NANO LETTERS
Electronic Structure of InP Quantum Rods: Differences between Wurtzite, Zinc Blende, and Different Orientations
2004 Vol. 4, No. 1 29-33
Jingbo Li and Lin-Wang Wang* Computational Research DiVision, Lawrence Berkeley National Laboratory, Berkeley, California 94720 Received September 26, 2003; Revised Manuscript Received October 31, 2003
ABSTRACT The electronic structures of zinc blende InP quantum rods are calculated using an atomistic pseudopotential method. The quantum rods in ref [001] and [111] growth directions are studied. We found dramatic differences in the valence bands between the wurtzite CdSe, zinc blende InP [111], and InP [001] quantum rods. We also found an unexpected Γ-X coupling in the conduction bands of the [111] direction quantum wires. Our calculated results await experimental confirmations.
Nanostructures with reduced dimensionality are both physically interesting and technologically important.1 Recently two fabrication methods have achieved shape control of semiconductor nanostructures. One of these methods synthesizes high-quality colloidal quantum rods (QRs) by controlling the growth kinetics in wet chemistry,2-5 while the other uses a vapor-liquid-solid mechanism to synthesize monodispersed nanowires.6 The electronic properties of QRs and quantum wires (QWs) are under intensive experimental study recently5-9 due to the advance of synthesis methods and the realization that the electronic properties can be tailored by the nanostructure shapes. However, the theoretical study for these shape effects is only beginning.10,11 Recently Hu et al.10 have calculated the electronic structure of CdSe QRs and found an interesting pz, pxy state crossing at the top of the valence band as a function of the rod aspect ratio (here pxyz denotes the polarization of the Bloch part of the wave function). In this work, we will use the same pseudopotential method to study the electronic structures of vapor-liquid-solid synthesized InP nanowires. Recent rapid advances in the experimental study of InP QRs6-9 present an urgency in the corresponding theoretical work. We will use the semiempirical pseudopotential method (SEPM),12,13 which was developed by fitting the ab initio local density approximation (LDA) results12,14 and the experimental band structures. Its wave function has a 99% overlap with the LDA wave functions. It has been successfully used by Fu et al.14 to study spherical InP quantum dots * Corresponding author. E-mail:
[email protected] 10.1021/nl034833+ CCC: $27.50 Published on Web 11/20/2003
© 2004 American Chemical Society
(QDs). The single-particle eigen states of the nanostructure are given by the following Schrodinger’s equation,
{
1 - ∇2 + 2
υR(|r - Rn,R|) ∑ n,R
}
ψi(x) ) iψi(x)
(1)
where υR(r) is the screened semiempirical pseudopotential of atom of type R at the position Rn,R. The pseudopotential υR(r) contains a local part and a nonlocal part that includes spin-orbit interaction. We have taken the SEPM υR(r) from ref 14, its band structure and effective masses agree well with experimental values. We study two types of QRs with their long axes along the [001] and [111] directions, respectively, (which we will call c-axis and z-direction in the following). Both types of QRs can be synthesized using the vapor-liquid-solid method.6 We use the experimental bulk lattice constants of a ) 5.826 Å. Following the practice in refs 12-14, we use effective “ligand potentials” near the surface atoms to passivate the dangling bonds of the surface atoms. Compared to ref 14, we have slightly modified the ligand potential for a better surface passivation.15 Equation 1 for thousand-atom QRs can be solved using the folded spectrum method.13 The purpose of this study is to see the evolution of the electronic structure when the system changes from spherical QDs to QRs, and then to QWs. The shape of the QRs is quantified by its aspect ratio. When the aspect ratio equals 1, it is a spherical QD with a In atom at its center. When the aspect ratio is infinity, it is a QW. Another purpose of this study is to compare the two different orientations of the QRs and to compare the zinc blende (ZB) InP results with the
Figure 1. Exciton energies of InP QDs as functions of diameter. Triangles and circles are experimental results.
