ARTICLE pubs.acs.org/JPCC
Honeycomb-Patterned Quantum Dots beyond Graphene Yan Xi, Mingwen Zhao,* Xiaopeng Wang, Shijie Li, Xiujie He, Zhenhai Wang, and Hongxia Bu School of Physics and State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, China ABSTRACT: Graphene quantum dots (QDs) hold great promises in spintronics. Here, we report our predictions of honeycomb-patterned QDs beyond graphene, on the basis of firstprinciples calculations and an extended Hubbard model. Our calculations showed that the electronic structures and spin-polarization of boron nitride (BN) and silicon carbide (SiC) QDs can be well tuned by controlling the shape and size of the QDs. Edge hydrogenation can not only greatly improve the stability but also diminish the spin-polarization of BNQDs. Triangular SiC-QDs have spin-polarized ground states, and the magnetic moments increase with the increase of QD size. Hexagonal SiC-QDs, however, possess spinunpolarized ground states whose energy gaps decrease with the increase of QD size. To understand the origins of the composition- and shape-dependent spin-polarization of these honeycomb-patterned QDs, we extended the single-orbital Hubbard model of graphene QDs by taking into account the onsite energy differences of the two sublattices. Our extended Hubbard model reproduces well the results of first-principles calculations and offers a simple model to predict the electronic structures of honeycomb-patterned QDs.
’ INTRODUCTION Quantum dots (QDs) with electrons confined in all physical dimensions open a tantalizing object for scientific researchers. Electron confinement in QDs leads to desired atom-like properties. The tunability of their electronic and optoelectronic properties controlled by size, shape, composition, and strain has wide applications in numerous fields, such as light-emitting diodes, quantum computing, solar cells, and so on. The fascinating properties of the recently discovered graphene14 have triggered intense studies of graphene quantum dots (G-QDs) constructed by cutting graphene in different patterns.514 First-principles calculations combined with a single orbital Hubbard model under a mean-field approximation (MFA) revealed that the G-QDs with triangular shape terminated by well-defined zigzag edges have a magnetic ground state with a finite net spin S = (NA NB)/2, where NA and NB are the number of atoms belonging to the two (A and B) sublattices.15,16 These features can be well interpreted using Lieb’s theorem for bipartite lattices.17 The spinpolarization of zigzag edges in G-QDs holds great promise for spintronics. However, the small gaps of larger G-QDs limit their optoelectronic applications. Searching for QDs of other materials possessing simultaneously large energy gap and magnetic ground state is thus desirable. Hexagonal boron nitride (h-BN), a structural analogue of graphite, has attracted special interest due to its superior thermal and chemical stabilities. In recent progress, two-dimensional (2D) BN sheets with a few atomic layers have been achieved via different approaches.1823 In contrast to zero-band gap graphene, the electron transfer between B and N atoms makes BN sheet wide-band gap semiconductors. Theoretical works predicted that when the BN sheet is cut into one-dimensional zigzag-edged nanoribbons, the spin-polarization of edge states is sensitive to the edge termination and passivation patterns.2427 r 2011 American Chemical Society
Analogously, boron nitride quantum dots (BN-QDs) with different shapes and sizes can be constructed from BN monolayer. Theoretical efforts have been paid to reveal the stability and electronic properties of these novel honeycomb QDs.28,29 Firstprinciples study showed that the triangular BN-QDs with bared zigzag edges exhibit magnetic ground states due to the spinpolarization of the unpaired electrons at the edges.28 However, the stability of the unpaired σ-electrons at the bared edges, especially under hydrogen-rich conditions, remains unclear. The role of edge hydrogenation on the electronic and magnetic properties of BN-QDs is crucial for their potential applications under ambient conditions. Additionally, silicon carbide (SiC) is another wide band gap semiconductor. The biocompatibility of silicon carbide quantum dots (SiC-QDs) makes them good candidates for nanostructured labels of biological molecules. First-principles calculations have predicted that when the