Crit. Rev. Solid
State Mater. Sci. 38, 57 (2013). Scott and H. Barnes, Geochim. Cosmochim. Acta 36, 1275 (1972). 3S. B. Qadri, E. F. Skelton, A. D. Dinsmore, J. Z. Hu, W. J. Kim, C. Nelson, and B. R. Ratna, J. Appl. Phys. 89, 115 (2001). 4E. Bellotti, K. F. Brennan, R. Wang, and P. P. Ruden, J. Appl. Phys. 82, 2961 (1997). 5S. Desgreniers, L. Beaulieu, and I. Lepage, Phys. Rev. B 61, 8726 (2000). 6M. Durandurdu, J. Phys. Chem. Solids 70, 645 (2009). 7W. Liu, Mater. Lett. 60, 551 (2006). 8Y. Zhao, Y. Zhang, H. Zhu, G. C. Hadjipanayis, and J. Q. Xiao, J. Am. Chem. Soc. 126, 6874 (2004). 2S.
J. Chem. Phys. 142, 114305 (2015) 9J.-Q. Sun, X.-P. Shen, K.-M. Chen, Q. Liu, and W. Liu, Solid State Commun.
147, 501 (2008). 10I. A. Banerjee, L. Yu, and H. Matsui, J. Am. Chem. Soc. 127, 16002 (2005). 11Y.
Pan, S. Qu, S. Dong, Q. Cui, W. Zhang, X. Liu, J. Liu, B. Liu, C. Gao, and G. Zou, J. Phys.: Condens. Matter 14, 10487 (2002). 12Z. Wang, L. L. Daemen, Y. Zhao, C. S. Zha, R. T. Downs, X. Wang, Z. L. Wang, and R. J. Hemley, Nat. Mater. 4, 922 (2005). 13Z. Li, B. Liu, S. Yu, J. Wang, Q. Li, B. Zou, T. Cui, Z. Liu, Z. Chen, and J. Liu, J. Phys. Chem. C 115, 357 (2011). 14S.-H. Wei and A. Zunger, Phys. Rev. B 60, 5404 (1999). 15J. Bardeen and W. Shockley, Phys. Rev. 80, 72 (1950). 16R. Resta, Phys. Rev. B 44, 11035 (1991). 17A. J. Williamson, L. W. Wang, and A. Zunger, Phys. Rev. B 62, 12963 (2000). 18O. L. Lazarenkova, P. von Allmen, F. Oyafuso, S. Lee, and G. Klimeck, Appl. Phys. Lett. 85, 4193 (2004). 19Y.-H. Li, X. G. Gong, and S.-H. Wei, Appl. Phys. Lett. 88, 042104 (2006). 20A. Janotti and C. G. Van de Walle, Phys. Rev. B 75, 121201 (2007). 21L.-W. Wang and A. Zunger, Phys. Rev. B 51, 17398 (1995). 22S. Baskoutas and G. Bester, J. Phys. Chem. C 114, 9301 (2010). 23Z. Zeng, C. S. Garoufalis, S. Baskoutas, and G. Bester, Phys. Rev. B 87, 125302 (2013). 24Z. Zeng, C. S. Garoufalis, and S. Baskoutas, J. Phys. Chem. C 118, 10502 (2014). 25Z. Nourbakhsh, J. Alloys Compd. 505, 698 (2010). 26A. D. Becke, J. Chem. Phys. 98, 5648 (1993). 27R. Dovesi, V. R. Saunders, C. Roetti, R. Orlando, C. M. Zicovich-Wilson, F. Pascale, B. Civalleri, K. Doll, N. M. Harrison, I. J. Bush, P. D’Arco, and M. Llunell, 06 User’s Manual, University of Torino, Torino, 2006. 28J. E. Jaffe and A. C. Hess, Phys. Rev. B 48, 7903 (1993). 29A. Lichanot, E. Apra, and R. Dovesi, Phys. Status Solidi B 177, 157 (1993). 30B. Welber, M. Cardona, C. K. Kim, and S. Rodriguez, Phys. Rev. B 12, 5729 (1975). 31G. A. Narvaez, G. Bester, and A. Zunger, Phys. Rev. B 72, 041307 (2005). 32E. Chang and G. Barsch, J. Phys. Chem. Solids 34, 1543 (1973). 33S. Ves, U. Schwarz, N. E. Christensen, K. Syassen, and M. Cardona, Phys. Rev. B 42, 9113 (1990). 34H. Morkoç and U. Özgür, Zinc Oxide (Wiley-VCH, 2009). 35M. Califano, G. Bester, and A. Zunger, Nano Lett. 3, 1197 (2003). 36D. Camacho and Y. M. Niquet, Physica E 42, 1361 (2010). 37H. Neumann, Cryst. Res. Technol. 39, 939 (2004). 38J. Hu, L.-s. Li, W. Yang, L. Manna, L.-w. Wang, and A. P. Alivisatos, Science 292, 2060 (2001). 39A. Franceschetti, H. Fu, L. W. Wang, and A. Zunger, Phys. Rev. B 60, 1819 (1999). 40R. Resta, Phys. Rev. B 16, 2717 (1977). 41J. Li and L.-W. Wang, Appl. Phys. Lett. 85, 2929 (2004). 42J. W. Luo, A. Franceschetti, and A. Zunger, Phys. Rev. B 79, 201301 (2009). 43M. Kuno, J. K. Lee, B. O. Dabbousi, F. V. Mikulec, and M. G. Bawendi, J. Chem. Phys. 106, 9869 (1997). 44G. Bester, S. Nair, and A. Zunger, Phys. Rev. B 67, 161306 (2003).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.129.120.3 On: Mon, 25 May 2015 23:02:53
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.129.120.3 On: Mon, 25 May 2015 23:02:53
THE JOURNAL OF CHEMICAL PHYSICS 142, 114305 (2015)
Excitonic optical properties of wurtzite ZnS quantum dots under pressure Zaiping Zeng,1 Christos S. Garoufalis,1 Sotirios Baskoutas,1,a) and Gabriel Bester2,b)
1 2
Materials Science Department, University of Patras, 26504 Patras, Greece Institut für Physikalische Chemie, Universität Hamburg, 20146 Hamburg, Germany
