ARTICLE pubs.acs.org/JPCC
Transition in the Optical Emission Polarization of ZnO Nanorods Sotirios Baskoutas†,‡ and Gabriel Bester*,† † ‡
Max-Planck-Institut f€ur Festk€ orperforschung, D-70569 Stuttgart, Germany Materials Science Department, University of Patras, 26504 Patras, Greece ABSTRACT: We study the electronic and optical properties of ZnO nanorods as a function of the aspect ratio using atomistic empirical pseudopotentials and configuration interaction for the excitonic states. We find a sharp crossover for the optical polarization, from an in-plane polarized luminescence, for an aspect ratio smaller than 3, to an out-of-plane (along c-axis) polarization for an aspect ratio larger than 3. This suggests the possibility to tailor the optical polarization by a change in morphology. We elucidate the underlying mechanism from an analysis of the valence band structure and highlight the importance of exciton correlations, which shift the crossover to a larger aspect ratio.
’ INTRODUCTION Quasi one-dimensional semiconductors, such as nanowires (NWs) and nanorods (NRs), may have optical and electrical properties favorable for device applications. Electrical transport could suffer from less scattering, and the polarized optical properties may be advantageous. Especially zinc oxide (ZnO) NWs and NRs have attracted much attention because of their large exciton binding energy and large oscillator strength—along with a certain ease in the growth1 and resilience of the material to be combined with others.2 Indeed, the literature on growth and characterization of ZnO nanostructures is overwhelming (see, e.g., ref 3). Similarly, the range of possible applications stretches from room-temperature lasing (demonstrated a decade ago,4 based on ZnO NRs), piezoelectric effects,5 and nano-optoelectronics6 to the recently suggested combination of ZnO NRs with graphene for photovoltaic applications.7 This multiplicity is in contrast to a very scarce theoretical literature on ZnO nanostructures. Some calculations exist on very small clusters,8,9 and there are only a few atomistic calculations on realistic heterostructures10 and spherical quantum dots11 but none on NRs. This, even though (or perhaps because) the valence band structure of bulk ZnO was controversial until recently, is due to a small and negative spinorbit splitting.12,13 This valence band peculiarity leads to a surprising highest occupied molecular orbital in quantum dots with orbital Pcharacter.11 The elongation along the c-axis, given in a NR, brings a new interesting component in the valence band structure, as it knowingly affects the energetic position of the wurtzite’s C-band. Hence, the collapse of emission to a single polarization axis (as already studied in detail for CdSe14), which makes these nanosystems ideal for many orientation-sensitive applications, has its origin in a rather intricate ordering of valence bands. r 2011 American Chemical Society
To capture all the relevant effects, we perform atomistic empirical pseudopotential calculations using our recently derived pseudopotential.11 The calculations include the effects of multiband coupling, multivalley coupling, spinorbit interaction, and excitonic correlation via configuration interaction. We will show that all these effects are important to obtain a qualitatively correct picture. We discuss the aspect ratio dependence of the optical band gap, the Stokes shift, and the polarization ratio. We find a good agreement with the available experimental data for the optical band gap. The optical polarization is found to exhibit a transition between in-plane and out-of-plane polarization at around an aspect ratio of 3. We show that this transition is strongly affected by exciton correlations as an electronic state with orbital P character becomes important above a certain elongation. The Stokes shift is shown to be nonmonotonic as a consequence of this mixing of configurations.