thickness of SiC sheet is reduced to a few atomic layers, layered honeycomb lattice becomes energetically preferable.30 Multiwalled SiC nanotubes, a structural analogue of carbon nanotubes, have been fabricated via the reactions between SiO and multiwalled carbon nanotubes.31 Motivated by this progress, intense studies have been paid to one-dimensional SiC nanoribbons and nanotubes.3241 However, to the best of our knowledge, theoretical work on the SiC-QDs with honeycomb lattice has never been reported so far. In view of the similarity between the valence electron configurations of C and Si atoms and their potential applications in various fields, firstprinciples study of SiC-QDs is quite necessary. Moreover, whether Lieb’s theorem for bipartite lattices works for these Received: June 17, 2011 Revised: August 5, 2011 Published: August 08, 2011 17743
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The Journal of Physical Chemistry C honeycomb-patterned QDs is an interesting issue worthy of further study. In this contribution, we performed first-principles calculations within density functional theory (DFT) to study the stability, electronic, and magnetic properties of honeycomb BN and SiCQDs of different shapes and sizes. We found that these two types of honeycomb QDs exhibit quite different electronic and magnetic properties due to the different valence electron configurations and electronegativities of component atoms. For BN-QDs, edge hydrogenation can not only improve greatly the stability but also diminish the spin-polarization. Triangular SiC-QDs have spin-polarized ground states, and the magnetic moments increase with the increase of QD size. Hexagonal SiC-QDs, however, possess spin-unpolarized ground states whose energy gaps decrease with the increase of QD size. Lieb’s theorem, which works for graphene G-QDs, is no longer valid for BN and SiCQDs. However, the electronic and magnetic properties of both BN and SiC-QDs can be well interpreted using an extended Hubbard model where the on-site energy differences of the component atoms are taken into account. This offers a simple model to describe the electronic structures of honeycombpatterned QDs. The diverse electric and magnetic properties of these QDs may find applications in building novel nanoscaled electronic devices.
’ METHODS AND COMPUTATIONAL DETAILS Our first-principle calculations were preformed using the plane wave basis Vienna ab initio simulation package known as VASP code,42,43 implementing the spin-polarized density functional theory (DFT) and projector augmented wave approach (PAW) pseudopotentials.44,45 The electron-correlation interaction was treated within the generalized gradient approximation (GGA) with the PerdewWang (PW91) exchange-correlation functional.46 The QDs were placed in a large supercell with vacuum region of 18 Å to exclude the mirror interactions between QDs. Only gamma point was considered in Brillouin zone integrations. During the structure optimizations, all of the atoms were fully relaxed using a conjugate gradient (CG) procedure until the HellmamFeynman forces acting on them were smaller than 0.02 eV/Å. The energy convergence was set to 104 eV. In the spin-polarization calculations, different initial spin configurations were adopted to get the magnetic moment of the ground states. To evaluate the plausibility of BN-QDs and SiC-QDs, we defined their formation energies (Eform) under different environments as: Ef orm ¼ Etot nR μR nβ μβ nH μH where Etot is the total energy of QDs, and nR, nβ, nH are the number of each component (R = B, Si; β = N, C) in the QDs. The chemical potentials μR, μβ, and μH depend on specifically employed atomic reservoirs. Here, we considered two limits corresponding to N-rich (or Si-rich) and B-rich (or C-rich) conditions, respectively. In the N-rich (or Si-rich) media, μN (or μSi) was calculated from nitrogen molecular in the gas phase (or silicon crystal), while under B-rich (or C-rich) conditions, a metallic boron crystal in the Rβ phase (or graphene) was used as the reservoir. In both cases, μR and μβ are linked by the thermodynamic constraint, μR + μβ = μRβ, where μRβ is the chemical potential of a BN (or SiC) unit in BN monolayer (or cubic SiC crystal).
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Figure 1. Optimized structure models of quantum dots (a) Tri-(NH)8BN-QD, (b) Hex-(BN)5-BN-QD, (c) Tri-(N)8-BN-QD, and (d) TriN8H3-BN-QD.
