Abstract

The size and edge-dependence of two-photon absorption (TPA) for rectangular graphene quantum dots (GQDs) is investigated theoretically in the framework of Dirac equation under hard wall boundary conditions. The TPA cross section associated with interband transitions around K point is derived and the transition selection rules are obtained. Results reveal that when the size of zigzag-edge M = 3M0 ± 1 (M0 is an integer), the GQD exhibits a semiconductor while for M = 3M0 it is metallic. For semiconducting rectangular GQDs, TPA is tuned by the sizes of both edges in GQDs and the armchair-edge dimension contributes more to TPA. While for metallic rectangular GQDs, zigzag-edge dimension affects TPA little and the position of absorption peak and the magnitude of the TPA coefficient are determined by the size of armchair-edge and the resonant enhancement occurs.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Recently, two-dimensional (2D) graphene has received tremendous attention due to its potential applications in light emitting, biomolecular sensing, cellular bioimaging and so on [1–4]. These potential applications arise from its particular massless linear band structure near K and K' points [5]. However, lack of bandgap limits its opto-electronic applications to some extent [6]. Recent improvements of different fabrication techniques, such as patterning epitaxially grown graphene [7], unzipping carbon nanotubes [8], lithographic techniques [9], stepwise organic synthesis [10], make the zero-dimensional graphene quantum dots (GQDs) available. Furthermore, these techniques allow precise control on size, edge and shape of resultant GQDs [10]. Owing to the quantum confinement effects, the bandgap emerges and electronic states appear discrete in GQDs. Also, above mentioned controllable geometrical factors let the bandgap and electronic states tuable. So GQDs-based nanostructures provide unprecedented opportunities for bioimaging, optical sensing, and biomedical applications because of their numbers of unique merits, including excellent photostability, small size, biocompatibility, highly tunable photoluminescence (PL) and multi-photon excitation property, and chemical inertness [10].

Numerical researches has been conducted on the optical properties of GQDs, in the context of both theory and experiment. Sung Kim et al measured the PL spectra of GQDs and presented the size-dependence of PL for circular-to-polygonal-shaped GQDs [11]. Liu et al reported two-photon induced fluorescence from N-doped GQDs [12]. The two-photon absorption (TPA) cross-section reaches 48000GM, which is 2 orders of magnitude larger than that of conventional organic dyes and is comparable to semi-QDs. As far as the theoretical studies are concerned, Ghaffarian et al investigated the dc Stark effect and the degenerate two-photon absorption of the small-size trigonal zigzag graphene nanoflakes, using the Hartree-Fock Su-Sheriffer-Heeger model and the sum-over-state method [13]. The visible light absorption of different sizes and shapes of GQDs and quantitative analysis on the contribution of each kind of edge atoms to the energy grade and the density of state has been investigated by Zhang, using the ab initio density functional theory method [14]. Jin et al discussed the uniaxial strain modulated electronic structure and optical absorption of a triangular zigzag GQD within the tight-binding approach [15].

In our previous work, we have explored the size-dependence of TPA in circular GQDs with the edge of armchair and zigzag respectively on the basis of electronic energy states obtained by solving the Dirac-Weyl equation under infinite-mass boundary condition [16,17]. The evident TPA absorption peaks appeared in infrared region is up to 1011 GM and the slight discrepancy of TPA spectra between the GQDs with two edges results from the surface state in GQDs with zigzag edge. Compared with the single-edged (either armchair or zigzag) circular GQDs, rectangular ones are an ideal object to investigate which edge plays a more important role to TPA, due to the fact that they are of two kinds of edges simultaneously. It is estimated that the edge effect will be more complicated and interesting. Up to now, however, we haven't seen the related reports on this topic.

In this work we study the size and edge-dependent TPA of rectangular GQDs. We calculate the TPA cross section for rectangular GQDs on the basis of electronic energy states obtained by solving the massless Dirac equation under hard wall boundary conditions [18]. The two-photon interband transitions from valence band to conduction band are involved in and the transition selection rules are obtained. We find that when the size of zigzag-edge M = 3M0 ± 1 (M0 is an integer), the rectangular GQD is always a semiconductor with a size-dependent bandgap. If the size of the two edges increases the same units respectively, the increase of armchair-edge dimension will brings a bigger increment of TPA than that of zigzag-edge. And for a rectangular GQD with M = 3M0, it is metallic with zero lowest energy state. The zigzag-edge dimension affects TPA little and the position of absorption peak and the magnitude of the TPA cross section is determined by the size of armchair-edge. The two-photon absorption transitions between (1, n) (n = 1, 2, 3 …) states play a dominant role since the energy levels of which are evenly spaced and resonance enhancement of two-photon absorption can easily occur. These theoretical analyses will provide guidance for the fabrication and application of optoelectronic devices.

2. Theory and calculation

The honeycomb lattice of the rectangular quantum dot made of monolayer graphene is schematically shown in Fig. 1. Unlike other shaped GQDs, the edges of the rectangular GQD are of two kinds simultaneously: zigzag at the top and bottom, and armchair at the left and right sides. The size of the rectangular graphene quantum dot determined by the units M and N along x- and y- axis respectively are LZZ = Ma,LAC=(N+13)32a, where a = 2.46Å is the lattice constant. Under the framework of tight-binding model, the envelop function of the electrons near the valley K = (−4π/3a, 0) and K' = (4π/3a, 0) obeys the following Dirac-like equation in momentum space [19]

HΨ=vF(0kxiky00kx+iky000000kx+iky00kxiky0)(ψAψBψA'ψB')=E(ψAψBψA'ψB')
where vF≈106 m/s is the Fermi velocity at the Dirac points [5]. And kx(y)=ix(y) is an operator to measure the momentum deviation from K or K' point. The total wavefunction Ψ(r) of the sublattice A and B are written by the four components of the spinor wavefunction in Eq. (1) as
{ΨA(r)=eiΚrψA(r)+eiΚ'rψA'(r)ΨB(r)=eiΚrψB(r)+eiΚ'rψB'(r).
The hard wall boundary conditions for both zigzag-edge and armchair-edge are adopted here [18,20]. At the zigzag edges
{ΨA(y=0)=0ΨB(y=LAC)=0,
and at the armchair edges

 

Fig. 1 (a) Schematic of a monolayer rectangular GQD, the black and red circles are two type of carbon atoms A and B. a and b are unit vectors. (b) The reciprocal lattice.

Download Full Size | PPT Slide | PDF

{ΨA(x=0)=ΨA(x=LZZ)=0ΨB(x=0)=ΨB(x=LZZ)=0.

These boundary conditions, which originate from the requirement that the electron probability amplitude at the hard wall must vanish have successfully been employed to work out the band structures of the graphene quantum dots with different edges [21,22]. Combining the above boundary conditions with Eq. (1), we can derive the electron eigen states in a rectangular GQD, which can be expressed as

(ψAψBψA'ψB')=(Ceikxxsin(kyy)Ceikxxsin(θk+kyy)Ceikxxsin(kyy)Ceikxxsin(θk+kyy))
whereθk = arctan(ky/kx), and C is the normalization constant determined by the normalization condition,

C=[2LZZLAC+LZZsin(2θk)/ky]1/2.