previously calculated wurtzite (WZ) CdSe results.5,10,11 We present the following findings. (1) Spherical QDs. The exciton energy in spherical QDs equals the single-particle energy gap CBM-VBM minus the Coulomb binding energy between the electron and the hole. Here, CBM stands for conduction band minimum and VBM stands for valence band maximum. We have screened the electron-hole Coulomb interaction by a distant dependent dielectric function -1(reh). We have used a generalized Penn’s model in ref 14 to obtain -1(reh) for a given QD. The calculated final exciton energies are shown in Figure 1 in comparison with experimental results from refs 16 and 17. Our calculated results agree very well with the experiment. This establishes the validity and quantitative accuracy of the current SEPM approach for the InP nanostructures. (2) The Lack of State Crossing in InP QRs. Figure 2a plots the state levels versus the aspect ratio for 1.9 nm diameter InP QRs in the [111] direction. Figure 2b shows the band structure of same diameter QWs. The 1.9 nm is close to the minimum diameter of InP nanowire, which can be grown using the vapor-liquid-solid method.6 The [111] InP QRs can be compared with the WZ CdSe QRs for their orientational similarity. However, unlike in the WZ CdSe QRs, where there is a pz and pxy state crossing at aspect ratio of 1.25,10 for the ZB InP QRs, the pz and pxy state splitting happens at aspect ratio of 1 (spherical QDs). This means that there will be no state crossing for any elongated ZB InP QRs. For ZB spherical QDs, the pz and pxy states form a doubly degenerated Γ8S3/2 VBM state,13 here S is the envelope function and 3/2 is the total angular momentum. This doublet state was found in previous QDs calculations for ZB crystal structure QDs.13 But in CdSe WZ structure, there are no doubly degenerated states, and the Γ8S3/2 state is split into one Γ9pxy state and one Γ7pz state. Much like at the Γ k-point of the bulk WZ band structure, here, due to positive crystal field splitting, the Γ9pxy state is above the Γ7pz state. However, for the nanowire, the xy-direction quantum confinement effective mass has light-hole component for pxy and has only heavy-hole component for pz. As a result, for QWs, the pz state is always above the pxy state. 30
Figure 2. (a) Energy levels of 1.9 nm diameter InP [111] direction QRs versus aspect ratio of the rod. The symbols are calculated results. (b) The band structure of 1.9 nm diameter InP [111] direction QWs. The conduction π band consisted of two almost degenerated states (a few meV apart). (c) The polarization ratio of the QR top of valence band state, where Qxyz are optical transition matrix elements of CB1 f VB1 for xy-polarization (Qxy) and z-polarization (Qz).
Therefore, when the aspect ratio changes from 1 to infinity, there must be a state crossing in between. Thus, the origin of the state crossing in the WZ CdSe QRs is due to the crystal field splitting in WZ crystal structure. Notice that, for negative spin-orbital splitting WZ materials (e.g., AlN), the two states crossing will occur at an aspect ratio smaller than 1. (3) Polarization of Photoluminescence (PL) Spectra. When the aspect ratio is small, the pz state in the above discussion is not purely z polarization, but has also some xy polarization components. This is demonstrated in Figure 2c, where the PL polarization of this pz VBM state is shown. For long QR or QW, the polarization is almost purely in the z-direction. The same is true for the [001] InP QR. In the WZ CdSe QRs, due to the state crossing, the PL polarization changes direction when the aspect ratio passes 1.25. In the InP QRs, the exciton ground state PL polarizations are always along the c-axis for all elongated QRs. This is a qualitative difference with significance in device design. Notice that, in ref 8, Wang et al. attributed their highly polarized PL spectra to the orientational dependence of the classical dielectric responses for their 10 nm diameter InP QWs. Here, we like to point out that, at least for small diameter QWs, the quantum confinement effects also produce strong polarization along the c-axis. (4) Anisotropy between the [111] and [001] Directions. The electronic structures for the [001] QR and WR are plotted in Figure 3. While the conduction bands of the [111] and [001] InP QRs are similar, their valence bands are completely different. The valence band energy levels in the Nano Lett., Vol. 4, No. 1, 2004
Figure 4. Wave function charge density of InP QRs with [001] growth direction. The diameter of rod is 1.9 nm. Red dots represent In-atom, and blue dots represent P-atom. (a) CB 1, (b) CB2, (c) CB3, (d) CB4, (e) VB1, (f) VB2, (g) VB3, and (h) VB4 state.
Figure 3. Same as Figure 2, but the InP QRs and QWs are in the [001] growth direction.