’ RESULTS AND DISCUSSION The QDs considered in this work have triangular (Tri) or hexagonal (Hex) shape with zigzag edges. All of the edge atoms are passivated by hydrogen atoms. Each atom (except hydrogen) is 3-fold coordinated, analogous to the case of 2D honeycomb lattice of graphene. These QDs are classified by the number of atomic hexagons along one edge, as shown in Figure 1. For instance, the BN-QD with triangular shape terminated by N-zigzag edges, each of which contains n atomic hexagons, is denoted as Tri-(NH)n-BN-QD, and this rule is followed throughout the work. The hexagonal BN-QD containing n atomic hexagons at each edge is referred to as Hex-(BN)n-BNQD. To evaluate the rule of hydrogen passivation on the stability of BN-QDs, we also calculated the triangular BN-QDs with bared or partially hydrogenated edges as proposed in ref 28, which are named as Tri-Nn-BN-QDs and Tri-NnH3-BN-QDs, respectively, as shown in Figure 1. In the partially hydrogenated model, only the atoms at the three corners are passivated by hydrogen atoms. A. Boron Nitride Quantum Dots. We first calculated the BNQDs, which have the edge atoms being fully passivated by hydrogen atoms. Structural optimization indicates that the atomic relaxation at the edges is very slight; that is, the BN bonds at the zigzag edges shrink less than 0.4% as compared to those in the center part. This is in contrast to the cases of bare-edged BNQDs, which involve severe distortion at edges.28 The nearest distances between H and edge atoms are 1.198 (BH) and 1.015 Å (NH), respectively, implying that H atoms are chemically attached to the QD edges. For the Tri-(BH)n-BN-QDs, the atoms distort slightly along the direction perpendicular to the basal plane, and the degree decreases with the increase of QD size. Similar results are also found in Tri-Nn-BN-QDs. Tri-(NH)n-BN-QDs have higher planar stability than do the Tri-NnH3-BN-QDs with unpaired electrons; that is, only the corner N atoms deviate from the basal plane. The planar feature of hexagonal BN-QDs is well preserved regardless of QD size. 17744
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Table 1. Formation Energies of BN Quantum Dots with Different Shapes, Sizes, and Hydrogen Passivation Patterns formation energy (eV/atom) configurations
n=3 n=4
n=5
n=6
n=7
n=8
Tri-(BH)n-BN B-rich 0.058
0.049
0.042
0.038
0.034
0.031
N-rich 0.220
0.212
0.205
0.196
0.187
0.178
Tri-(NH)n-BN B-rich 0.160 0.108 0.102 0.0957 0.089 0.0845 N-rich 0.006 0.054 0.061 0.062 0.063 0.064 Tri-Nn-BN Tri-NnH3-BN Hex-(BN)n-BN
B-rich 1.332
1.218
1.094
0.969
0.915
0.844
N-rich 1.095
0.980
0.868
0.788
0.715
0.656
B-rich 1.114
1.090
0.976
0.909
0.848
0.791
N-rich 0.976
0.824
0.764
0.707
0.655
0.626
0.078
0.107
0.051
To compare the stability of the BN-QDs referred above, we calculated the formation energies using first-principles calculations and listed them in Table 1. From this table, we can see the following features: (1) The hydrogen-passivated BN-QDs, Tri-(NH)n-BN-QDs, and Tri-(BH)n-BN-QDs are energetically more favorable than those with partially hydrogenated or bared edges under ambient conditions (B-rich or N-rich). (2) Tri-(BH)n-BN-QDs are more stable than Tri-(NH)n-BN-QDs under B-rich conditions, whereas Tri-(NH)n-BN QDs become energetically preferable under N-rich conditions. (3) The formation energies of Hex-(BN)n-BN-QDs are comparable to those of the Tri-(BH)n-BN-QDs (or Tri-(NH)n-BN-QDs) under B-rich (or N-rich) conditions. These features clearly indicate that edge hydrogenation can improve greatly the energetic stability of the QDs as compared to those with bared edges, because the unpaired σ-electrons are energetically disadvantageous for the stabilization of BN-QDs. This is also in good agreement with the results of ref 29. It is noteworthy that the negative formation energies of the Tri-(NH)n-BN-QDs imply the high plausibility under N-rich conditions. We then calculated the spin-polarization and electronic structures of these BN-QDs. We found that the QDs with bared or partially hydrogenated edges possess a very large magnetic moment, for example, 21 μB and 18 μB for Tri-N8-BN-QDs and Tri-N8H3-BN-QDs, respectively. The later date is in good agreement with that reported in ref 28. However, for both Tri-(NH)n-BN-QDs and Tri-(BH)n-BN-QDs, edge hydrogen passivation diminishes the spin-polarization completely; that is, their ground states have zero magnetic moment, as shown in Figure 2. In view of the great energy advantage of hydrogenpassivated BN-QDs over those with bared or partially hydrogenated edges, fabricating stable magnetic BN-QDs is challenging under ambient conditions. The energy gap between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbitals (LUMO) of Tri-(NH)n-BN-QDs and Tri-(BH)n-BN-QDs was also studied as shown in Table 2. In contrast to G-QDs, these BN-QDs have a wide HOMOLUMO gap, which varies gradually around the band gap of BN monolayer (4.85 eV) with the increase of QD size. The HOMOLUMO gap of Tri-(BH)nBN-QDs is narrower than that of Tri-(NH)n-BN-QDs. Previous theoretical study has revealed that, similar to the case of graphene nanostructures, the electronic properties of BN monolayer and hydrogenated BN nanoribbons are mainly dominated by the 2pz orbitals of B and N atoms.36 The net spin S of triangular G-QDs
Figure 2. Left panel: (a and b) Electronic KohnSham energy levels of BN-QDs in the region near the Fermi level. Solid (or dot) lines indicate occupied (or unoccupied) states, respectively. Right panel: Total spin resolved electron density of states of the BN-QDs. Red (or blue) lines represent the energy levels of spin-up (or spin-down) branch. Dash line indicates the Fermi level.