The corresponding eigen energy of Eq. (1) is given by

E=±vF|k|=±vFkx2+ky2
where the ± signs correspond to the conduction and valence band, respectively. It seems that the energy has the same linear dispersion relation as graphene. However, the allowed kx and ky around K point need to satisfy the following quantized condition
{(K0+kx)LZZ=mkπθk+kyLAC=nkπ
where mk and nk are integers, which give the quantized electron states. So it is convenient to denote the electron state by the symbol (m, n), where m and n stands for the orders of kx and ky. For example, for M = 12, the lowest kx can be zero when mk = 16. So we sign (1, n) for the states for mk = 16.

In most experiments, TPA is measured in terms of the absorption cross-section σ2 which is related to the two-photon transition rate W(2) by the formulas in our previous publication [16,17]. Unlike circular GQDs, the matrix elements of one-photon transition from initial state (m0, n0) to intermediate state (m1, n1) in K valley for electron-photon interaction H = (evF/c)A·σ is deduced as

Hv,iint=m1,n1|(evF/c)Aσ|m0,n0=C0C1eAvFcLACLZZ2δm1,m0×{e[sin(n0πn1π+θk1)sinθk0n0πθk0n1π+θk1sin(n0π+n1πθk1)sinθk0n0πθk0+n1πθk1]+e+[sin(n1πn0π+θk0)sinθk1n1πθk1n0π+θk0sin(n1π+n0πθk0)sinθk1n1πθk1+n0πθk0]}

The transition matrix element for rectangular GQDs is in proportion to the size of GQDs LAC and LZZ, while in circular QDs it is independent on the size [16]. The transition matrix element will not be zero only when the quantum numbers of the initial and final electron satisfying ∆mk = 0. This is the selection rule for a two-photon absorption transition in rectangular graphene quantum dots.

3. Result and discussion

The calculation of the reliable numerical values from the two-photon absorption transition requires knowledge of the interaction Hamiltonian matrix elements among all the eigen states of the GQDs and summations over all the energy bands. Here we perform the TPA calculations associated with interband transitions from valence band to conduction band for GQDs with the shape of rectangular. In our numerical calculations, we set ℏγ = 60meV (corresponding to relaxation time 10fs), and dielectric constant at the light frequency εω = 3 [23]. For the sake of convenience, only the states around K point are considered.

Since two-photon absorption cross section is dependent on the energy states of carriers, we display the energy spectrum first. Figure 2 shows the energy levels of rectangular GQDs with five different sizes, in order to demonstrate the size dependence of the energy state. The energy states for rectangular GQDs are quite sensitive to the size of zigzag- and armchair-edge dimensions. It is obvious that the density of energy state will increase, whether vary the size LZZ with a fixed LAC or vary the size LAC with a fixed LZZ. And the bandgap between conduction band and valence band decreases when the size increases. We can find that with the increasing of size for armchair-edge, LAC, i.e. N = 5, 10, 20, while zigzag-edged size LZZ with M fixed at 10, the quantized kx keeps the same. That is to say, in this case the varying of energy sates only results from ky. This can be explained by Eq. (8), which tell us kx is only a function of LZZ or M, and ky depends on both M and N. So in Figs. 2(d), 2(b) and 2(e), even though N is fixed, kx and ky both contribute to the increasement of density of state.

 

Fig. 2 Energy spectra of rectangular GQDs with different sizes as a function of kx.

Download Full Size | PPT Slide | PDF

In Fig. 3, we plot the curves of three energy states (1, 1), (1, 2) and (1, 3) of rectangular GQDs with varied LZZ or M and a fixed LAC for N = 10. Something special appears when M = 3M0 with M0 an integer. First of all, when M = 3M0, the rectangular GQD is metallic since kx and ky can be both zero then the lowest energy state (1, 1) is zero. While for M = 3M0 ± 1 it is always a semiconductor since kx and ky cannot be both zero. Secondly, the states (1, n) remain unchanged provided M = 3M0 with fixed N even though M is changed. Deriving from Eq. (8), when kx = 0 we can deduce ky = 0 or ky = (l + 1/2)π/LAC with l an integer. So, the energy difference between states (1, 1) and (1, 2) is ℏvFπ/(2LAC) while the energy differences between other neighboring states are ℏvFπ/LAC. These equidistant energy levels provide the possibility of resonance for the two-photon absorption transitions.

 

Fig. 3 Three energy states (1, 1), (1, 2) and (1, 3) of rectangular GQDs with varied M and a fixed N = 10.

Download Full Size | PPT Slide | PDF

Figure 4 shows the size-dependent TPA spectra of rectangular GQDs with non-3M0 sizes, calculated by the TPA model presented in the previous section. It is found that the TPA cross section for these sized rectangular GQDs is in magnitude of 10−51 m4s/photon, which is approximately four orders lower than that of circular GQDs [17]. This can be explained by the TPA resonant enhancement in circular GQDs arising from their tiny changed energy level spaces. We choose two ways to vary the sizes: increasing M by 3 units with fixed N = 20 and increasing N by 3 units with fixed M = 20. It is found that there is a red shift for the absorption peaks with the increase of the GQDs’ size, both of the two ways. This is owing to the fact that as the consequence of quantum size effect, energy differences become smaller when size increases, which can be seen in Fig. 2. Also, with the increase of size, the magnitude of TPA coefficient increases too. The bigger the GQD is, the lager the density of state is. Therefore, more transitions can occur. This property is in consistent with the case of conventional semiconductor QDs [24]. In order to investigate the contribution of edges, i.e. zigzag and armchair to the TPA, we compared Figs. 4(a) with 4(b). It is seen that the amplitude increasement of TPA cross section when N increases every 3 units is nearly 3 times of that for M. We can conclude that the size of armchair-edge in a rectangular grapheme quantum dot contributes more to the two-photon absorption than that of zigzag-edge dimension.

 

Fig. 4 Two-photon absorption spectra for non-3M0 sized GQDs plotted as a function of incident photon energy.

Download Full Size | PPT Slide | PDF

We also demonstrate the TPA spectra for 3M0-sized metallic rectangular GQDs with M = 9, 12, 15, 18 and N = 15, 20, 25 in Fig. 5. With the varying of M and a fixed N, the magnitude of TPA will not change remarkably and the absorption peak will not shift. In order to explore the reason of this anomaly, we plot the TPA spectrum contributed only by the transitions between the states when kx = 0, i.e. (1, n) states. It can be seen that the dominant portion of TPA is contributed from the transitions between the (1, n) states, where resonant transitions can occur due to the uniformly-spaced energy levels. It has been mentioned above that once the size of armchair-edged dimension LAC, i.e. N, is fixed, the energy level spaces between the (1, n) states are confirmed to be ℏvFπ/(2LAC) or ℏvFπ/LAC. So the position of absorption peak will be estimated at ℏω = ℏvFπ/(2LAC) and ℏvFπ/LAC around. If we alter the size of armchair-edged dimension LAC, i.e. N, there is a red shift for the absorption peaks with the increase of N. And the magnitude of the TPA peaks increases significantly because of more resonant transitions involved.