(Received 15 December 2014; accepted 27 February 2015; published online 17 March 2015) By means of atomistic empirical pseudopotentials combined with a configuration interaction approach, we have studied the optical properties of wurtzite ZnS quantum dots in the presence of strong quantum confinement effects as a function of pressure. We find the pressure coefficients of quantum dots to be highly size-dependent and reduced by as much as 23% in comparison to the bulk value of 63 meV/GPa obtained from density functional theory calculations. The many-body excitonic effects on the quantum dot pressure coefficients are found to be marginal. The absolute gap deformation potential of quantum dots originates mainly from the energy change of the lowest unoccupied molecular orbital state. Finally, we find that the exciton spin-splitting increases nearly linearly as a function of applied pressure. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4914473] Zinc sulfide (ZnS), as one of the most important members in the family of group II-VI wide-gap semiconductors, has been extensively investigated because of its potential applications in flat-panel display, light-emitting diodes (LEDs), infrared windows, electroluminescence, sensors, lasers, and photocatalysis due to its diverse range of possible structures and morphologies, and superior chemical and thermal stabilities (see Ref. 1 and references therein). In bulk form, ZnS usually stabilizes into a cubic zinc-blende (ZB) structure (B3 phase in Struckturbericht notation) at standard temperature and pressure. It transforms into meta-stable hexagonal wurtzite (WZ) structure (B4 phase in Struckturbericht notation) at temperature above 1020 ◦C under ambient pressure.2 The WZ phase is a high temperature phase, but shows a stronger luminescence than the ZB phase, which makes it more favorable in many applications.3 Furthermore, in comparison to the ZB phase, ZnS with the WZ phase presents a higher ionization transition rate, and therefore, has a higher optical gain.4 High pressure can change the structure and morphology of a material and further modify its optical properties. The existing experimental and theoretical studies5,6 on the highpressure behaviour of WZ ZnS are focused on the phase transformations. It is well established that wurtzite ZnS experiences a phase transition into a rocksalt (RS) phase around 12 GPa.5 When the physical dimensions of the bulk WZ ZnS are reduced to be comparable to the Bohr excitonic radius, ZnS quantum dots (QDs) with significant confinement are produced. This type of nanostructure can be fabricated at low temperature (∼400 ◦C [Ref. 3], ∼280 ◦C [Ref. 7], ∼150 ◦C [Ref. 8], ∼110 ◦C [Ref. 9]) and even at room temperature,10 i.e., significantly lower than for bulk material (>1020 ◦C as mentioned before), thus reducing the cost of processing. Furthermore, WZ ZnS QDs exhibit a pressure dependence which is entirely different from the bulk case. As reported by Qadri et al.,3 WZ ZnS QDs with an average size of 25.3 nm a)Electronic mail: [email protected] b)Electronic mail: [email protected]
0021-9606/2015/142(11)/114305/5/$30.00
transform into the ZB phase at a pressure smaller than 0.5 GPa. However, reducing the size of the QD leads to an increase in the compressibility and the WZ to ZB transformation occurs at 11.5 GPa for QDs with an average size of 6 nm [Ref. 11]. Further, high-pressure experiments have also been performed to study the phase transformations of WZ ZnS nanobelts12 and very recently nanorods.13 Although valuable insights have been gained about the pressure tolerance for ZnS nanostructures in the WZ phase, information concerning the pressure-dependent optical properties of WZ ZnS QDs remains missing, both theoretically and experimentally. The pressure coefficient a p for an interband transition (e.g., Γ9v → Γ7c for direct gap WZ ZnS) is defined as14 ap =
dEg , dp
(1)