Table 2. HOMOLUMO Gap of BN- and SiC-QDs of Different Sizes and Shapes HOMOLUMO gap (eV) configurations
n=3
n=4
n=5
n=6
n=7
n=8
Tri-(BH)n-BN
5.107
5.084
4.996
4.890
4.826
4.756
Tri-(NH)n-BN
5.102
5.282
5.311
5.144
5.082
4.957
Hex-(BN)n-BN
5.904
4.785
4.739
BN monolayer
4.85
obeys the so-called Lieb’s law, that is, S = (NA NB)/2, where NA and NB are the number of atoms belonging to the two (A and B) sublattices.15,16 However, the net spins of both Tri-(NH)n-BN-QDs and Tri-(BH)n-BN-QDs are zero regardless of the imbalance between the two sublattices. The failure of Lieb’s law in triangular BN-QDs can be attributed to two factors: (1) In BN-QDs, the 2pz orbitals of the two sublattices have different on-site energies. Electrons with opposite spins prefer to reside on N atoms with low on-site energy, which is unfavorable for spin-polarization. (2) The number of 2pz (π) electrons in Tri-(NH)n-BN-QDs or Tri-(BH)n-BN-QDs is different from that of the corresponding G-QDs. Therefore, the half-filling occupation criterion in Lieb’s law is not satisfied in BN-QDs. In the last part of this work, we will employ an extended Hubbard model where the two factors were taken into account to reproduce the electronic properties BN-QDs. 17745
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B. Silicon Carbon Quantum Dots. Analogously, we calculated the formation energies and electronic structures of SiCQDs using first-principles calculations. In view of the great energetic advantage of hydrogenated QDs over those with bared or partially hydrogenated edges, we only considered hydrogenated SiC-QDs in the present work. Structural optimization results in slight atomic relaxation at the edges. The SiC bonds at the zigzag edges shrink less than 1.8% as compared to those in the center part. The CH and SiH distances at the edges are approximately 1.092 and 1.448 Å, very close to the values of SiC nanoribbons of 1.09 and 1.49 Å.32 The planar configurations are well preserved in the SiC-QDs under study. The energetic favorability of planar configuration over the buckling ones arises
Table 3. Formation Energies of SiC-QDs with Different Shapes and Sizesa formation energy (eV/atom) n=3
configurations
n=5
n=6
n=7
n=8
Tri-(CH)n-SiC
C-rich
0.441
0.456
0.467
0.471
0.473
0.479
Tri-(SiH)n-SiC
Si-rich C-rich
0.443 0.470
0.459 0.488
0.468 0.498
0.473 0.503
0.476 0.507
0.482 0.509
Si-rich
0.468
0.486
0.496
0.500
0.504
0.507
0.407
0.437
0.456
Hex-SiC a
n=4
The edge atoms are passivated by hydrogen atoms.