 

Fig. 5 TPA spectra for 3M0-sized metallic rectangular GQDs with M = 9, 12, 15, 18 and N = 15, 20, 25. The dashed line is the TPA spectra contributed from the transitions between (1, n).

Download Full Size | PPT Slide | PDF

4. Conclusion

In conclusion, by solving the Dirac equation analytically under hard wall boundary conditions, we have obtained the energy spectrum of a rectangular GQD. It seems that the energy has the same linear dispersion relation as graphene but it is quantized. The bigger the sizes of the rectangular GQDs are, the smaller the spaces of the energy levels are. On the basis of energy levels and electronic states, the calculation for the size-dependent two-photon absorption cross section associated with interband transitions have been conducted. When the size of zigzag-edge M = 3M0 ± 1 (M0 is an integer), the rectangular GQD is always a semiconductor with a size-dependent band gap. The size-dependence is coincident with the other conventional semiconductor QDs, namely a red shift and magnitudes increasing accompany the increase of GQD’s size. Further, if the size of the two edges increases the same units respectively, the increase of armchair-edge dimension will brings a bigger increment of TPA than that of zigzag-edge. And for a rectangular GQD with M = 3M0, it is metallic with zero lowest energy state. The zigzag-edge dimension affects TPA little and the position of absorption peak and the magnitude of the TPA coefficient are determined by the size of armchair-edge. Resonant enhancement of two-photon absorption occurs within the two-photon absorption transitions between (1, n) as the energy levels of which are evenly spaced. These theoretical analyses are of much importance to multi-photon fluorescence imaging and optoelectronic applications of GQDs.

Funding

National Natural Science Foundation of China (NSFC) (11304275 and 11764047); Applied Basic Research Foundation of Yunnan Province (2017FB009); Innovative Talents of Science and Technology Plan Projects of Yunnan Province (2014HB010).

References and links

1. P. Suvarnaphaet and S. Pechprasarn, “Graphene-based materials for biosensors: a review,” Sensors 17(10), 2161 (2017).

2. H. Zhang, S. Virally, Q. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012). [CrossRef]   [PubMed]  

3. J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016). [CrossRef]   [PubMed]  

4. Y. Xu, Z. Wang, Z. Guo, H. Huang, Q. Xiao, H. Zhang, and X. F. Yu, “Solvothermal synthesis and ultrafast photonics of black phosphorus quantum dots,” Adv. Opt. Mater. 4(8), 1223–1229 (2016). [CrossRef]  

5. A. K. Geim, “Graphene: status and prospects,” Science 324(5934), 1530–1534 (2009). [CrossRef]   [PubMed]  

6. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004). [CrossRef]   [PubMed]  

7. C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312(5777), 1191–1196 (2006). [CrossRef]   [PubMed]  

8. D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda, A. Dimiev, B. K. Price, and J. M. Tour, “Longitudinal unzipping of carbon nanotubes to form graphene nanoribbons,” Nature 458(7240), 872–876 (2009). [CrossRef]   [PubMed]  

9. Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438(7065), 201–204 (2005). [CrossRef]   [PubMed]  

10. X. T. Zheng, A. Ananthanarayanan, K. Q. Luo, and P. Chen, “Glowing graphene quantum dots and carbon dots: properties, syntheses, and biological applications,” Small 11(14), 1620–1636 (2015). [CrossRef]   [PubMed]  

11. S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012). [CrossRef]   [PubMed]  

12. Q. Liu, B. Guo, Z. Rao, B. Zhang, and J. R. Gong, “Strong two-photon-induced fluorescence from photostable, biocompatible nitrogen-doped graphene quantum dots for cellular and deep-tissue imaging,” Nano Lett. 13(6), 2436–2441 (2013). [CrossRef]   [PubMed]  

13. M. Ghaffarian and F. Ebrahimi, “The linear and nonlinear optical properties of trigonal zigzag graphene nanoflakes,” Phys. Scr. 88(2), 025703 (2013). [CrossRef]  

14. R. Zhang, S. Qi, J. Jia, B. Torre, H. Zeng, H. Wu, and X. Xu, “Size and refinement edge-shape effects of graphene quantum dots on UV-visible absorption,” J. Alloys Compd. 623(6), 186–191 (2015).

15. F. Qi and G. Jin, “Strain sensing and far-infrared absorption in strained graphene quantum dots,” J. Appl. Phys. 114(7), 073509 (2013). [CrossRef]  

16. X. Feng, X. Li, Z. Li, and Y. Liu, “Size-dependent two-photon absorption in circular graphene quantum dots,” Opt. Express 24(3), 2877–2884 (2016). [CrossRef]   [PubMed]  

17. X. Feng, Z. Li, X. Li, and Y. Liu, “Giant Two-photon Absorption in Circular Graphene Quantum Dots in Infrared Region,” Sci. Rep. 6(6), 33260 (2016). [CrossRef]   [PubMed]  

18. T. Fang, A. Konar, H. Xing, and D. Jena, “Mobility in semiconducting graphene nanoribbons: Phonon, impurity, and edge roughness scattering,” Phys. Rev. B 78(20), 205403 (2008). [CrossRef]  

19. T. Ando and T. Nakanishi, “Impurity Scattering in Carbon Nanotubes Absence of Back Scattering,” J. Phys. Soc. Jpn. 67(67), 1704–1713 (1998). [CrossRef]  

20. J. Qian, M. Dutta, and M. A. Stroscio, “Phonon bottleneck effects in rectangular graphene quantum dots,” J. Comput. Electron. 11(3), 293–301 (2012). [CrossRef]  

21. B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard, “Spin qubits in graphene quantum dots,” Nat. Phys. 3(3), 192–196 (2007). [CrossRef]  

22. P. G. Silvestrov and K. B. Efetov, “Quantum dots in graphene,” Phys. Rev. Lett. 98(1), 016802 (2007). [CrossRef]   [PubMed]  

23. F. C. Salomão, E. Lanzoni, C. Costa, C. Deneke, and E. B. Barros, “Determination of high frequency dielectric constant and surface potential of graphene oxide and influence of humidity by KPFM,” Languir 31(41), 11339–11343 (2015). [CrossRef]   [PubMed]  

24. X. Feng and W. Ji, “Shape-dependent two-photon absorption in semiconductor nanocrystals,” Opt. Express 17(15), 13140–13150 (2009). [CrossRef]   [PubMed]  