’ METHOD Using the pseudopotential for ZnO we have recently derived,11 we obtain the single-particle energies and states for both conduction and valence bands, using the plane-wave pseudopotential method15 for NRs with different aspect ratios. The Hamiltonian we are using for the single-particle states has the form ^ ¼ 1∇2 þ H 2
½vR ðr R Rn Þ þ ^vSO ∑ R nR
ð1Þ
where n is an atomic index; R specifies the atom type; and ^vSO R is the nonlocal spinorbit operator. The screened atomic pseudopotentials vR (with R = Zn, O) are centered at each atomic position, Received: May 9, 2011 Revised: July 1, 2011 Published: July 14, 2011 15862
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The Journal of Physical Chemistry C and their superposition generates the crystal potential. The parameters are optimized to reproduce the known band structure and the bulk properties of ZnO, and the results are given in ref 11. The surface passivation is approximated by a high band gap artificial material, as practiced successfully previously.11,16 The corresponding envelope functions of the single-particle states in both valence and conduction bands are obtained by projecting the fast oscillating atomic wave functions onto the bulk Bloch states at each unit cell (according to eq 6 of ref 11). This effectively smears out the atomistic oscillations and leads to an envelope function that can be displayed easily. This procedure also allow us to obtain the Bloch function character of each NR state and attribute them A-, B-, or C-band (or a mixture of them) parentage. The excitonic wave functions are expanded in terms of single-substitution Slater
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determinants constructed from the single-particle wave functions of electrons and holes. The corresponding many-body Hamiltonian is solved within the framework of configuration interaction (CI) as well as in the so-called single configuration (SC) approximation. At this level of theory the intraconfiguration Coulomb and exchange matrix elements are fully included, but the interaction between different configurations is neglected.17 At the SC level, correlations are neglected. Our computational limitations allow us to include in the CI treatment 10 states from the valence band and 4 states from the conduction band. For the screened Coulomb interaction we used the phenomenological isotropic and uniform model proposed by Resta.18 Very recently, more sophisticated models for the Coulomb and exchange interaction have been proposed.19 The optical dipole matrix elements are calculated within the dipole approximation, and the oscillator strength was calculated using Fermi’s golden rule. A review of the method can be found in ref 10.
’ RESULTS AND DISCUSSION
Figure 1. (a) Geometry of our smallest nanorod Zn378O336 with F = 1 and D = 2.2 nm (L = 2.2 nm). (b) Geometry of our largest nanorod Zn1806O1764 with F = 5 and D = 2.2 nm (L = 11.0 nm).
Electronic Properties. We define six different ZnO NRs of diameter D = 2.2 nm and various lengths L covering the aspect ratios F from 1 to 5. The numbers of atoms for the respective structures are: Zn378O336 (L = 2.2 nm), Zn504O462 (L = 3 nm), Zn840O798 (L = 5 nm), Zn1302O1260 (L = 8 nm), Zn1638O1596 (L = 10 nm), and Zn1806O1764 (L = 11 nm). A pictorial representation of our largest and smallest structure is given in Figure 1. The results for the projected envelope functions of the first four electron states e0,1,2,3 and first four hole states h0,1,2,3 for D = 2.2 nm are given in Figure 2 for different aspect ratios. To characterize the symmetry of the wave functions, we use the
Figure 2. Envelope functions for the first four electron (e0,1,2,3) and first four hole (h0,1,2,3) states for different aspect ratios (1.00, 1.36, 2.27, 3.63, 4.54, 5.00) and a constant diameter D = 2.2 nm. The white isosurfaces enclose 75% of the state density. 15863
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Table 1. Character of the First Four Envelope Functions for Electrons and Holes for D = 2.2 nm and Various Aspect Ratiosa
We use the notation (ω,ζ) with ω describing the number of nodes one encounters while moving across the xyplane and ζ describing the number of nodes moving along z.
a
Figure 3. Analysis of the Bloch function character of the first four hole states h0,1,2,3 for different aspect ratios, with D = 2.2 nm. The colors red, blue, and green correspond to A-, B-, and C-bands, respectively.
notation (ω,ζ), where the indices ω and ζ represent the number of nodes encountered by moving across the xy plane and along z, respectively. The indices can be S, P, D, etc., where S represents the form of the wave function without node, P with one node etc. The results are summarized in Table 1. We can see that for F g 2.27 all the electron wave functions have the form (S,ζ), as expected from a truly 1D-system. Only e2 and e3 for small aspect ratios develop an in-plane node. The situation for the hole states is more complicated, and we analyze them by projection onto bulk ZnO bands (see eq 5, ref 11, for details). The results are given in Figure 3 for the first four hole states, where the contribution of the A-, B-, and C-bands is given in percent.