from the tendency to reduce dipole moment along the direction perpendicular to the basal plane.47,48 The formation energies of SiC-QDs are listed in Table 3. It is clear that Tri-(CH)n-SiC-QDs are more stable than Tri-(SiH)nSiC-QDs of same size. The formation energy of these QDs varies slightly with the increase of QD size. We also calculated the formation energy of the SiC monolayer and found that it is about 0.486 eV/atom, which is higher than those of Tri-(CH)n-SiCQDs and hexagonal SiC-QDs. Therefore, hexagonal and Tri-(CH)n-SiC-QDs may have higher plausibility than the SiC monolayer. First-principles calculations indicate that the ground states of SiC-QDs exhibit magnetic properties similar to those of G-QDs. Triangular SiC-QDs have spin-polarized ground states, whereas the hexagonal SiC-QDs have zero magnetic moment. It is noteworthy that the Tri-(CH)n-SiC-QDs and Tri-(SiH)n-SiCQDs have different spin-resolved energy level alignments near the Fermi level, as shown in Figure 3a and b. For example, the spin-up branch of Tri-(CH)4-SiC-QD has a HOMOLUMO gap of 2.77 eV, while there is small HOMOLUMO gap of 0.62 eV in the spin-down branch. The Fermi level is close to the HOMO of both branches. The electron density of states (PDOS) projected onto Si and C atoms indicates that the energy levels near the Fermi level are contributed by the C 2pz orbitals of the edge atoms. This is also consistent with the spin density distribution as shown in Figure 3c. For Tri-(SiH)4-SiC-QD, however, the Fermi level is close to the LUMO of spin-up and
Figure 3. Left panel: (a and b) Electronic KohnSham energy levels of BN-QDs in the region near the Fermi level. Solid (or dot) lines indicate occupied (or unoccupied) states. Right panel: Partial spin polarized density of states corresponding to Tri-(CH)4-SiC and Tri-(SiH)4-SiC, respectively. Red and blue lines represent the energy level of spin-up and spin-down branches, respectively. Dash line is the Fermi level. (c and d) Spatial distribution of spin density of Tri-(CH)4-SiC-QD and Tri-(SiH)4-SiC-QD. Yellow and blue bubbles represent the isosurfaces of spin-up and spin-down density. 17746
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Table 4. HOMOLUMO Gaps of SiC-QDs of Different Sizes and Shapesa HOMOLUMO gap (eV) configurations
a
Figure 4. The schematic representation of π-electron distribution of (a) Tri-(NH)4-BN-QD and (b) Tri-(BH)4-BN-QD. We assume that the π-electrons occupy the N sites because N atom has higher electronegativity than B atom. The hopping effect of π-electrons between adjacent N and B sites is not shown, because hopping effects of spin-up and spin-down electrons compensate each other and thus have no contribution to the spin-polarization. (c) Magnetism moments of triangular SiC-QDs with Si or C edges as a function of sizes represented by n, which were obtained using the extended Hubbard model (Hubbard) and density functional theory (DFT). (d,e) The schematic representations of the spin density distribution of Tri-(CH)4-SiC-QD and Tri-(SiH)4-SiC-QD given by the extended Hubbard model. Red and blue balls represent the spin-up and spin-down, respectively. Hydrogen atoms at the edges are not displayed in (a), (b), (d), and (e).
spin-down branches. The electron levels near the Fermi level arise mainly from the Si 3pz orbitals of edge atoms, and most of the net spin density is distributed along the edges, as shown in Figure 3d. Such difference is related to the charge redistribution in the region near the edges. For Tri-(CH)n-SiC-QDs, the edge C atoms have less π electron occupation as compared to those in the center part, because each edge C atom connects to only two Si atoms. Therefore, the edges states near the HOMO contributed mainly by the C 2pz orbitals cannot by fully filled. For Tri-(SiH)n-SiC-QDs, however, the edge Si atoms have more π electron occupation as compared to those in the center part, and the Fermi level is pinned at the states contributed by Si 3pz orbitals. It is interesting to see that with the increase of QD size, the magnetic moment of triangular SiC QDs increases, but the magnitudes are smaller than those predicted by Lieb’s law, as shown in Figure 4. For large size triangular QDs (n > 6), Tri-(CH)n-SiC-QDs have larger magnetic moment than Tri-(SiH)n-SiC QDs. This is related to the different charge states of the edge atoms in the two types of QDs. Because of the computation limitation of present first-principles calculations, we only studied three hexagonal SiC QDs corresponding to Hex-(SiC)n-SiC QDs with n = 3, 4, and 5. We found that the HOMOLUMO gap of these QDs decreases with the increasing n (see Table 4), and their edge states are spinunpolarized. This is in contrast to the case of SiC nanoribbons
n=3
n=4
n=5
n=6
n=7
n=8
Tri-(CH)n-SiC
0.339
0.351
0.298
0.301
0.293
0.267
Tri-(SiH)n-SiC
0.415
0.387
0.231
0.204
0.175
0.090
Hex-(SiC)n-SiC SiC monolayer
2.606 2.58
2.051
1.454
All of the edge Si or C atoms are passivated by hydrogen atoms.