References

  • View by:
  • |
  • |
  • |

  1. P. Suvarnaphaet and S. Pechprasarn, “Graphene-based materials for biosensors: a review,” Sensors 17(10), 2161 (2017).
  2. H. Zhang, S. Virally, Q. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012).
    [Crossref] [PubMed]
  3. J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
    [Crossref] [PubMed]
  4. Y. Xu, Z. Wang, Z. Guo, H. Huang, Q. Xiao, H. Zhang, and X. F. Yu, “Solvothermal synthesis and ultrafast photonics of black phosphorus quantum dots,” Adv. Opt. Mater. 4(8), 1223–1229 (2016).
    [Crossref]
  5. A. K. Geim, “Graphene: status and prospects,” Science 324(5934), 1530–1534 (2009).
    [Crossref] [PubMed]
  6. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
    [Crossref] [PubMed]
  7. C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312(5777), 1191–1196 (2006).
    [Crossref] [PubMed]
  8. D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda, A. Dimiev, B. K. Price, and J. M. Tour, “Longitudinal unzipping of carbon nanotubes to form graphene nanoribbons,” Nature 458(7240), 872–876 (2009).
    [Crossref] [PubMed]
  9. Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438(7065), 201–204 (2005).
    [Crossref] [PubMed]
  10. X. T. Zheng, A. Ananthanarayanan, K. Q. Luo, and P. Chen, “Glowing graphene quantum dots and carbon dots: properties, syntheses, and biological applications,” Small 11(14), 1620–1636 (2015).
    [Crossref] [PubMed]
  11. S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
    [Crossref] [PubMed]
  12. Q. Liu, B. Guo, Z. Rao, B. Zhang, and J. R. Gong, “Strong two-photon-induced fluorescence from photostable, biocompatible nitrogen-doped graphene quantum dots for cellular and deep-tissue imaging,” Nano Lett. 13(6), 2436–2441 (2013).
    [Crossref] [PubMed]
  13. M. Ghaffarian and F. Ebrahimi, “The linear and nonlinear optical properties of trigonal zigzag graphene nanoflakes,” Phys. Scr. 88(2), 025703 (2013).
    [Crossref]
  14. R. Zhang, S. Qi, J. Jia, B. Torre, H. Zeng, H. Wu, and X. Xu, “Size and refinement edge-shape effects of graphene quantum dots on UV-visible absorption,” J. Alloys Compd. 623(6), 186–191 (2015).
  15. F. Qi and G. Jin, “Strain sensing and far-infrared absorption in strained graphene quantum dots,” J. Appl. Phys. 114(7), 073509 (2013).
    [Crossref]
  16. X. Feng, X. Li, Z. Li, and Y. Liu, “Size-dependent two-photon absorption in circular graphene quantum dots,” Opt. Express 24(3), 2877–2884 (2016).
    [Crossref] [PubMed]
  17. X. Feng, Z. Li, X. Li, and Y. Liu, “Giant Two-photon Absorption in Circular Graphene Quantum Dots in Infrared Region,” Sci. Rep. 6(6), 33260 (2016).
    [Crossref] [PubMed]
  18. T. Fang, A. Konar, H. Xing, and D. Jena, “Mobility in semiconducting graphene nanoribbons: Phonon, impurity, and edge roughness scattering,” Phys. Rev. B 78(20), 205403 (2008).
    [Crossref]
  19. T. Ando and T. Nakanishi, “Impurity Scattering in Carbon Nanotubes Absence of Back Scattering,” J. Phys. Soc. Jpn. 67(67), 1704–1713 (1998).
    [Crossref]
  20. J. Qian, M. Dutta, and M. A. Stroscio, “Phonon bottleneck effects in rectangular graphene quantum dots,” J. Comput. Electron. 11(3), 293–301 (2012).
    [Crossref]
  21. B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard, “Spin qubits in graphene quantum dots,” Nat. Phys. 3(3), 192–196 (2007).
    [Crossref]
  22. P. G. Silvestrov and K. B. Efetov, “Quantum dots in graphene,” Phys. Rev. Lett. 98(1), 016802 (2007).
    [Crossref] [PubMed]
  23. F. C. Salomão, E. Lanzoni, C. Costa, C. Deneke, and E. B. Barros, “Determination of high frequency dielectric constant and surface potential of graphene oxide and influence of humidity by KPFM,” Languir 31(41), 11339–11343 (2015).
    [Crossref] [PubMed]
  24. X. Feng and W. Ji, “Shape-dependent two-photon absorption in semiconductor nanocrystals,” Opt. Express 17(15), 13140–13150 (2009).
    [Crossref] [PubMed]

2017 (1)

P. Suvarnaphaet and S. Pechprasarn, “Graphene-based materials for biosensors: a review,” Sensors 17(10), 2161 (2017).

2016 (4)

J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
[Crossref] [PubMed]

Y. Xu, Z. Wang, Z. Guo, H. Huang, Q. Xiao, H. Zhang, and X. F. Yu, “Solvothermal synthesis and ultrafast photonics of black phosphorus quantum dots,” Adv. Opt. Mater. 4(8), 1223–1229 (2016).
[Crossref]

X. Feng, X. Li, Z. Li, and Y. Liu, “Size-dependent two-photon absorption in circular graphene quantum dots,” Opt. Express 24(3), 2877–2884 (2016).
[Crossref] [PubMed]

X. Feng, Z. Li, X. Li, and Y. Liu, “Giant Two-photon Absorption in Circular Graphene Quantum Dots in Infrared Region,” Sci. Rep. 6(6), 33260 (2016).
[Crossref] [PubMed]

2015 (3)

X. T. Zheng, A. Ananthanarayanan, K. Q. Luo, and P. Chen, “Glowing graphene quantum dots and carbon dots: properties, syntheses, and biological applications,” Small 11(14), 1620–1636 (2015).
[Crossref] [PubMed]

R. Zhang, S. Qi, J. Jia, B. Torre, H. Zeng, H. Wu, and X. Xu, “Size and refinement edge-shape effects of graphene quantum dots on UV-visible absorption,” J. Alloys Compd. 623(6), 186–191 (2015).

F. C. Salomão, E. Lanzoni, C. Costa, C. Deneke, and E. B. Barros, “Determination of high frequency dielectric constant and surface potential of graphene oxide and influence of humidity by KPFM,” Languir 31(41), 11339–11343 (2015).
[Crossref] [PubMed]

2013 (3)

F. Qi and G. Jin, “Strain sensing and far-infrared absorption in strained graphene quantum dots,” J. Appl. Phys. 114(7), 073509 (2013).
[Crossref]

Q. Liu, B. Guo, Z. Rao, B. Zhang, and J. R. Gong, “Strong two-photon-induced fluorescence from photostable, biocompatible nitrogen-doped graphene quantum dots for cellular and deep-tissue imaging,” Nano Lett. 13(6), 2436–2441 (2013).
[Crossref] [PubMed]

M. Ghaffarian and F. Ebrahimi, “The linear and nonlinear optical properties of trigonal zigzag graphene nanoflakes,” Phys. Scr. 88(2), 025703 (2013).
[Crossref]

2012 (3)

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

H. Zhang, S. Virally, Q. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012).
[Crossref] [PubMed]

J. Qian, M. Dutta, and M. A. Stroscio, “Phonon bottleneck effects in rectangular graphene quantum dots,” J. Comput. Electron. 11(3), 293–301 (2012).
[Crossref]

2009 (3)

X. Feng and W. Ji, “Shape-dependent two-photon absorption in semiconductor nanocrystals,” Opt. Express 17(15), 13140–13150 (2009).
[Crossref] [PubMed]