[001] QRs are much denser than in the [111] QRs. The top of valence band pz, pxy state splitting from aspect ratio 1 is very gradual in the [001] QRs; they are within ∼25 meV until the aspect ratio of 1.6 (see vertical arrow in Figure 3a). This situation in the QR is related to the band structure in QW, where the top of valence band σz and σxy states are close together for khz > 0.15 a.u. (see vertical arrow in Figure 3b). (Note that σ denotes an s-like envelope function in the xy cross section, while π is a p-like envelope function. The subscripts z and xy indicate the polarization direction in the Bloch part of the wave function.) The closeness of the Γ hpoint σz and σxy states in [001] QW, and the consequential high density of state in [001] QRs can be explained as follows. The lateral confinement of the σz state is due to the heavy-hole effective mass in the lateral plane, which is 0.44 me (me is the free electron effective-mass) for the [001] QW, and 0.94 me for the [111] QW. As a result, the confinement for the [111] σz is much weaker, hence its energy much higher (see Figures 2b and 3b). For the σxy state, its lateral confinement has both the heavy hole and light hole components. The light hole masses for different orientations are all around 0.14 me. Thus, judging by the effective mass alone, σxy of the [111] QW should have a smaller confinement, hence higher energy than the [001] counterpart, in contrary to the results shown in Figure 2b and Figure 3b. The answer lies in the atomic features of the boundary conditions and the wave functions. As the pxy Bloch state has nodes in the (100) and (010) directions, it is much easier for it to be truncated by the boundary of the C4V symmetry crossing section of the [001] QW, compared to the C6V symmetry crossing section of the [111] QW. This is illustrated in the insets of Figure 2b and Figure 3b, where the wave function squares are shown on the cross sections. As a result, the [111] QW σxy state has effectively a smaller cross section, hence larger confinement and lower eigen energy. Thus the shifts of both the σz and σxy states have contributed to their closeness in the [001] QW. Note that, the shift of the σz state Nano Lett., Vol. 4, No. 1, 2004
Figure 5. Same as Figure 4, but the InP QRs are in the [111] growth direction.
cannot be explained by a spherical k.p model, while the shift of σxy state cannot be explained even by a nonspherical k.p model (atomic details are needed here). The σ′xy states shown in Figure 2b and Figure 3b come from the bulk spinorbit split-off band, they have a very small pz component, and behave much like the first σxy states. The π states are just the next envelope states following the σ states. (5) QR Wave Functions. The wave function square |Ψi(r)|2 of InP QRs in the [001] and [111] directions are shown in Figure 4 and Figure 5, respectively. The diameter of the rod is 1.9 nm in both figures. In Figure 4 the aspect ratio of the rod is 5.9 and the number of atom is 1251, while in Figure 5 the aspect ratio is 6.3 and the number of atom is 1247. The conduction band (CB) and valence band (VB) states are denoted as CBn and VBn respectively, where n-1 is the number of the nodes in the z direction. Except for VB4 in the [001] direction, all the other states have a σ-type envelope function in the xy cross section. As a result, the states can be described by simple sine wave functions in the z-direction as in a simple effective mass model. For VB4, it has a xy polarization in the Bloch part of the wave function, and is related to the σxy band in the [001] QW as shown in Figure 3b. (6) Intervalley Coupling. The single-particle wave function Ψi can be projected into bulk Bloch functions unk(r)eikr of band index n and the supercell-allowed reciprocal vector k within the first Brillouin-zone. After summing up the band index n, one gets a projection function. This projection function Pi(k) is very useful in understanding the origin of the nanostructure wave functions in terms of the bulk states. Figure 6 shows Pi(k) for CB1 states of the two InP QRs 31
Figure 7. The change of state energies measured from the top of valence band as funcitons of external pressure for InP QWs with [111] growth direction. There are two very close states for Γ h 9(X).
Figure 6. Spectral analysis of CB1 state of InP QRs growth along (a) [001], and (b) [111] directions. The real space wave functions of CB1 states are shown in Figure 4a and Figure 5a, respectively. Pk is plotted as a function of kz. The sum of all points equals 1.
