pubs.acs.org/JPCL
Exciton Dissociation in CdSe/CdTe Heterostructure Nanorods Shuzhi Wang* and Lin-Wang Wang Computational Research Division, Lawrence Berkeley National Laboratory, One Cyclotron Road, Mail Stop 50 F, Berkeley, California 94720, United States
ABSTRACT Type-II heterostructure nanorods hold good prospects for efficient charge separation in nano solar cells. Here we employed local density approximation (LDA) quality plane wave pseudopotential methods to study exciton dissociation in CdSe/CdTe collinear nanorods. We corrected the LDA band gap by approximating GWequations, and studied the correlation effect with configuration interaction methods. The calculated binding energy and radiative decay time of the charge transfer excitons agree well with experiments. The thermally activated escaping time is estimated to be shorter than the radiative recombination time, indicating the possibility of exciton dissociation if the nonradiative channel is ignored. SECTION Nanoparticles and Nanostructures
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anostructured inorganic semiconductors have been widely proposed to be used in photovoltaic and photoelectrochemical (PV/PEC) cells because of the many desirable properties.1,2 The optical, electronic, and transport properties of nanocrystals have strong dependence on morphology and can be finely tuned to meet various requirements. The quantity and requirement for the purity of materials in nano solar cells can be greatly reduced to lower manufacturing cost,1,2 which has become an important factor to be considered in order to satisfy the terawatt scale global energy need.3 Compared to thin film solar cells, however, the charge separation in a nano cell can be a major issue. In semiconductors, the built-in electric field in a p-n junction is usually used for the photogenerated exciton dissociation. In nanosystems, however, it is difficult to achieve consistent doping to form a strong p-n junction.4 Instead, type-II heterojunctions are often employed to separate charges, where the driving force is provided by the band offsets between the two materials, much as in organic photovoltaics.1,5 Recently, many such heterostructure nanosystems have been synthesized, such as ZnO/ZnS core/shell nanorods,6 CdSe/CdTe tetrapods,7 and CdSe/CdTe collinear nanorods.8 The optical and electronic properties of heterostructured nanocrystals, such as band offsets,9 charge carrier dynamics,10 and electron-hole recombination,11,12 have been studied via various experimental techniques. The detailed physical process of exciton dissociation in type-II nano heterojunctions, however, is not fully understood yet. When a photogenerated exciton in either side of the heterojunction diffuses to the interface, the electron or the hole will transfer to the other side to form a so-called “charge transfer'' (CT)exciton (cf. Figure 1).8 The electron and hole in this exciton will be localized and bounded near the interface by Coulomb attraction. A fast dissociation of the CTexciton is
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then required to fully separate the electron and hole before they recombine either radiatively or nonradiatively. Thus the quantum efficiency of nano solar cells depends on the rates of these two processes, which will be the focus of this letter. In order to study the exciton recombination and dissociation rates, we employed ab initio quality plane wave pseudopotential methods to study CTexcitons in CdSe/CdTe collinear heterostructure nanorods. Theoretically, we would also like to address the following questions: How large is the correlation effect in a nanostructure exciton? Is a single electron-hole configuration good enough in describing an exciton in a nanostructure? We have studied three CdSe/CdTe heterostructure nanorods in a wurtzite crystal structure containing 13 000-30 000 atoms, as shown in Figure 2. The size of our model nanorods;32 nm in length, and 3.2, 4.2, and 5.2 nm in diameter, respectively;is comparable to those synthesized in experiments.8,13 Following experimental results that show the nonstoichiometry of Cd/SeþTe ratios8 and the preference of binding of Cd surface atoms by organic ligands,14,15 we terminate the surface of the nanorods by Cd atoms only, which are then passivated by pseudohydrogen atoms of 1.5 e, representing an ideal ligand passivation situation.16 The valence band offset (VBO) and conduction band offset (CBO) of the two materials in their natural lattice constants are calculated to be 0.54 and 0.28 eV (cf. Figure 1), respectively, which agree well with previous calculation results.17,18 These results are obtained from a superlattice calculation with an averaged lattice constant, and then the deformation potential
Received Date: October 17, 2010 Accepted Date: December 7, 2010 Published on Web Date: December 13, 2010
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DOI: 10.1021/jz101423s |J. Phys. Chem. Lett. 2011, 2, 1–6
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ever, if the conduction band minimum (CBM) and valence band maximum (VBM) of eq 1 were used, it would not have taken into account the wave function localization caused by the electron-hole Coulomb interaction as shown in Figure 1. In order to account for this localization effect, the Coulomb potential has to be included in the single-particle Kohn-Sham equations, as follows:21,22 1 - r2 þ VLDA þ C ðrÞ þ PðrÞ - Vv ðrÞ ψc ðrÞ ¼ εc ψc ðrÞ ð2Þ 2 1 - r2 þ VLDA þ C ðrÞ - PðrÞ þ Vc ðrÞ ψv ðrÞ ¼ εv ψv ðrÞ ð3Þ 2 R Here VLDAþC(r)=V þ VCnonlocal(r), Vv(r)= |ψv(r0 )|2W(r,r0 )R tot(r) 0 0 2 0 0 dr and Vc(r)= |ψc(r )| W(r,r )dr are the screened Coulomb potential due to the hole and electron wave functions ψv and ψc, respectively, which can be solved using r 3 [ε(r)rVv(c)(r)] = 4π|ψv(c)(r)|2. These two equations need to be solved selfconsistently between ψv and ψc, much like in a two-particle Hartree-Fock equation. Because the Coulomb terms are included in both eqs 2 and 3, in order to avoid double counting, the Coulomb energy between ψv and ψc has to be added to the difference of theirReigen energies to obtain the exciton energy, Eexc = εc - εv þ |ψv(r)|2W(r,r0 )|ψc(r0 )|2drdr0, in this single configuration approximation. The exciton binding energy Eb can then be obtained as Eb =Eh þ Ee - Eexc, where Eh is the energy of the nanorod with only a hole in the VBM, and Ee is the energy with only an electron in the CBM as solved in eq 1. Thus, Eh þ Ee corresponds to the energy of the system when the electron and hole are far away and the exciton is dissociated. Our nanorods fall in the regime of strong quantum confinement in the x and y directions, where the system size is smaller than the bulk exciton radius, but not in the z direction. As a consequence, the correlation effect might be significant in the z direction. To study the many-body correlation effect beyond the above single configuration self-consistent calculations, we diagonalize a configuration interaction (CI) Hamiltonian20-22 with exciton configurations constructed from the single particle eigen states of eqs 2 and 3 We note that in the case of one exciton configuration, this approach reduces to the exciton Bethe-Salpeter equation (BSE).23 However, the CI method is more general and can be used to study multiexciton processes, e.g., the Auger effect24 and multiexciton generation in a quantum dot.25 We used 20 and 10 edge states of the valence band (VB) and conduction band (CB), respectively, each of which has a double Kramers spin degeneracy. Thus, there are 800 basis functions Φvn,cm(rh,re) = ψvn(rh)ψcm(re) in the CI basis set. Two basis functions Φv1,c1 and Φv2,c2 belong to the same “configuration'' if ψv1 and ψv2, as well as ψc1 and ψc2, are degenerate: εv1=εv2 and εc1=εc2. The 20 and 10 band edge states cover the z-direction “envelope function'' oscillations (nodal structure) down to a scale smaller than the exciton length in the z direction (as to be shown later in Figure 5). Thus they should describe well the z-direction correlation effect. The CI matrix is then calculated and diagonalized to obtain the P v P Nc (R) exciton energies and wave functions Ψ(R)= Nn=1 m=1 Cvn,cm Φvn,cm, where R is the exciton quantum number. Here, Nv and Nc are the numbers of the VB and CB states including the Kramers degeneracy.
Figure 1. Schematic plot of the type-II band alignment in CdSe/ CdTe heterostructure nanorods. The band offsets are calculated with CdSe and CdTe natural lattice constants.
Figure 2. Atomic configurations of the three model nanorods. CdTe is on the left, and CdSe is on the right.
effects19 for individual bands are added to get the natural lattice constant results. The single-particle wave functions and energies of the model nanorod systems are calculated using the plane-wave norm-conserving pseudopotential method and a charge patching method,32 under the local density approximation (LDA) of the density function theory. There is a well-known problem for the band gap calculated by LDA. The GWapproximation of the many-body perturbation theory is often used to solve this problem. However, direct GW calculations are computationally too demanding for systems with more than 100 atoms. When the LDA exchange correlation potential μxc(F(r)) is used to replace the GW self-energy term Σ(r,r0,εi), there are a short-range error, which is corrected by the LDA þ C (correction) approach (modifying the s, p, and d nonlocal pseudopotentials to fit the experimental bulk band gap), and a long-range error, which can be approximated by a classical image polarization potential.21 The modified single-particle Kohn-Sham equation approximating the GW equation is 1 2 C - r þ Vtot ðrÞ þ Vnonlocal ðrÞ ( PðrÞ ψi ðrÞ ¼ εi ψi ðrÞ ð1Þ 2 where the total potential Vtot(r) is calculated from theR charge density F(r) using the LDA formula Vtot(r) = Vion(r) þ (F(r0 ))/ (|r-r0 |)d3r0 þ μxc(r), and VCnonlocal(r) is the nonlocal part of the pseudopotential with the LDA þ C correction including the spin-orbit interactions. After solving eq 1 for the single particle eigen functions, we can include the electron-hole Coulomb, exchange, and correlation interactions to obtain exciton wave functions and energies. The simplest approximation to the exciton wave function is to use a single electron-hole configuration. How-
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DOI: 10.1021/jz101423s |J. Phys. Chem. Lett. 2011, 2, 1–6
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Figure 4. Exciton energy Eexc (filled) and exciton binding energy Eb (open) of the three CdSe/CdTe nanorods. Figure 3. Exciton energy spectrum calculated at the single particle, single configuration, and CI levels. In the bottom panel, energy levels of the ground exciton state are magnified and shown with the degeneracy and the mixing coefficients.
The calculated exciton energy spectrum for the 3.2-nmdiameter rod is shown in Figure 3 at three different levels of approximations. The single particle exciton levels including the Coulomb interaction (denoted as “single particle'' in the figure) are obtained from eqs 2 and 3. The central column shows the energy levels with only the intraconfiguration Coulomb and exchange interaction (here, due to Kramers doubling, each configuration has four spin states in its basis set). This amounts to retaining only the diagonal blocks of the Hamiltonian matrix, which correspond to matrix elements Hvn cm ,vn,cm between basis functions belonging to the same configuration. The full CI spectrum is shown in the right column. The difference between columns 1 and 2 is due to exchange splitting because the exchange interaction is not included in eqs 2 and 3. From Figure 3, we can see that the effect of exchange (cf. column “single cfg'') is very small in the whole spectrum, partly because of the small spatial overlap between the electron and hole. The ground-state exciton level is split by the exchange interaction into two doublet states with 0.1 meV energy difference. Including the correlation effect in full CI leads to slightly larger splitting, and the groundstate energies drop by ∼0.3 meV. The extent of configuration mixing canP be quantified by the mixing coefficient defined as R(R) = 1 - vn,cm|Cvn,cm(R)|2, where the sum runs over only the four spin states within one configuration R. For the ground exciton state, the mixing is very small (