D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda, A. Dimiev, B. K. Price, and J. M. Tour, “Longitudinal unzipping of carbon nanotubes to form graphene nanoribbons,” Nature 458(7240), 872–876 (2009).
[Crossref] [PubMed]

A. K. Geim, “Graphene: status and prospects,” Science 324(5934), 1530–1534 (2009).
[Crossref] [PubMed]

2008 (1)

T. Fang, A. Konar, H. Xing, and D. Jena, “Mobility in semiconducting graphene nanoribbons: Phonon, impurity, and edge roughness scattering,” Phys. Rev. B 78(20), 205403 (2008).
[Crossref]

2007 (2)

B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard, “Spin qubits in graphene quantum dots,” Nat. Phys. 3(3), 192–196 (2007).
[Crossref]

P. G. Silvestrov and K. B. Efetov, “Quantum dots in graphene,” Phys. Rev. Lett. 98(1), 016802 (2007).
[Crossref] [PubMed]

2006 (1)

C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312(5777), 1191–1196 (2006).
[Crossref] [PubMed]

2005 (1)

Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438(7065), 201–204 (2005).
[Crossref] [PubMed]

2004 (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

1998 (1)

T. Ando and T. Nakanishi, “Impurity Scattering in Carbon Nanotubes Absence of Back Scattering,” J. Phys. Soc. Jpn. 67(67), 1704–1713 (1998).
[Crossref]

Ananthanarayanan, A.

X. T. Zheng, A. Ananthanarayanan, K. Q. Luo, and P. Chen, “Glowing graphene quantum dots and carbon dots: properties, syntheses, and biological applications,” Small 11(14), 1620–1636 (2015).
[Crossref] [PubMed]

Ando, T.

T. Ando and T. Nakanishi, “Impurity Scattering in Carbon Nanotubes Absence of Back Scattering,” J. Phys. Soc. Jpn. 67(67), 1704–1713 (1998).
[Crossref]

Ashrafi, M.

J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
[Crossref] [PubMed]

Bae, S.

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

Bao, Q.

J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
[Crossref] [PubMed]

H. Zhang, S. Virally, Q. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012).
[Crossref] [PubMed]

Barros, E. B.

F. C. Salomão, E. Lanzoni, C. Costa, C. Deneke, and E. B. Barros, “Determination of high frequency dielectric constant and surface potential of graphene oxide and influence of humidity by KPFM,” Languir 31(41), 11339–11343 (2015).
[Crossref] [PubMed]

Berger, C.

C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312(5777), 1191–1196 (2006).
[Crossref] [PubMed]

Brown, N.

C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312(5777), 1191–1196 (2006).
[Crossref] [PubMed]

Bulaev, D. V.

B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard, “Spin qubits in graphene quantum dots,” Nat. Phys. 3(3), 192–196 (2007).
[Crossref]

Burkard, G.

B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard, “Spin qubits in graphene quantum dots,” Nat. Phys. 3(3), 192–196 (2007).
[Crossref]

Chen, P.

X. T. Zheng, A. Ananthanarayanan, K. Q. Luo, and P. Chen, “Glowing graphene quantum dots and carbon dots: properties, syntheses, and biological applications,” Small 11(14), 1620–1636 (2015).
[Crossref] [PubMed]

Choi, H. J.

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

Choi, S. H.

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

Conrad, E. H.

C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312(5777), 1191–1196 (2006).
[Crossref] [PubMed]

Costa, C.

F. C. Salomão, E. Lanzoni, C. Costa, C. Deneke, and E. B. Barros, “Determination of high frequency dielectric constant and surface potential of graphene oxide and influence of humidity by KPFM,” Languir 31(41), 11339–11343 (2015).
[Crossref] [PubMed]

de Heer, W. A.

C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312(5777), 1191–1196 (2006).
[Crossref] [PubMed]

Deneke, C.

F. C. Salomão, E. Lanzoni, C. Costa, C. Deneke, and E. B. Barros, “Determination of high frequency dielectric constant and surface potential of graphene oxide and influence of humidity by KPFM,” Languir 31(41), 11339–11343 (2015).
[Crossref] [PubMed]

Dhanabalan, S. C.

J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
[Crossref] [PubMed]

Dimiev, A.

D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda, A. Dimiev, B. K. Price, and J. M. Tour, “Longitudinal unzipping of carbon nanotubes to form graphene nanoribbons,” Nature 458(7240), 872–876 (2009).
[Crossref] [PubMed]

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Dutta, M.

J. Qian, M. Dutta, and M. A. Stroscio, “Phonon bottleneck effects in rectangular graphene quantum dots,” J. Comput. Electron. 11(3), 293–301 (2012).
[Crossref]

Ebrahimi, F.

M. Ghaffarian and F. Ebrahimi, “The linear and nonlinear optical properties of trigonal zigzag graphene nanoflakes,” Phys. Scr. 88(2), 025703 (2013).
[Crossref]

Efetov, K. B.

P. G. Silvestrov and K. B. Efetov, “Quantum dots in graphene,” Phys. Rev. Lett. 98(1), 016802 (2007).
[Crossref] [PubMed]

Fang, T.

T. Fang, A. Konar, H. Xing, and D. Jena, “Mobility in semiconducting graphene nanoribbons: Phonon, impurity, and edge roughness scattering,” Phys. Rev. B 78(20), 205403 (2008).
[Crossref]

Feng, X.

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

First, P. N.

C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312(5777), 1191–1196 (2006).
[Crossref] [PubMed]

Geim, A. K.

A. K. Geim, “Graphene: status and prospects,” Science 324(5934), 1530–1534 (2009).
[Crossref] [PubMed]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Ghaffarian, M.

M. Ghaffarian and F. Ebrahimi, “The linear and nonlinear optical properties of trigonal zigzag graphene nanoflakes,” Phys. Scr. 88(2), 025703 (2013).
[Crossref]

Godbout, N.

Gong, J. R.

Q. Liu, B. Guo, Z. Rao, B. Zhang, and J. R. Gong, “Strong two-photon-induced fluorescence from photostable, biocompatible nitrogen-doped graphene quantum dots for cellular and deep-tissue imaging,” Nano Lett. 13(6), 2436–2441 (2013).
[Crossref] [PubMed]

Grigorieva, I. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Guo, B.

Q. Liu, B. Guo, Z. Rao, B. Zhang, and J. R. Gong, “Strong two-photon-induced fluorescence from photostable, biocompatible nitrogen-doped graphene quantum dots for cellular and deep-tissue imaging,” Nano Lett. 13(6), 2436–2441 (2013).
[Crossref] [PubMed]

Guo, Z.

Y. Xu, Z. Wang, Z. Guo, H. Huang, Q. Xiao, H. Zhang, and X. F. Yu, “Solvothermal synthesis and ultrafast photonics of black phosphorus quantum dots,” Adv. Opt. Mater. 4(8), 1223–1229 (2016).
[Crossref]

Hass, J.

C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312(5777), 1191–1196 (2006).
[Crossref] [PubMed]

Higginbotham, A. L.

D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda, A. Dimiev, B. K. Price, and J. M. Tour, “Longitudinal unzipping of carbon nanotubes to form graphene nanoribbons,” Nature 458(7240), 872–876 (2009).
[Crossref] [PubMed]

Hong, B. H.