where Eg is the band gap and p is the applied pressure. The deformation potentials ai read14 dEi , (2) dlnV where i = c, v, and g indicates the conduction band (c), valence band (v), gap (g) deformation potentials, respectively, Ei is the corresponding energy, and V is the volume of the crystal. These quantities are very important physical properties which are introduced to describe the electron-phonon interaction and the strain-induced energy level shift in a crystal,15,16 and they are the crucial factors in assessing quantum confinement for holes and separately for electrons in semiconductors and related nanostructures.17–19 Plenty of effort has been made to derive these quantities for materials with diamond and ZB structures14,19 and typical III-V and II-VI compounds with WZ phase20 using first principles calculations. However, such information is missing for bulk WZ ZnS.6 Moreover, it is still an open question how the deformation potential(s) and the pressure coefficient(s) behave when the physical dimensions of the WZ ZnS structure reaches the nanoscale. In the present work, we perform density functional theory (DFT) calculations on bulk WZ ZnS. This delivers the value of the
142, 114305-1
ai =
© 2015 AIP Publishing LLC
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.129.120.3 On: Mon, 25 May 2015 23:02:53
114305-2
Zeng et al.
J. Chem. Phys. 142, 114305 (2015)
pressure coefficient, which is crucial for the construction of the pressure-dependent empirical pseudopotentials for WZ ZnS. The calculations for WZ ZnS QDs are performed using the atomistic empirical pseudopotential method (EPM),21–24 followed by a configuration interaction (CI) approach accounting for the excitonic effects. Deformation potentials and pressure coefficients for bulk wurtzite ZnS. Only few reports with regard to the study of the pressure behaviour of bulk WZ ZnS are available in the literature and are mainly based on DFT within the local density approximation (LDA) or generalized gradient approximation (GGA) (see, for example, Refs. 6 and 25). However, the band gaps of bulk WZ ZnS obtained from these calculations (∼1.89 eV by DFT/LDA and ∼2.12 eV by DFT/GGA [Ref. 25]) are often significantly underestimated in comparison to the well-known experimental value (∼3.75 eV [see Ref. 24 and references therein]). For this reason, we perform DFT calculations based on the hybrid nonlocal exchange-correlation functional of Becke and Lee, Yang and Parr (B3LYP) (Ref. 26). All calculations were performed with the Crystal06 code27 using the basis sets of Refs. 28 and 29 for the Zn and S atoms, respectively. Starting from the relaxed structure, we compress the cell in order to achieve the desired pressure and continue with a fixed volume relaxation. In all calculations, a 2 × 2 × 2 supercell and a 8 × 8 × 8 MonkhorstPack grid are adopted while the convergence threshold for RMS gradient of the structural relaxation was set to 3 × 10−4 atomic unit. We note that the pressure values throughout the work are determined by using the Murnaghan equation of state,30,31 ′
P = (B/B′)[(V0/V)B − 1],
(3)
where B = 75.8 GPa and B = 4.41 (Ref. 32) are the experimental bulk modulus and its first pressure derivative, respectively. V0 and V are the volumes of the supercell in the absence of pressure and under applied pressure, respectively. We perform the calculations on bulk ZnS within a pressure range of 0–10 GPa with a step of 2 GPa. At zero pressure, our DFT/B3LYP results give a band gap 3.62 eV for bulk WZ ZnS, which is in a good agreement with the experimental value of 3.75 eV (see Ref. 24 and references therein) and significantly better than other DFT results.25 Under pressure, both the conduction band minimum (CBM) and valence band maximum (VBM) shift to higher energy, but the shift of the CBM is dominant. The gap deformation potential dE ag = d ln gV = −4.67 eV, which is in a good agreement with the experimental value for bulk ZB ZnS (∼ − 5.0 eV [see Ref. 33]). The obtained gap values as a function of pressure are fitted according to the following quadratic function: ′