where the edge states of the zigzag edges are spin-polarized resulting in half-metallicity of the nanoribbons.32,36 The appearance of finite HOMOLUMO gap can be attributed to the quantum confinement effects of electron motion in the hexagonal SiC-QDs. The large HOMOLUMO gap (>2.0 eV) of small size hexagonal SiC-QDs is unfavorable for spin-polarization of edge states. It is expected that for large size hexagonal SiC-QDs, the HOMOLUMO gap will come to close, and the spinpolarization of the zigzag edges will appear. Overall, the tunable electronic structures and spin-polarization offer a promising way to tune the electronic and magnetic properties of honeycombpatterned SiC-QDs by controlling their shapes and sizes. C. Extended Hubbard Model for Honeycomb-Patterned QDs. To address the electronic and magnetic properties of BN and SiC QDs, we employed an extended Hubbard model where the on-site energy difference between B and N (or Si and C) atoms was taken into account. The contributions from σ-orbitals in the basal plane are excluded because the electronic structures near the Fermi level of these materials are mainly dominated by the pz orbitals.36 The extended Hubbard Hamiltonian under a mean-field-approximation (MFA) reads: H ¼
hniσ icþ ∑iσ εi cþiσ ciσ h∑ijiσ tijðcþiσ cjσ þ ciσ cþjσ Þ þ U ∑ iσ ciσ i, σ
Here, the εi denotes the on-site energy of an electron at site i, and tij is the hopping integral connecting sites i and j. c+iσ and ciσ are the creation and annihilation operators of electron with spin σ (σ = v,V) in site i. The term U represents the Coulomb repulsion between the electrons at a same site i. hniσ iis the expectation value of particle number operator in σ spin ((niσ = (niσ+niσ)), ̅ ̅ ̅ ̅ which can be given using the expression: hniσ i ¼
∑k jRikσ j2 f ðEkσ Þ
Here, Ekσ and Rikσ are the eigenvalues and eigenvectors of the ̅ ̅ Hamiltonian, and f(E kσ) is the FermiDirac distribution func̅ The major differences between the tion of electrons in Ekσ level. ̅ present Hamiltonian and that adopted in ref 15 lie in (1) the different on-site energies of electron on the two sublattices are taken into account (the first term); and (2) FermiDirac distribution function, which includes temperature effects, is adopted to calculate Æniσæ. The eigenvalues and eigenvectors of ̅ the Hamiltonian can be obtained in a self-consistent way. Under nearest-neighbor approximation, the parameters in this Hamiltonian given by fitting DFT results are εB εN = 4.57 eV, εSi εC = 2.85 eV, tBN = 1.95 eV, tSiC = 1.42 eV, and U = 2.2 eV, respectively.36 We first calculated the electronic structures of triangular BNQDs with fully hydrogenated edges using the extended Hubbard 17747
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The Journal of Physical Chemistry C Hamiltonian. It is noteworthy that the Tri-(NH)n-BN-QDs and Tri-(BH)n-BN-QDs with same size have different π (pz) electrons. For example, Tri-(NH)4-BN-QD has 36 π-electrons, whereas Tri-(BH)4-BN has 30 π-electrons, as shown in Figure 4a and b, both of which differ from the corresponding value (33 π-electrons) of G-QDs with the same size, which satisfies the half-filling rule of triangular G-QDs. Such differences will definitely affect the third term of the Hamiltonian. Our calculations indicate the ground states of Tri-(NH)n-BN and Tri-(BH)n-BN QDs are spin-unpolarized, in good agreement with the results of first-principles calculations. Moreover, the extended Hubbard model shows that the spin-compensation feature of triangular BN-QDs is mainly dominated by the number of π-electrons rather than on-site energy differences between the two sublattices. The number of π-electrons in either Tri-(NH)n-BN-QDs or Tri-(BH)n-BN-QDs does not satisfy the half-filling rule of triangular G-QDs.1517 The more (or less) π-electrons of Tri-(NH)n-BNQDs (or Tri-(BH)n-BN-QDs) as compared to the corresponding G-QD diminish the spin-polarization, as shown in Figure 4a and b. This may be the common features of IIIV group compound QDs of the same morphologies, such as triangular GaN-QDs and AlN-QDs. Silicon and carbon atoms have similar valence electron configurations. The spin-polarization of triangular SiC-QDs is therefore expected because the half-filling rule is satisfied. Indeed, our calculations on the basis of the extended Hubbard mode clearly indicate that the ground states of triangular SiC-QDs are spinpolarized. Moreover, the extended Hubbard model reproduced well the variation trend of magnetic moment as a function of QD size, as shown in Figure 4c. The magnetic moments, as well as the spin distribution given by the extended Hubbard Hamiltonian, agree well with those of first-principles calculations, as shown in Figure 4d and e. The deviation from Leib’s law is due to the πelectron occupation distribution over the energy levels with closed energies near the Fermi level at finite temperature, as shown in Figure 3. This also confirms that the π-electronmediated spin-polarization in triangular honeycomb-patterned QDs is dominated mainly by the number of π-electrons. The onsite energy difference between the two sublattices affects the energy level alignment near the Fermi level and thus the magnitude of magnetic moment at finite temperature. Overall, once the half-filling rule of π-electrons is satisfied in triangular honeycomb-patterned QDs, spin-polarized ground states are expected, and the magnitudes of magnetic moments depend on the QD size and the on-site energy difference between the two sublattices.