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

Huang, H.

Y. Xu, Z. Wang, Z. Guo, H. Huang, Q. Xiao, H. Zhang, and X. F. Yu, “Solvothermal synthesis and ultrafast photonics of black phosphorus quantum dots,” Adv. Opt. Mater. 4(8), 1223–1229 (2016).
[Crossref]

Hwang, E.

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

Hwang, S. W.

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

Jena, D.

T. Fang, A. Konar, H. Xing, and D. Jena, “Mobility in semiconducting graphene nanoribbons: Phonon, impurity, and edge roughness scattering,” Phys. Rev. B 78(20), 205403 (2008).
[Crossref]

Ji, W.

Jia, J.

R. Zhang, S. Qi, J. Jia, B. Torre, H. Zeng, H. Wu, and X. Xu, “Size and refinement edge-shape effects of graphene quantum dots on UV-visible absorption,” J. Alloys Compd. 623(6), 186–191 (2015).

Jiang, D.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Jin, G.

F. Qi and G. Jin, “Strain sensing and far-infrared absorption in strained graphene quantum dots,” J. Appl. Phys. 114(7), 073509 (2013).
[Crossref]

Kim, C. O.

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

Kim, M. K.

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

Kim, P.

Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438(7065), 201–204 (2005).
[Crossref] [PubMed]

Kim, S.

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

Ko, G.

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

Kockaert, P.

Konar, A.

T. Fang, A. Konar, H. Xing, and D. Jena, “Mobility in semiconducting graphene nanoribbons: Phonon, impurity, and edge roughness scattering,” Phys. Rev. B 78(20), 205403 (2008).
[Crossref]

Kosynkin, D. V.

D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda, A. Dimiev, B. K. Price, and J. M. Tour, “Longitudinal unzipping of carbon nanotubes to form graphene nanoribbons,” Nature 458(7240), 872–876 (2009).
[Crossref] [PubMed]

Lanzoni, E.

F. C. Salomão, E. Lanzoni, C. Costa, C. Deneke, and E. B. Barros, “Determination of high frequency dielectric constant and surface potential of graphene oxide and influence of humidity by KPFM,” Languir 31(41), 11339–11343 (2015).
[Crossref] [PubMed]

Li, P.

J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
[Crossref] [PubMed]

Li, S.

J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
[Crossref] [PubMed]

Li, T.

C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312(5777), 1191–1196 (2006).
[Crossref] [PubMed]

Li, X.

X. Feng, Z. Li, X. Li, and Y. Liu, “Giant Two-photon Absorption in Circular Graphene Quantum Dots in Infrared Region,” Sci. Rep. 6(6), 33260 (2016).
[Crossref] [PubMed]

X. Feng, X. Li, Z. Li, and Y. Liu, “Size-dependent two-photon absorption in circular graphene quantum dots,” Opt. Express 24(3), 2877–2884 (2016).
[Crossref] [PubMed]

C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312(5777), 1191–1196 (2006).
[Crossref] [PubMed]

Li, Z.

X. Feng, X. Li, Z. Li, and Y. Liu, “Size-dependent two-photon absorption in circular graphene quantum dots,” Opt. Express 24(3), 2877–2884 (2016).
[Crossref] [PubMed]

X. Feng, Z. Li, X. Li, and Y. Liu, “Giant Two-photon Absorption in Circular Graphene Quantum Dots in Infrared Region,” Sci. Rep. 6(6), 33260 (2016).
[Crossref] [PubMed]

Liu, Q.

Q. Liu, B. Guo, Z. Rao, B. Zhang, and J. R. Gong, “Strong two-photon-induced fluorescence from photostable, biocompatible nitrogen-doped graphene quantum dots for cellular and deep-tissue imaging,” Nano Lett. 13(6), 2436–2441 (2013).
[Crossref] [PubMed]

Liu, Y.

X. Feng, X. Li, Z. Li, and Y. Liu, “Size-dependent two-photon absorption in circular graphene quantum dots,” Opt. Express 24(3), 2877–2884 (2016).
[Crossref] [PubMed]

X. Feng, Z. Li, X. Li, and Y. Liu, “Giant Two-photon Absorption in Circular Graphene Quantum Dots in Infrared Region,” Sci. Rep. 6(6), 33260 (2016).
[Crossref] [PubMed]

Lomeda, J. R.

D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda, A. Dimiev, B. K. Price, and J. M. Tour, “Longitudinal unzipping of carbon nanotubes to form graphene nanoribbons,” Nature 458(7240), 872–876 (2009).
[Crossref] [PubMed]

Loss, D.

B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard, “Spin qubits in graphene quantum dots,” Nat. Phys. 3(3), 192–196 (2007).
[Crossref]

Luo, K. Q.

X. T. Zheng, A. Ananthanarayanan, K. Q. Luo, and P. Chen, “Glowing graphene quantum dots and carbon dots: properties, syntheses, and biological applications,” Small 11(14), 1620–1636 (2015).
[Crossref] [PubMed]

Marchenkov, A. N.

C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312(5777), 1191–1196 (2006).
[Crossref] [PubMed]

Massar, S.

Mayou, D.

C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312(5777), 1191–1196 (2006).
[Crossref] [PubMed]

Mccoubrey, K.

J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
[Crossref] [PubMed]

Morozov, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Mu, H.

J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
[Crossref] [PubMed]

Nakanishi, T.

T. Ando and T. Nakanishi, “Impurity Scattering in Carbon Nanotubes Absence of Back Scattering,” J. Phys. Soc. Jpn. 67(67), 1704–1713 (1998).
[Crossref]

Naud, C.

C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312(5777), 1191–1196 (2006).
[Crossref] [PubMed]

Novoselov, K. S.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Park, J. H.

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

Pechprasarn, S.

P. Suvarnaphaet and S. Pechprasarn, “Graphene-based materials for biosensors: a review,” Sensors 17(10), 2161 (2017).

Ping, L. K.

Ponraj, J. S.

J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
[Crossref] [PubMed]

Price, B. K.

D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda, A. Dimiev, B. K. Price, and J. M. Tour, “Longitudinal unzipping of carbon nanotubes to form graphene nanoribbons,” Nature 458(7240), 872–876 (2009).
[Crossref] [PubMed]

Qi, F.

F. Qi and G. Jin, “Strain sensing and far-infrared absorption in strained graphene quantum dots,” J. Appl. Phys. 114(7), 073509 (2013).
[Crossref]

Qi, S.

R. Zhang, S. Qi, J. Jia, B. Torre, H. Zeng, H. Wu, and X. Xu, “Size and refinement edge-shape effects of graphene quantum dots on UV-visible absorption,” J. Alloys Compd. 623(6), 186–191 (2015).

Qian, J.

J. Qian, M. Dutta, and M. A. Stroscio, “Phonon bottleneck effects in rectangular graphene quantum dots,” J. Comput. Electron. 11(3), 293–301 (2012).
[Crossref]

Rao, Z.