We notice that hole states with a mixed band character, such as h2 for aspect ratio 1, tend to be ω = P type, as we have found previously for the case of spherical quantum dots.11 Wave function with dominant either A-, B-, or C-band contribution tends to be of (S,ζ) type. For F g 3.63, the hole wave functions h0,h1 have a large contribution from the C-band and have (S,ζ) symmetry, similar to to electron states. This is the characteristic of a state derived from a single band. To see that hole states can acquire a simple singleband character, we show in Figure 4 the results for a very large aspect ratio of F = 10. With this aspect ratio, the hole states are nearly pure C-band states (not shown) and show the simple (S,ζ) sequence. In Figure 5 we study the aspect ratio dependence of the first ten hole-state eigenvalues for D = 2.2 nm NRs and connect the states according to their symmetry. As we can see, the A- and Bbands are largely insensitive to the aspect ratio in comparison to the C-band states which “feel” the elongation and rise above the other single hole states for F g 3.63. This is simply the consequence of the Bloch function character of the C-band in wurtzite. It has orbital pz character and is sensible to confinement in the zdirection. We therefore see a crossing at around F = 1.5 of the strongly F-dependent C-band states with the nearly F-independent A- and B-band states. Reference 20 shows a smooth polarization transition for InAs nanorods. Optical Properties. While the literature on ZnO NRs in the intermediate and weak confinement is extensive,2128 the literature on ZnO NRs with radii smaller than or equal to the exciton Bohr radius (1.4 nm) and showing quantum confinement effects is recent and rather limited.2932 The optical band gap for varying aspect ratio is given in Figure 6(a) at three different levels of theory: at the single particle (SP) level (gray), at the SC level including electronhole exchange interactions (red), and at the correlated CI level (black). We can see that the band gap dependence strongly depends on the level of theory applied. At the SP level, the band gap variation seems to plateau only around our largest aspect ratio of 5. Adding Coulomb and exchange integrals at the SC level (red) leads to a nearly Findependent picture with a gap variation confined to the region 10 < F < 2.5. Adding correlations at the CI level (black), we recover an optical gap that varies up to our largest F. These trends can be easily understood: The Coulomb and exchange integrals (difference between gray and red) are larger for smaller structures and lead to a larger reduction of band gap for such structures. Correlations (difference between red and black) become more relevant for larger, elongated structures. The value we have obtained with full CI is in a good agreement with the experimental value 3.53 ( 0.05 eV deduced from absorption measurements at room temperature30,31 for NRs with D = 2.2 nm and L = 40 nm. 15864
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Figure 4. As Figure 3 but for a large aspect ratio F = 10 and 1 nm diameter.
Figure 5. Energy of the first ten hole states versus aspect ratio for ZnO nanorods with 2.2 nm diameter. The lines connect states which are of the same symmetry (ω,ζ). The (red, blue, green) lines connect states with dominant (A-, B-, C)-band character, respectively. HOMO stands for the highest occupied molecular orbital.
In Figure 6(b) we show the Stokes shift at room temperature that we define as the energetic difference between the lowest exciton state and the first bright state for different aspect ratios. The Stokes shift presents a nonmonotonic behavior in the case of full CI and a monotonic behavior in the single configuration case. A monotonic behavior has been predicted for InAs NRs,20 while a nonmonotonic behavior was measured and calculated for CdSe NRs.14 The reason for our nonmonotonic dependence will be given in the next paragraph. In Figure 6(c) we show the polarization ratio β at room temperature defined as )
I I^ I þ I^
)
β¼
ð2Þ
)
where I is the emission intensity of light polarized parallel to the c-axis and I^ is the intensity of light polarized perpendicular to it. The values of β as a function of F for both full CI and SC differ significantly. We observe an abrupt change of polarization between 1 < F < 2 for SC, while it is observed between 2.5 < F < 3.5 for CI. Optical polarization was measured6,33,34 for ZnO NRs and found to be along the wire axis, in agreement with our results. For such polarization along the c-axis, the ratio of the modal gain to the material gain increases and hence improves the
Figure 6. (a) Optical band gap in the full CI scheme (black line), in the SC including exchange (red line), and in the single-particle gap (dashed line) for various aspect ratios and 2.2 nm diameter NRs. (b) Stokes shift. (c) Linear polarization ratio from eq 2.