Eg = E0g + bp + cp2,
(4)
which gives a linear and quadratic pressure coefficients b = 63.2 meV/GPa and c = −1.48 meV/GPa2, respectively. These two values are in a very good agreement with the DFT/LDA results (b = 65.3 meV/GPa and c = −1.0 meV/ GPa2, see Ref. 6) and also with the experimental values of bulk ZB ZnS (b = 63.5 meV/GPa and c = −1.31 meV/GPa2, see Ref. 33). These are expected from the fact that the nearest-neighbor tetrahedral environment is very similar in
both ZB and WZ structures.14 Based on DFT/B3LYP, similar calculations have been done for WZ ZnO. We obtain a pressure coefficient a p = 20 meV/GPa, which is close to the experimental value 24.7 meV/GPa.34 Deformation potentials and pressure coefficients of wurtzite ZnS quantum dots. After knowing detailed information about the bulk WZ ZnS, we turn our attentions to the WZ ZnS QDs. We have studied QDs with three different sizes with diameter D = 3.5 nm, 4 nm, and 4.5 nm, respectively. QDs with these sizes can be experimentally fabricated by using the synthesis methods proposed in Refs. 3, 7, and 8. Our smallest QD (e.g., with D = 3.5 nm) has 1148 atoms without the surface passivation, thus being far beyond the capability of modern ab initio calculations. The QDs are cut from the bulk material with approximately spherical shape by using the experimental structure parameters (see Ref. 24), taking into account the atomistic nature of the structure. The dangling bonds at the surface of the QDs are passivated by high-band-gap artificial material, as successfully practiced previously.22–24,35 The pressure is created by suitably compressing the lattice constants in both the in-plane and outof-plane directions. The geometry optimization is performed by using Keating’s valence force field model for non-ideal WZ materials36 in order to minimize the total elastic energy, taking into account the bond-stretching, bond-bending, and bondangle coupling interactions. The two bond stretching constants (α and α ′) and two bond bending constants ( β and β ′) are taken as α = 38.98 N/m, α ′ = 42.38 N/m, β = 64.99 N/m, β ′ = 69.68 N/m, respectively. It gives the elastic constants, C11 = 124.05 GPa, C12 = 57.41 GPa, C13 = 50.03 GPa, C33 = 138.55 GPa, C44 = 31.85 GPa, C66 = 33.32 GPa, and bulk modulus B = 77.9 GPa, which reproduce almost exactly the corresponding experimental values,32,37 C11 = 124.2 GPa, C12 = 60.1 GPa, C13 = 45.5 GPa, C33 = 140.0 GPa, C44 = 28.6 GPa, C66 = 32.0 GPa, and B = 75.8 GPa, respectively. The relaxed structure is then taken as the input geometry. The crystal potential of the system is generated as a superposition of the atomic empirical pseudopotentials which are strain-dependent and given as follows:17 vα (r, ε) = vαeq (r, P = 0)[1 + γα Tr(ε)],
(5)
vαeq (r, P
where α is an atomic index, = 0) is the atomic empirical pseudopotential at zero pressure, which has been chosen as exponential type and well-tested. The parameters are V −V given in Ref. 24. Tr(ε) = V 0 is the local hydrostatic strain, 0 and γα is a fitting parameter. In our calculations, γα is taken as 4.11 for α = Zn and 0 for α = S, respectively, which reproduce exactly the pressure coefficient obtained from our DFT/B3LYP results (e.g., 63.2 meV/GPa for the bulk material). The conduction band and valence band deformation potentials are ac = −4.86 eV and a v = −0.13 eV, respectively. The resultant gap deformation ag = −4.73 eV, which agrees very well with the previous mentioned DFT results (e.g., ag = −4.67 eV) and the corresponding experimental value for bulk ZB ZnS (e.g., ag = −5.0 eV). With the crystal potential specified, the single-particle electronic states and energies are calculated by the plane-wave atomistic EPM.21–24,38 The many-body excitonic properties are calculated via CI.39 The excitonic wave functions are expanded in terms of single-substitution
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.129.120.3 On: Mon, 25 May 2015 23:02:53
114305-3
Zeng et al.