Q. Liu, B. Guo, Z. Rao, B. Zhang, and J. R. Gong, “Strong two-photon-induced fluorescence from photostable, biocompatible nitrogen-doped graphene quantum dots for cellular and deep-tissue imaging,” Nano Lett. 13(6), 2436–2441 (2013).
[Crossref] [PubMed]

Salomão, F. C.

F. C. Salomão, E. Lanzoni, C. Costa, C. Deneke, and E. B. Barros, “Determination of high frequency dielectric constant and surface potential of graphene oxide and influence of humidity by KPFM,” Languir 31(41), 11339–11343 (2015).
[Crossref] [PubMed]

Shin, D. H.

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

Shin, D. Y.

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

Silvestrov, P. G.

P. G. Silvestrov and K. B. Efetov, “Quantum dots in graphene,” Phys. Rev. Lett. 98(1), 016802 (2007).
[Crossref] [PubMed]

Sim, S.

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

Sinitskii, A.

D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda, A. Dimiev, B. K. Price, and J. M. Tour, “Longitudinal unzipping of carbon nanotubes to form graphene nanoribbons,” Nature 458(7240), 872–876 (2009).
[Crossref] [PubMed]

Sone, C.

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

Song, Z.

C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312(5777), 1191–1196 (2006).
[Crossref] [PubMed]

Stormer, H. L.

Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438(7065), 201–204 (2005).
[Crossref] [PubMed]

Stroscio, M. A.

J. Qian, M. Dutta, and M. A. Stroscio, “Phonon bottleneck effects in rectangular graphene quantum dots,” J. Comput. Electron. 11(3), 293–301 (2012).
[Crossref]

Suvarnaphaet, P.

P. Suvarnaphaet and S. Pechprasarn, “Graphene-based materials for biosensors: a review,” Sensors 17(10), 2161 (2017).

Tan, Y. W.

Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438(7065), 201–204 (2005).
[Crossref] [PubMed]

Thakur, S.

J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
[Crossref] [PubMed]

Torre, B.

R. Zhang, S. Qi, J. Jia, B. Torre, H. Zeng, H. Wu, and X. Xu, “Size and refinement edge-shape effects of graphene quantum dots on UV-visible absorption,” J. Alloys Compd. 623(6), 186–191 (2015).

Tour, J. M.

D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda, A. Dimiev, B. K. Price, and J. M. Tour, “Longitudinal unzipping of carbon nanotubes to form graphene nanoribbons,” Nature 458(7240), 872–876 (2009).
[Crossref] [PubMed]

Trauzettel, B.

B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard, “Spin qubits in graphene quantum dots,” Nat. Phys. 3(3), 192–196 (2007).
[Crossref]

Virally, S.

Wang, Y.

J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
[Crossref] [PubMed]

Wang, Z.

Y. Xu, Z. Wang, Z. Guo, H. Huang, Q. Xiao, H. Zhang, and X. F. Yu, “Solvothermal synthesis and ultrafast photonics of black phosphorus quantum dots,” Adv. Opt. Mater. 4(8), 1223–1229 (2016).
[Crossref]

Wu, H.

R. Zhang, S. Qi, J. Jia, B. Torre, H. Zeng, H. Wu, and X. Xu, “Size and refinement edge-shape effects of graphene quantum dots on UV-visible absorption,” J. Alloys Compd. 623(6), 186–191 (2015).

Wu, X.

C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312(5777), 1191–1196 (2006).
[Crossref] [PubMed]

Xiao, Q.

Y. Xu, Z. Wang, Z. Guo, H. Huang, Q. Xiao, H. Zhang, and X. F. Yu, “Solvothermal synthesis and ultrafast photonics of black phosphorus quantum dots,” Adv. Opt. Mater. 4(8), 1223–1229 (2016).
[Crossref]

Xing, H.

T. Fang, A. Konar, H. Xing, and D. Jena, “Mobility in semiconducting graphene nanoribbons: Phonon, impurity, and edge roughness scattering,” Phys. Rev. B 78(20), 205403 (2008).
[Crossref]

Xu, X.

R. Zhang, S. Qi, J. Jia, B. Torre, H. Zeng, H. Wu, and X. Xu, “Size and refinement edge-shape effects of graphene quantum dots on UV-visible absorption,” J. Alloys Compd. 623(6), 186–191 (2015).

Xu, Y.

Y. Xu, Z. Wang, Z. Guo, H. Huang, Q. Xiao, H. Zhang, and X. F. Yu, “Solvothermal synthesis and ultrafast photonics of black phosphorus quantum dots,” Adv. Opt. Mater. 4(8), 1223–1229 (2016).
[Crossref]

Xu, Z. Q.

J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
[Crossref] [PubMed]

Yang, S. B.

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

Yu, X. F.

Y. Xu, Z. Wang, Z. Guo, H. Huang, Q. Xiao, H. Zhang, and X. F. Yu, “Solvothermal synthesis and ultrafast photonics of black phosphorus quantum dots,” Adv. Opt. Mater. 4(8), 1223–1229 (2016).
[Crossref]

Yuan, J.

J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
[Crossref] [PubMed]

Zeng, H.

R. Zhang, S. Qi, J. Jia, B. Torre, H. Zeng, H. Wu, and X. Xu, “Size and refinement edge-shape effects of graphene quantum dots on UV-visible absorption,” J. Alloys Compd. 623(6), 186–191 (2015).

Zhang, B.

Q. Liu, B. Guo, Z. Rao, B. Zhang, and J. R. Gong, “Strong two-photon-induced fluorescence from photostable, biocompatible nitrogen-doped graphene quantum dots for cellular and deep-tissue imaging,” Nano Lett. 13(6), 2436–2441 (2013).
[Crossref] [PubMed]

Zhang, H.

J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
[Crossref] [PubMed]

Y. Xu, Z. Wang, Z. Guo, H. Huang, Q. Xiao, H. Zhang, and X. F. Yu, “Solvothermal synthesis and ultrafast photonics of black phosphorus quantum dots,” Adv. Opt. Mater. 4(8), 1223–1229 (2016).
[Crossref]

H. Zhang, S. Virally, Q. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012).
[Crossref] [PubMed]

Zhang, R.

R. Zhang, S. Qi, J. Jia, B. Torre, H. Zeng, H. Wu, and X. Xu, “Size and refinement edge-shape effects of graphene quantum dots on UV-visible absorption,” J. Alloys Compd. 623(6), 186–191 (2015).

Zhang, Y.

J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
[Crossref] [PubMed]

Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438(7065), 201–204 (2005).
[Crossref] [PubMed]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Zheng, X. T.