’ CONCLUSIONS In summary, our calculations indicate that edge hydrogenation can not only improve the energetic stability but also diminish the spin-polarization of triangular BN-QDs. Under N-rich conditions, triangular BN-QDs terminated by hydrogen-passivated N atoms are energetically more favorable than those terminated by hydrogen-passivated B atoms, whereas the energetic order reserves under B-rich conditions. With the increase of QD size, the HOMO LUMO gap of triangular BN-QDs varies slightly around the band gap of BN monolayer. For SiC-QDs, however, the triangular QDs have spin-polarized ground states, and the magnetic moment increases with the increase of QD size. For hexagonalshaped BN-QDs and SiC-QDs, the ground states are spinunpolarized, and the HOMOLUMO gap decreases with the
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increase of QD size. The size- and shape-dependent electronic structures and spin-polarization of BN-QDs and SiC-QDs can be explained in term of an extended Hubbard model where the onsite energy differences and temperature effects are taken into account. The spin-polarized features of triangular honeycombpatterned QDs are mainly dominated by the number of π-electrons. The on-site energy difference between the two sublattices affects the energy level alignment near the Fermi level and thus the magnitude of magnetic moment at finite temperature.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT This work is supported by the National Natural Science Foundation of China (No. 10974119), the Natural Science Fund for Distinguished Young Scholars of Shandong Province (No. JQ201001), and the Independent Innovation Foundation of Shandong University (IIFSDU, No. 2009JQ003). ’ REFERENCES (1) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. Science 2004, 306, 666. (2) Gein, A. K.; Novoselov, K. S. Nat. Mater. 2007, 6, 183. (3) Lee, C.; Wei, X. D.; Kysar, J. W.; Home, J. Science 2008, 321, 385. (4) Balandin, A. A.; Ghosh, S.; Bao, W. Z.; Calizzo, I.; Teweldebrhan, D.; Miao, F.; Lau, C. N. Nano Lett. 2008, 8, 902. (5) Bunch, J. S.; Yaish, Y.; Brink, M.; Bolotin, K.; McEuen, P. L. Nano Lett. 2005, 5, 287. (6) Vazquez de Parga, A. L.; Calleja, F.; Borca, B.; Passeggi, M. C. G., Jr.; Hinarejos, J. J.; Guinea, F.; Miranda, R. Phys. Rev. Lett. 2008, 100, 056807. (7) Hod, O.; Barone, V.; Scuseria, G. E. Phys. Rev. B 2008, 77, 035411. (8) Ezawa, M. Phys. Rev. B 2007, 76, 245415. (9) Ezawa, M. Phys. Rev. B 2008, 77, 155411. (10) G€ucl€u, A. D.; Potasz, P.; Voznyy, O.; Korkusinski, M. Hawrylak, P. Phys. Rev. Lett. 2009, 103, 246805. (11) Potasz, P.; G€ucl€u, A. D.; Hawrylak, P. Phys. Rev. B 2010, 81, 033403. (12) Kuc, A.; Heine, T.; Seifert, G. Phys. Rev. B 2010, 81, 085430. (13) Silva, A. M.; Pires, M. S.; Freire, V. N. J. Phys. Chem. C 2010, 14, 17472. (14) He, H. Y.; Zhang, Y.; Pan, B. C. J. Appl. Phys. 2010, 107, 114322. (15) Fernandez-Rossier, J.; Palacios, J. J. Phys. Rev. Lett. 2007, 99, 177204. (16) Wang, W. L.; Meng, S.; Kaxiras, E. Nano Lett. 2008, 8, 241. (17) Lieb, E. H. Phys. Rev. Lett. 1989, 62, 1201. (18) Corso, M.; Auw€arter, W.; Muntwiler, M.; Tamai, A.; Greber, T.; Osterwalder, J. Science 2004, 303, 217. (19) Laskowski, R.; Blaha, P.; Gallauner, T.; Schwarz, K. Phys. Rev. Lett. 2007, 98, 106802. (20) Auw€arter, W.; Suter, H. U.; Sachdev, H.; Greber, T. Chem. Mater. 