Slater determinants constructed from the single-particle wave functions of electrons and holes. The corresponding manybody Hamiltonian is solved in the CI scheme. The Coulomb and exchange integrals are screened by the position dependent and size-dependent screening function proposed by Resta,40 which gives a physically smooth transition from short range (unscreened) to long range (screened).39 The values of the parameters used in Resta model for WZ ZnS are given in Ref. 24. It should be noted that the dielectric screening is assumed to be pressure independent, and four states from the conduction band and ten states from the valence band (without accounting for the spin degree of freedom) are included in the CI calculations. We have performed calculations for pressure smaller than or equal to 2 GPa, which is far below the experimentally obtained WZ to ZB phase transformation point (∼11.5 GPa) for similarly sized QD (∼6 nm) (Ref. 11). The calculations are performed at single-particle level (without Coulomb, exchange and correlation effects) and at CI level (with Coulomb, exchange and correlation effects), and the results are depicted in Figs. 1(a) and 1(b), respectively. We find that the gaps of the QDs increase nearly linearly with increasing
J. Chem. Phys. 142, 114305 (2015)
the applied pressure for all the QD sizes. A quadratic fit (according to Eq. (4)) of the calculated single-particle gaps gives the pressure coefficient a p = 49.16 meV/GPa, 51.37 meV/GPa, and 53.03 meV/GPa for D = 3.5 nm, 4 nm, and 4.5 nm, respectively. In comparison to the bulk sample (a p = 63.2 meV/GPa), the pressure coefficients for QDs are significantly lowered, by as much as 22% for our smallest QD. Both the conduction band (e.g., ac ) and valence band (e.g., a v ) deformation potentials for the QDs, as shown in Table I, are significantly smaller than the bulk values. In comparison to a v , ac is much larger in absolute value and it has a dominant contribution to the gap deformation potential ag . Furthermore, Table I shows that QD deformation potentials are highly size-dependent and they are decreased (in absolute value) with enhancing the quantum confinement. For our smallest QD, the gap deformation potential is reduced by as much as 22%. Yet, the signs of all the deformation potentials remain the same as those in the bulk sample. To understand the physical origin of the size-dependence of deformation potentials, we follow the model proposed by Li and Wang41 and analyze the QD deformation potentials by using the bulk deformation potentials at different k-points. For simplicity but without loss of generality, we will take ac of our smallest QD as an example in the following discussion. First, we project the QD wave function of the LUMO state into the bulk Bloch functions {unk eik r } of band index n and supercell allowed reciprocal vector k within the first Brillouin zone. After summing over the bands, one can get the k-spectral ikr 2 weight Wc (k), where Wc (k) = ∞ ⟩| . We n=1 |⟨ψLUMO(r)|unke find that the QD LUMO state originates mainly from the lowest bulk conduction band. The k-spectral weight Wc (k) at Γ point only contributes around 5% of the total weight and the vast majority of the weight comes from the off-Γ k-points. This remains true when the pressure is applied, although the weight at Γ point is slightly increased. Consequently, one can approximately write the eigenenergy of the QD LUMO state as the following:41 E LU MO = Wc (k)E(k), (6) k
where E(k) is the energy of the lowest bulk conduction band at k point. Using the above equation, the QD conduction band TABLE I. Conduction band (a c ), valance band (a v ), electronic gap (a g ), and excitonic gap (a gX ) deformation potentials (in unit of eV) for bulk WZ ZnS and its quantum dots obtained from the EPM calculations. The superscript A, B, C, and X indicates the bulk A-band, B-band, C-band, and exciton, respectively. Quantum dots Diameter (nm) 3.5 4.0 4.5
ac
av
ag
a gX
−3.72 −3.89 −4.02
−0.038 −0.046 −0.057
−3.68 −3.84 −3.96
−3.63 −3.79 −3.92
Bulk FIG. 1. Single-particle gap (a) and optical band gap (b) as a function of the applied pressure for three WZ ZnS quantum dots with sizes 3.5 nm, 4 nm, and 4.5 nm in diameter. Each line represents a quadratic fit (according to Eq. (4)) to the calculated gap values.