X. T. Zheng, A. Ananthanarayanan, K. Q. Luo, and P. Chen, “Glowing graphene quantum dots and carbon dots: properties, syntheses, and biological applications,” Small 11(14), 1620–1636 (2015).
[Crossref] [PubMed]

ACS Nano (1)

S. Kim, S. W. Hwang, M. K. Kim, D. Y. Shin, D. H. Shin, C. O. Kim, S. B. Yang, J. H. Park, E. Hwang, S. H. Choi, G. Ko, S. Sim, C. Sone, H. J. Choi, S. Bae, and B. H. Hong, “Anomalous behaviors of visible luminescence from graphene quantum dots: interplay between size and shape,” ACS Nano 6(9), 8203–8208 (2012).
[Crossref] [PubMed]

Adv. Opt. Mater. (1)

Y. Xu, Z. Wang, Z. Guo, H. Huang, Q. Xiao, H. Zhang, and X. F. Yu, “Solvothermal synthesis and ultrafast photonics of black phosphorus quantum dots,” Adv. Opt. Mater. 4(8), 1223–1229 (2016).
[Crossref]

J. Alloys Compd. (1)

R. Zhang, S. Qi, J. Jia, B. Torre, H. Zeng, H. Wu, and X. Xu, “Size and refinement edge-shape effects of graphene quantum dots on UV-visible absorption,” J. Alloys Compd. 623(6), 186–191 (2015).

J. Appl. Phys. (1)

F. Qi and G. Jin, “Strain sensing and far-infrared absorption in strained graphene quantum dots,” J. Appl. Phys. 114(7), 073509 (2013).
[Crossref]

J. Comput. Electron. (1)

J. Qian, M. Dutta, and M. A. Stroscio, “Phonon bottleneck effects in rectangular graphene quantum dots,” J. Comput. Electron. 11(3), 293–301 (2012).
[Crossref]

J. Phys. Soc. Jpn. (1)

T. Ando and T. Nakanishi, “Impurity Scattering in Carbon Nanotubes Absence of Back Scattering,” J. Phys. Soc. Jpn. 67(67), 1704–1713 (1998).
[Crossref]

Languir (1)

F. C. Salomão, E. Lanzoni, C. Costa, C. Deneke, and E. B. Barros, “Determination of high frequency dielectric constant and surface potential of graphene oxide and influence of humidity by KPFM,” Languir 31(41), 11339–11343 (2015).
[Crossref] [PubMed]

Nano Lett. (1)

Q. Liu, B. Guo, Z. Rao, B. Zhang, and J. R. Gong, “Strong two-photon-induced fluorescence from photostable, biocompatible nitrogen-doped graphene quantum dots for cellular and deep-tissue imaging,” Nano Lett. 13(6), 2436–2441 (2013).
[Crossref] [PubMed]

Nanotechnology (1)

J. S. Ponraj, Z. Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
[Crossref] [PubMed]

Nat. Phys. (1)

B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard, “Spin qubits in graphene quantum dots,” Nat. Phys. 3(3), 192–196 (2007).
[Crossref]

Nature (2)

D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda, A. Dimiev, B. K. Price, and J. M. Tour, “Longitudinal unzipping of carbon nanotubes to form graphene nanoribbons,” Nature 458(7240), 872–876 (2009).
[Crossref] [PubMed]

Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438(7065), 201–204 (2005).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (1)

T. Fang, A. Konar, H. Xing, and D. Jena, “Mobility in semiconducting graphene nanoribbons: Phonon, impurity, and edge roughness scattering,” Phys. Rev. B 78(20), 205403 (2008).
[Crossref]

Phys. Rev. Lett. (1)

P. G. Silvestrov and K. B. Efetov, “Quantum dots in graphene,” Phys. Rev. Lett. 98(1), 016802 (2007).
[Crossref] [PubMed]

Phys. Scr. (1)

M. Ghaffarian and F. Ebrahimi, “The linear and nonlinear optical properties of trigonal zigzag graphene nanoflakes,” Phys. Scr. 88(2), 025703 (2013).
[Crossref]

Sci. Rep. (1)

X. Feng, Z. Li, X. Li, and Y. Liu, “Giant Two-photon Absorption in Circular Graphene Quantum Dots in Infrared Region,” Sci. Rep. 6(6), 33260 (2016).
[Crossref] [PubMed]

Science (3)

A. K. Geim, “Graphene: status and prospects,” Science 324(5934), 1530–1534 (2009).
[Crossref] [PubMed]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312(5777), 1191–1196 (2006).
[Crossref] [PubMed]

Sensors (1)

P. Suvarnaphaet and S. Pechprasarn, “Graphene-based materials for biosensors: a review,” Sensors 17(10), 2161 (2017).

Small (1)

X. T. Zheng, A. Ananthanarayanan, K. Q. Luo, and P. Chen, “Glowing graphene quantum dots and carbon dots: properties, syntheses, and biological applications,” Small 11(14), 1620–1636 (2015).
[Crossref] [PubMed]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1 (a) Schematic of a monolayer rectangular GQD, the black and red circles are two type of carbon atoms A and B. a and b are unit vectors. (b) The reciprocal lattice.
Fig. 2
Fig. 2 Energy spectra of rectangular GQDs with different sizes as a function of kx.
Fig. 3
Fig. 3 Three energy states (1, 1), (1, 2) and (1, 3) of rectangular GQDs with varied M and a fixed N = 10.
Fig. 4
Fig. 4 Two-photon absorption spectra for non-3M0 sized GQDs plotted as a function of incident photon energy.
Fig. 5
Fig. 5 TPA spectra for 3M0-sized metallic rectangular GQDs with M = 9, 12, 15, 18 and N = 15, 20, 25. The dashed line is the TPA spectra contributed from the transitions between (1, n).

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

HΨ= v F ( 0 k x i k y 0 0 k x +i k y 0 0 0 0 0 0 k x +i k y 0 0 k x i k y 0 )( ψ A ψ B ψ A ' ψ B ' )=E( ψ A ψ B ψ A ' ψ B ' )
{ Ψ A (r)= e iΚr ψ A (r)+ e iΚ'r ψ A ' (r) Ψ B (r)= e iΚr ψ B (r)+ e iΚ'r ψ B ' (r) .
{ Ψ A (y=0)=0 Ψ B (y= L AC )=0
{ Ψ A (x=0)= Ψ A (x= L ZZ )=0 Ψ B (x=0)= Ψ B (x= L ZZ )=0 .
( ψ A ψ B ψ A ' ψ B ' )=( C e i k x x sin( k y y) C e i k x x sin( θ k + k y y) C e i k x x sin( k y y) C e i k x x sin( θ k + k y y) )
C= [ 2 L ZZ L AC + L ZZ sin(2 θ k )/ k y ] 1/2 .
E=± v F | k |=± v F k x 2 + k y 2
{ ( K 0 + k x ) L ZZ = m k π θ k + k y L AC = n k π
H v,i int = m 1 , n 1 |(e v F /c)Aσ| m 0 , n 0 = C 0 C 1 eA v F c L AC L ZZ 2 δ m 1 , m 0 ×{ e [ sin( n 0 π n 1 π+ θ k1 )sin θ k0 n 0 π θ k0 n 1 π+ θ k1 sin( n 0 π+ n 1 π θ k1 )sin θ k0 n 0 π θ k0 + n 1 π θ k1 ] + e + [ sin( n 1 π n 0 π+ θ k0 )sin θ k1 n 1 π θ k1 n 0 π+ θ k0 sin( n 1 π+ n 0 π θ k0 )sin θ k1 n 1 π θ k1 + n 0 π θ k0 ]}

Metrics