performance of these NRs as waveguides.35 We are not aware of results for NRs with small aspect ratio. To understand the change in the polarization and Stokes shift, we calculated the absorption spectra with and without correlations and analyzed the results in terms of dominant configurations in Figure 7. From the singleparticle picture (Figure 5) we expect a crossing of the in- and outof-plane polarized excitons, when the C-band hole states (green line) dive below the A- and B-band states (red, blue). This happens for F between 1 and 1.3. Including the Coulomb interaction, but neglecting correlations (Figure 7a), leads to a 15865
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mixed state of (0,0) and (1,1) excitons. For the (1,1) exciton, the symmetry of both electron and hole is (S,P), with a node along the NR axis. This strong configuration mixing leads to a large correlation energy and a strong red shift. This mixing is also responsible for the gap dependence in Figue 6(a). The crux is that this favorable mixing is only possible when the electron e1 state with (S,P) symmetry (see Figure 5) is close enough in energy to e0 with (S,S) symmetry. This ceases to be the case for F < 3.63 when the electron (S,P) states are repelled by confinement and move up in energy. Without the ability to lower its energy by correlation with the (1,1) state, the c-polarized (0,0) state (red in Figure 7(b)) sharply increases in energy and stops being the band edge exciton. So, correlation has significantly moved the polarization switch to a higher aspect ratio, as seen in Figure 6(c). We do not know about experiments for ZnO at this aspect ratio but note that theoretical work on CdSe14 had predicted, neglecting correlations, a crossover at an aspect ratio somewhat lower than the experimental observation.14 We could speculate that correlations would bring theory and experiment in better agreement. Theoretical work on InAs NRs20 showed a much smoother transition than in the ZnO case to a value of polarization around 85%, so lower than the 100% polarization in ZnO. Our results of a nonmonotonic Stoke shift (Figure 6(b)) can now be understood from the nature of the correlated exciton states: The Stoke shift is qualitatively different for the heavily coupled (0,0)(1,1) state (red in Figue 7) at F < 2.27 than for the in-plane polarized exciton (black in Figure 7) for F < 3.63.
Figure 7. Oscillator strengths for the transition |0> to |X> in 2.2 nm diameter ZnO nanorods for different aspect ratios. The number on top of the absorption peaks refers to the dominant single-particle levels involved in the transitions (e,h). The symbol M denotes a highly mixed state. Transitions polarized along the c-axis are shown in red, while the transitions polarized in-plane are shown in black. (a) Spectra obtained by SC with exchange interaction. (b) Spectra obtained by full CI.
similar picture. For F > 1.36, the lowest exciton state is polarized along the wire axis (c-polarized) as indicated by the red color. At F = 1.36, the c-polarized exciton (red peak labeled (0,0)) moves energetically just above the in-plane polarized transitions (black peak labeled (0,0)). This change in the band edge is pictured in Figure 6(c) as a change of polarization. The Stokes shift, defined in Figue 6(b) as the exchange splitting of the lowest transition, is smooth as a function of F since it originates for all F’s from (0,0) excitons with (S,S) symmetry, for both electron and hole. Including correlations changes the situation drastically. In Figure 7(b) we see that for F > 2.27 the lowest exciton is a
’ CONCLUSIONS In the present work, we study the electronic and optical properties of ZnO nanorods as a function of the aspect ratio within the atomistic empirical pseudopotential framework and configuration interaction for realistic structures of several thousand atoms. We analyze the states according to their envelope function symmetry and their Bloch function character using a projection technique. We find that when states are derived from a single band, A-, B-, or C-, the symmetry of the single-particle state is (S,ζ), where ζ, describing the number of nodes along the NR’s axis, can take the values S, P, D, and F for our NRs. The emergence of an in-plane node, as for states with (P,S) symmetry, requires bulk band mixing. We find electron and hole states with this (P,S) symmetry with mixed A- plus B-band character. Analyzing the symmetry of the hole states and connecting them as a function of aspect ratio (Figure 5), we find a strong dependence of C-band derived hole states upon elongation, while A- and B-band derived states are nearly elongationindependent. This qualitative difference leads to a crossing of the single-particle hole states at an aspect ratio of around 1.2. In the next step we take electronhole Coulomb and exchange interactions into account and build single-configuration electronhole pairs (uncorrelated but Coulomb coupled). The oscillator strength reflects the symmetry of the hole states, and we observe a crossing of in-plane and out-of-plane polarized transitions. These originate from the crossing of the single-particle hole states, but this crossing is shifted to an aspect ratio of around 2. In the last step of sophistication we add correlation to build the true excitons via configuration interaction. We observe a surprising large effect on the optical properties. The transition between the two different polarizations is now shifted even further, to an aspect ratio of around 3. We explain this unusual behavior through the mixing of two configurations we label as (0,0) and 15866
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The Journal of Physical Chemistry C (1,1). The configuration (1,1), where electron and hole state have P-character along the c-axis, is energetically too remote to mix with (0,0) for small aspect ratios (