ac −4.86
a vA
a vB
aC v
ag
−0.13
−0.13
−0.37
−4.73
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.129.120.3 On: Mon, 25 May 2015 23:02:53
114305-4
Zeng et al.
J. Chem. Phys. 142, 114305 (2015) dE
deformation potential (e.g., ac = d LUMO ln V ) can be calculated as41 dWc (k) dE(k) ac = [ E(k) + Wc (k) ], (7) d ln V d ln V k where V is the volume of the QD. It can be seen from Eq. (7) that ac has two terms and the latter one is dominant. Comparing to the bulk sample, the absolute value of this term (e.g., Wc (k) dE(k) d ln V ) decreases in the QD since the bulk deformation potential (e.g., dE(k) d ln V ) at off-Γ points is smaller than the value at Γ-point (see Fig. 2). This in turn explains why the deformation potentials of the WZ ZnS QDs are smaller than the bulk values. The weight of the off-Γ points increases with enhancing the quantum confinement. This explains why the deformation potentials decrease with decreasing the QD size. Comparing to the single-particle gaps, the optical band gaps are considerably decreased for all the pressures, especially for smaller QDs, as expected (see Fig. 1). The pressure coefficients at CI level appear to be slightly smaller than their single-particle counterparts and they are 48.47 meV/GPa, 50.73 meV/GPa, and 52.43 meV/GPa for D = 3.5 nm, 4 nm, and 4.5 nm, respectively. It turns out that considering the excitonic effects only leads to a further slight decrease in the pressure coefficient, even for our smallest QD (only ∼1.4%). Similar behaviour has also been found for the gap deformation potential (see Table I). In other words, the excitonic effects seem to have only marginal influence on the pressure coefficient and the gap deformation potential, even for WZ ZnS QDs with sizes comparable to the Bohr radius. Finally, except the valence band deformation potential for bulk A-band, we also present in Table I the corresponding ones for bulk B-band and C-band for possible comparisons with future experiments. Pressure-dependent exciton spin-splitting (SS) and fine structure splitting (FSS) of wurtzite ZnS quantum dots. For all the QDs studied herein within the pressure range 0–2 GPa,
FIG. 2. Conduction band deformation potential in absolute value |a c | of bulk WZ ZnS in the k-space obtained by EPM. The wave vector is in unit of 2π a in the direction of Γ (0,0,0) to M (0,0.5,0), and it is in unit of 2π in the direction c of Γ (0,0,0) to A (0,0,0.5), where a and c are the lattice constants of bulk WZ ZnS.