shown in Figures 4 and 5. As we can see, there are strong intervalley coupling in the conduction band. The single Γ point contribution is only about 28.6% in [001] QRs and 24.5% in [111] QRs. For the [001] QRs, the strongly coupled state is at the X point as expected. However, surprisingly, the strongly coupled state in the [111] QRs is not at L point but rather at X point. This is true for all the long [111] QRs. This is surprising because not only is the L point in the [111] direction, but also the bulk L point in InP has lower energy than the X point. This strong Gamma-X coupling in the long [111] QR can be understood by the situation in QW. In a [111] QW, the periodicity of the primary cell in the z-direction has a length of x3a. As a result, the L point h point of the (with |k| ) x3π/a) does not fold in to the Γ primary cell, thus does not couple to the Γ h point CB1 of the QWs. On the other hand, the X points with projections of kz ) 2π/x3a in the [111] direction will be folded in to the Γ h point of the primary cell, thus to be coupled to the CB1 state. (7) Pressure Dependence and Anticrossing. We have calculated the pressure dependence of energy levels of InP [111] QWs with 1.9 nm diameter. The results are shown in Figure 7. At p ) 0 GPa, the four lowest conduction states at khz ) 0 are one Γ h 7(Γ) state, two very close Γ h 9(X) states, and one Γ h 7(X) state; here the symbol inside the bracket indicates the folded-in k points. As the pressure increases, the two Γ h7 states will anticross at about 7.8 GPa. The anticrossing gap E(Γ,X) is about 53 meV. For 4.5 nm diameter QWs, this anticrossing gap drops to 26 meV. 32
In summary, we have used a plane-wave semiempirical pseudopotential method to investigate the electronic structures of surface-passivated InP QRs and QWs. We found the following surprising results, which can be tested in future experiments. (i) Unlike in the WZ CdSe QRs, for the ZB [111] InP QRs, there is no state crossing at the top of the valence band and, consequently, there is no change in the PL polarization as a function of the aspect ratio. This can be tested directly by PL experiments similar to that in ref 5 as demonstrated recently for small III-V ZB QRs and QWs.18,19 (ii) The [111] and [001] InP QRs are dramatically different in their valence bands; the [001] QW has much larger density of states. This can be checked from the experimental absorption spectra after both [111] and [001] QRs and QWs can be synthesized. Especially, polarized absorption spectra could be used to resolve the onset of the σxy states. (iii) There is an unexpected Γ-X coupling in the [111] InP QWs, instead of the naively expected Γ-L coupling. This can be tested experimentally by high-pressure experiments which monitor the PL intensity and polarization changes. The X point derived state and L point derived state have very different pressure coefficients, which can be used experimentally to reveal the nature of the conduction band states after the state crossing as shown in Figure 7. Acknowledgment. The authors thank Professor A. Paul Alivisatos for introducing this problem. This work is supported by the U.S. Department of Energy, under Contract No. DE-AC03-76SF00098. This research used the resources of the National Energy Research Scientific Computing Center. Supporting Information Available: Energy level data for [001] and [111] InP QRs. This material is available free of charge via the Internet at http://pubs.acs.org. References (1) Alivisatos, A. P. Science 1996, 271, 933. (2) Peng, X.; Manna, L.; Yang, W. D.; Wickham, J.; Scher, E.; Kadavanich, A.; Alivisatos, A. P. Nature 2000, 404, 59. Nano Lett., Vol. 4, No. 1, 2004
(3) Bruchez, M.; Moronne, M.; Gin, P.; Weiss, S.; Alivisatos, A. P. Science 1998, 281, 2013. (4) Huynh, W. U.; Dittmer, J. J.; Alivisatos, A. P. Science 2002, 295, 2425. (5) Hu, J.; Li, L. S.; Yang, W.; Manna, L.; Wang, L. W.; Alivisatos, A. P. Science 2001, 292, 2060. (6) Duan, X.; Lieber, C. M. AdV. Mater. 2000, 12, 298. (7) Gudiksen, M. S.; Wang, J.; Lieber, C. M. J. Phys. Chem. B 2002, 106, 4036. Gudiksen, M. S.; Wang, J.; Lieber, C. M. J. Phys. Chem. B 2001, 105, 4062. (8) Wang, J.; Gudiksen, M. S.; Duan, X.; Cui, Y.; Lieber, C. M. Science 2001, 293, 1455. (9) Duan, X.; Huang, Y.; Cui, Y.; Wang, J.; Lieber, C. M. Nature 2001, 409, 66. (10) Hu, J.; Wang, L. W.; Li, L. S.; Yang, W.; Alivisatos, A. P. J. Phys. Chem. B 2002, 106, 2447. (11) Li, J.; Wang, L. W. Nano Lett. 2003, 3, 101. Li, J.; Wang, L. W. Nano Lett. 2003, 3, 1357. (12) Wang, L. W.; Zunger, A. Phys. ReV. B 1996, 53, 9579.
Nano Lett., Vol. 4, No. 1, 2004
(13) Wang, L. W.; Zunger, A. J. Phys. Chem. B 1998, 102, 6449. (14) Fu, H.; Zunger, A. Phys. ReV. Lett. 1998, 80, 5397. Fu, H.; Zunger, A. Phys. ReV. B 1997, 56, 1496. Fu, H.; Zunger, A. Phys. ReV. B 1997, 55, 1642, and references therein. (15) The passivation Gaussian ligand potential parameters are taken from ref 14. The atom-ligand distance is 1/4 of the original bond length, except for In atoms with two dangling bonds, where the distance is 1/2 of the original bond length. (16) Micic, O. I.; Curtis, C. J.; Jones, K. M.; Sprague, J. R.; Nozik, A. J. J. Phys. Chem. 1994, 98, 4966. (17) Guzelian, A. A.; Katari, J. E. B.; Kadavanich, A. V.; Banin, U.; Hamad, K.; Juban, E.; Alivisatos, A. P.; Wolters, R. H.; Arnold, C. C.; Health, J. R. J. Phys. Chem. 1996, 100, 7212. (18) Yu, H.; Li, J.; Loomis, R. A.; Wang, L. W.; Buhro, W. E. Nature Mater. 2003, 2, 517. (19) Kan, S.; Mokari, T.; Rothenberg, W.; Banin, U. Nature Mater. 2003, 2, 155.
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