2004, 16, 343. (21) M€uller, F.; St€owe, K.; Sachdev, H. Chem. Mater. 2005, 17, 3464. € Zettl, A. Appl. Phys. Lett. (22) Pacile, D.; Meyer, J. C.; Girit, C. O.; 2008, 92, 133107. (23) M€uller, F.; H€ufer, S.; Sachdev, H. Surf. Sci. 2009, 603, 425. (24) Nakamura, J.; Nitta, T.; Natori, A. Phys. Rev. B 2005, 72, 205429. (25) Zheng, F. W.; Zhou, G.; Liu, Z. R.; Wu, J.; Duan, W. H.; Gu, B. L.; Zhang, S. B. Phys. Rev. B 2008, 78, 205415. 17748
dx.doi.org/10.1021/jp2057157 |J. Phys. Chem. C 2011, 115, 17743–17749
The Journal of Physical Chemistry C
ARTICLE
(26) Barone, V.; Peralta, J. E. Nano Lett. 2008, 8, 2210. (27) Zhang, Z. H.; Guo, W. L. Phys. Rev. B 2008, 77, 075403. (28) Du, A. J.; Chen, Y.; Zhu, Z. H.; Amal, R.; Lu, G. Q.; Smith, S. C. J. Am. Chem. Soc. 2009, 131, 17354. (29) Liu, Y.; Bhowmick, S.; Yakobson, B. I. Nano Lett. 2011, 11, 3113. (30) Freeman, C. L.; Claeyssens, F.; Allan, N. L.; Harding, J. H. Phys. Rev. Lett. 2006, 96, 066102. (31) Sun, X. H.; Li, C. P.; Wong, W. K.; Wong, N. B.; Lee, C. S.; Lee, S. T.; Teo, B. K. J. Am. Chem. Soc. 2002, 124, 14464. (32) Sun, L.; Li, Y. F.; Li, Z. Y.; Li, Q. X.; Zhou, Z.; Chen, Z. F.; Yang, J. L.; Hou, J. G. J. Chem. Phys. 2008, 129, 174114. (33) Lou, P.; Lee, J. Y. J. Phys. Chem. C 2009, 113, 12637. (34) Lou, P.; Lee, J. Y. J. Phys. Chem. C 2010, 114, 10947. (35) Lou, P.; Lee, J. Y. J. Phys. Chem. C 2009, 113, 21213. (36) Zhao, K.; Zhao, M. W.; Wang, Z. H.; Fan, Y. C. Physica E 2010, 43, 440. (37) Zhao, M. W.; Xia, Y. Y.; Li, F.; Zhang, R. Q.; Lee, S. T. Phys. Rev. B 2005, 71, 085312. (38) Zhao, M. W.; Xia, Y. Y.; Zhang, R. Q.; Lee, S. T. J. Chem. Phys. 2005, 122, 214707. (39) Li, F.; Xia, Y. Y.; Zhao, M. W.; Liu, X. D.; Huang, B. D.; Yang, Z. H.; Ji, Y. J.; Song, C. J. Appl. Phys. 2005, 97, 104311. (40) Wang, H.; Li, X. D.; Kim, T. S.; Kim, D. P. Appl. Phys. Lett. 2005, 86, 173104. (41) Nicolas, K.; Cuong, P. H.; Gabrielle, E.; Valerie, K.; Marc, J. L. Carbon 2003, 41, 2131. (42) Kresse, G.; Furthm€uller, J. Phys. Rev. B 1996, 54, 11169. (43) Kresse, G.; Furthm€uller, J. Comput. Mater. Sci. 1996, 6, 15. (44) Bl€ochl, P. E. Phys. Rev. B 1994, 50, 17953. (45) Kresse, G.; Joubert, D. Phys. Rev. B 1999, 59, 1758. (46) Perdew, J. P.; Wang, Y. Phys. Rev. B 1992, 45, 13244. (47) Zhang, X. J.; Zhang, H. Y.; He, T.; Zhao, M. W. J. Appl. Phys. 2010, 108, 064317. (48) Bekaroglu, E.; Topsakal, M.; Cahangirov, S.; Ciraci, S. Phys. Rev. B 2010, 81, 075433.
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dx.doi.org/10.1021/jp2057157 |J. Phys. Chem. C 2011, 115, 17743–17749