the lowest exciton stems mainly from the HOMO-LUMO transition, where the HOMO state exhibits an S-type envelope function and originating from the bulk A-band (>88%), while the LUMO state also presents an S-type envelope but has a pure parentage from bulk Γ7c band (CBM) (∼93%). This exciton is the so-called A-exciton, which consists of two degenerate “dark” states and two singly degenerate, in-plane polarized “bright” states. The energy difference between the first “dark” and the first “bright” states is defined as the exciton spin-splitting. This splitting is usually caused by the electron-hole exchange interaction.42 Its pressure dependence for various QD sizes is shown in Fig. 3. We find that the exciton spin-splitting is in the range of meV and appears to be a decreasing function of the QD diameter, as expected. Similar behaviour has been experimentally found for WZ CdSe QDs which have the same “dark” exciton mechanism as our WZ ZnS QDs (e.g., spin-forbidden [Ref. 43]). Remarkably, for a specific QD, the exciton spin-splitting increases nearly linearly with applied pressure. For our smallest QD, the increase can be as much as 7% at pressure 2 GPa. This nearly linear pressure-dependence of the exciton spin-splitting is attributed to the enhancement of the electron-hole wave function overlap by the applied pressure. Finally, we fit the pressure-dependent exciton SS by the following quadratic function: SS SS SS 2 ESS P = E0 + b p + c p ,
(8)
SS where ESS 0 is the spin-splitting at zero pressure. b is the linear coefficient (in unit of meV/GPa), which is 0.15, 0.1, and 0.07 for dot sizes, 3.5 nm, 4 nm, and 4.5 nm, respectively. cSS is the quadratic coefficient (in unit of meV/GPa2), which are −0.016, −0.011, and −0.0078 for the previously mentioned three QDs. Another important optical property we study in the present work is the exciton fine structure splitting (FSS), referring to the splitting of the first two “bright” exciton states. The origin of this splitting is attributed to the combined effects of the atomistic symmetry, spin-orbit interaction, and the electronhole exchange interaction.44 The FSS is usually in the range of 10−6 eV (e.g., µeV). We find that the FSS is a decreasing
FIG. 3. Exciton spin-splitting as a function of the applied pressure for three WZ ZnS QDs with diameter D = 3.5 nm, 4 nm, and 4.5 nm. Each solid line represents a quadratic fit to the calculated values of the exciton spin-splitting.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.129.120.3 On: Mon, 25 May 2015 23:02:53
114305-5
Zeng et al.
function of the QD size, irrespectively of the applied pressure. However, only a very marginal increase is observed when the pressure is applied. For example, the FSS is 1.07 µeV for our largest QD at zero pressure and it is increased by only 0.14 µeV at pressure 2 GPa. To summarize, we have studied the pressure-dependent optical properties of WZ ZnS colloidal QDs in the presence of strong quantum confinement effects by using the empirical pseudopotential method and configuration interaction approach. The pressure coefficients of the QDs are substantially lower than their bulk counterpart obtained by performing DFT/B3LYP calculations. For our smallest QD, the pressure coefficient is reduced by as much as ∼23% in comparison to the bulk sample. The excitonic effects are found to have only marginal effects on the pressure coefficients. The gap deformation potential is dominated by the conduction band deformation potential in both the bulk sample and QDs. However, in comparison to the bulk sample, the absolute gap deformation potential is smaller and it is decreased by enhancing the quantum confinement. This is analyzed by using the bulk deformation potential at off-Γ points. The exciton spin-splitting increases nearly linearly with the applied pressure. It is enhanced by as much as 7% at a pressure of 2 GPa for our smallest QD. Concerning the exciton fine structure splitting, only a very slight increase is observed when pressure is applied. This work was supported by the project “Thales” under Contract No. MIS-380252. 1X. Wang, H. Huang, B. Liang, Z. Liu, D. Chen, and G. Shen, Crit. Rev. Solid
State Mater. Sci. 38, 57 (2013). Scott and H. Barnes, Geochim. Cosmochim. Acta 36, 1275 (1972). 3S. B. Qadri, E. F. Skelton, A. D. Dinsmore, J. Z. Hu, W. J. Kim, C. Nelson, and B. R. Ratna, J. Appl. Phys. 89, 115 (2001). 4E. Bellotti, K. F. Brennan, R. Wang, and P. P. Ruden, J. Appl. Phys. 82, 2961 (1997). 5S. Desgreniers, L. Beaulieu, and I. Lepage, Phys. Rev. B 61, 8726 (2000). 6M. Durandurdu, J. Phys. Chem. Solids 70, 645 (2009). 7W. Liu, Mater. Lett. 60, 551 (2006). 8Y. Zhao, Y. Zhang, H. Zhu, G. C. Hadjipanayis, and J. Q. Xiao, J. Am. Chem. Soc. 126, 6874 (2004). 2S.
J. Chem. Phys. 142, 114305 (2015) 9J.-Q. Sun, X.-P. Shen, K.-M. Chen, Q. Liu, and W. Liu, Solid State Commun.
147, 501 (2008). 10I. A. Banerjee, L. Yu, and H. Matsui, J. Am. Chem. Soc. 127, 16002 (2005). 11Y.
Pan, S. Qu, S. Dong, Q. Cui, W. Zhang, X. Liu, J. Liu, B. Liu, C. Gao, and G. Zou, J. Phys.: Condens. Matter 14, 10487 (2002). 12Z. Wang, L. L. Daemen, Y. Zhao, C. S. Zha, R. T. Downs, X. Wang, Z. L. Wang, and R. J. Hemley, Nat. Mater. 4, 922 (2005). 13Z. Li, B. Liu, S. Yu, J. Wang, Q. Li, B. Zou, T. Cui, Z. Liu, Z. Chen, and J. Liu, J. Phys. Chem. C 115, 357 (2011). 14S.-H. Wei and A. Zunger, Phys. Rev. B 60, 5404 (1999). 15J. Bardeen and W. Shockley, Phys. Rev. 80, 72 (1950). 16R. Resta, Phys. Rev. B 44, 11035 (1991). 17A. J. Williamson, L. W. Wang, and A. Zunger, Phys. Rev. B 62, 12963 (2000). 18O. L. Lazarenkova, P. von Allmen, F. Oyafuso, S. Lee, and G. Klimeck, Appl. Phys. Lett. 85, 4193 (2004). 19Y.-H. Li, X. G. Gong, and S.-H. Wei, Appl. Phys. Lett. 88, 042104 (2006). 20A. Janotti and C. G. Van de Walle, Phys. Rev. B 75, 121201 (2007). 21L.-W. Wang and A. Zunger, Phys. Rev. B 51, 17398 (1995). 22S. Baskoutas and G. Bester, J. Phys. Chem. C 114, 9301 (2010). 23Z. Zeng, C. S. Garoufalis, S. Baskoutas, and G. Bester, Phys. Rev. B 87, 125302 (2013). 24Z. Zeng, C. S. Garoufalis, and S. Baskoutas, J. Phys. Chem. C 118, 10502 (2014). 25Z. Nourbakhsh, J. Alloys Compd. 505, 698 (2010). 26A. D. Becke, J. Chem. Phys. 98, 5648 (1993). 27R. Dovesi, V. R. Saunders, C. Roetti, R. Orlando, C. M. Zicovich-Wilson, F. Pascale, B. Civalleri, K. Doll, N. M. Harrison, I. J. Bush, P. D’Arco, and M. Llunell, 06 User’s Manual, University of Torino, Torino, 2006. 28J. E. Jaffe and A. C. Hess, Phys. Rev. B 48, 7903 (1993). 29A. Lichanot, E. Apra, and R. Dovesi, Phys. Status Solidi B 177, 157 (1993). 30B. Welber, M. Cardona, C. K. Kim, and S. Rodriguez, Phys. Rev. B 12, 5729 (1975). 31G. A. Narvaez, G. Bester, and A. Zunger, Phys. Rev. B 72, 041307 (2005). 32E. Chang and G. Barsch, J. Phys. Chem. Solids 34, 1543 (1973). 33S. Ves, U. Schwarz, N. E. Christensen, K. Syassen, and M. Cardona, Phys. Rev. B 42, 9113 (1990). 34H. Morkoç and U. Özgür, Zinc Oxide (Wiley-VCH, 2009). 35M. Califano, G. Bester, and A. Zunger, Nano Lett. 3, 1197 (2003). 36D. Camacho and Y. M. Niquet, Physica E 42, 1361 (2010). 37H. Neumann, Cryst. Res. Technol. 39, 939 (2004). 38J. Hu, L.-s. Li, W. Yang, L. Manna, L.-w. Wang, and A. P. Alivisatos, Science 292, 2060 (2001). 39A. Franceschetti, H. Fu, L. W. Wang, and A. Zunger, Phys. Rev. B 60, 1819 (1999). 40R. Resta, Phys. Rev. B 16, 2717 (1977). 41J. Li and L.-W. Wang, Appl. Phys. Lett. 85, 2929 (2004). 42J. W. Luo, A. Franceschetti, and A. Zunger, Phys. Rev. B 79, 201301 (2009). 43M. Kuno, J. K. Lee, B. O. Dabbousi, F. V. Mikulec, and M. G. Bawendi, J. Chem. Phys. 106, 9869 (1997). 44G. Bester, S. Nair, and A. Zunger, Phys. Rev. B 67, 161306 (2003).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.129.120.3 On: Mon, 25 May 2015 23:02:53
