## Abstract

We investigate theoretically the size-dependence of two-photon absorption (TPA) for circular graphene quantum dots (GQDs) on the basis of electronic energy states obtained by solving the Dirac-Weyl equation analytically under infinite-mass boundary condition. The analytical expressions for TPA coefficient are derived with an arbitrary size-distribution and the transition selection rules are obtained. Results reveal that the intraband transitions in conduction band and valence band contribute much more to TPA than interband transitions. The energy spectrum and TPA peaks are tuned by the size of GQDs.

© 2016 Optical Society of America

## 1. Introduction

Graphene, which is composed of a one-atom-thick two-dimensional honeycomb lattice of carbon atoms, has attracted intense attention due to its extraordinary physical and chemical properties, such as high intrinsic mobility, thermal stability, and electrical conductivity [1–3]. However, owing to the lack of a bandgap, its electronic and opto-electronic applications have been limited to some extent [4]. Recent advances in material process technology have enabled the preparation of graphene quantum dots (GQDs) with desired size, edge and shape [5,6]. Due to the quantum confinement and edge effect, the bandgap in GQDs is tunable by changing process parameters. Also, GQDs have advantages over conventional semiconductor QDs in terms of low toxicity, reduced photobleaching, good solubility and high biocompatibility. Therefore, GQDs have great potentials in light emitting, biomolecular sensing and cellular bioimaging applications [7–9]. In these applications, two-photon absorption (TPA) is an effective nonlinear optical process in laser excitation, owing to the advantages of longer excitation wavelength, deeper penetration depth and higher spatial resolution [10].

There have been tremendous advances in theoretical investigations into electronic structure of GQDs with kinds of shapes (such as circular, hexagonal, triangular and so on) under different boundary condition (zigzag, armchair, infinite-mass) utilizing the tight-binding model or continuum approach respectively [11–13]. As far as the optical properties are concerned, most researches focused on linear optical properties. J. Peng *et al* experimentally measured both the linear absorption spectra range from UV to visible and PL spectra of several different sized GQDs. And they found that the bandgap and photoluminescence can be tailored through varying the size of GQDs [14]. R. P. Choudhary *et al* conducted the measurements of UV-vis absorption, PL and infrared spectra in few layered GQDs. The shift in the emission spectrum and disappearance of n→π* transition in the absorption spectrum on reduction of the ablated samples confirmed the formation of GQDs [15]. The interband linear optical absorption spectrum in a circular GQD in the presence of an external magnetic field perpendicular to the dot was theoretically obtained both for infinite-mass and zigzag boundary condition by M. Grujić’s group [16]. Also, it is reported that the green upconversion photoluminescence of the uniform graphene oxide quantum disks (diameter = 4 nm) at ~563nm was observed by a femto-second Ti: sapphire pulse laser [17]. Up to now, however, no studies have been reported on the TPA of GQDs, in the context of both theory and experiment.

Here we report a simple analytical theory on the size-dependent TPA of circular GQDs on the basis of electronic energy states obtained by solving the Dirac-Weyl equation under infinite-mass boundary condition (IMBC), which would be a better choice if the spontaneous edge reconstruction could be suppressed by any means [18]. The analytical expressions for TPA coefficient are derived for the circular GQDs with an arbitrary size-distribution function and the transition selection rules are obtained. Both the interband transitions and intraband transition in conduction band and valence band are involved in. We find, for the first time, that the absorption peak associated with intraband transitions is much higher than that of interband transitions, as a result of the slight difference of energy level spaces in either conduction band or valence band. Also, the intraband TPA resonance occurs with lower incident photon energy. Due to the quantum size effect, there is a red shift for the absorption peak and the magnitudes of the TPA coefficient increases with the increase of GQD’s radius. These theoretical analyses are of much importance to the applications based on two-photon fluorescence imaging as well as academic interest.

## 2. Theory

#### 2.1 Electron energy levels and wave functions

We consider an isolated circular quantum dot with radius *R* made of monolayer graphene (Fig. 1). The Dirac-Weyl Hamiltonian for low-energy electron states in GQD, in the absence of external field, reads

*Hψ*(

*r*,

*θ*) =

*Ε*(

*r*,

*θ*) in cylindrical coordinates with the wave function being a two-component spinor,

*ψ*(

*r*,

*θ*) = [

*ψ*

_{1}(

*r*,

*θ*),

*ψ*

_{2}(

*r*,

*θ*)]

^{T}.

*v*is the Fermi velocity, and

_{F}**σ = (**σ

*, σ*

_{x}

_{y}**)**are Pauli matrices, which takes into account contributions of two different graphene sublattices. The parameter

*τ*takes the two values ± 1 distinguished the two valleys

*K*and

*K*. We assume that a mass-related potential energy

^{’}*V*(

*r*) is coupled to the Hamiltonian [12]. The mass in the dot is zero, but trends to infinity at the edge of the dot, i.e.

*V*(

*r*) = 0 for

*r*<

*R*, and

*V*(

*r*) → ∞ for

*r*≥

*R*. Therefore the Klein tunneling effect at the interface between the internal and external regions of the dot can be avoided and, carriers will be confined. This infinite-mass boundary condition can be introduced in the Dirac equation by defining

*ψ*

_{2}(

*R*,

*θ*)/

*ψ*

_{1}(

*R*,

*θ*) =

*iτe*. Due to its simplicity, this type of boundary condition has been used in the study of circular graphene dots and rings [12,19,20]. Many theoretical researches in graphene as well as conventional semiconductor quantum dots also adopt this boundary condition which is verified by the experimental data [21]. Since the operator for the total angular momentum is conserved quantity, [

^{iθ}*H*,

*J*] = 0, the two-component wave function has the form:

_{z}*χ*

_{1}(

*r*) in

*K*valley,

*χ*

_{2}(

*r*), we can getwhere

*D*is the normalization constant,

_{m}*ε*is related to the carrier energy

*E*by$\epsilon =E/\hslash {v}_{F}={x}_{m}^{n}/R$, where ${x}_{m}^{n}$the

*n*th root of the

*m*th-order Bessel function. It is obvious that the electron energies are dependent on the size of the dot. We show below that the states in the circular GQD are labeled by the orbital quantum number

*m*and the principal quantum number

*n*in

*K*(

*K’*) valley. Therefore, it is convenient to denote them by the symbol (

*m*,

*n*). In discussing the various properties of the energy spectrum in the continuum model we use the notation ${E}_{\tau ,m,n}^{p}$, where

*p*=

*e*(

*h*) denotes electron (hole) eigenvalues. The infinite-mass boundary condition gives the relation $\tau {J}_{m}(\epsilon R)={J}_{m+1}(\epsilon R).$ Together with the properties${J}_{m}(x)={(-1)}^{m}{J}_{-m}(x)$ and ${J}_{m}(x)={(-1)}^{m}{J}_{m}(-x)$ which are obeyed by the Bessel functions, we derived several interesting properties: (1) unlike zigzag boundary condition, the electron-hole symmetry under IMBC is destroyed [18]. A similar symmetry is expressed as ${E}_{\pm 1,m,n}^{e}=-{E}_{\pm 1,-(m+1),n}^{h}$; (2) energy spectrum for intervalley is represented as ${E}_{+1,m,n}^{e,h}={E}_{-1,-(m+1),n}^{e,h}$; (3) these two relations above further indicate intervalley electron-hole symmetry ${E}_{\pm 1,m,n}^{e}=-{E}_{\mp 1,-(m+1),n}^{h}$

_{.}

#### 2.2 Two-photon absorption coefficient

Two-photon absorption is a process wherein two photons are absorbed simultaneously from the initial state through one intermediate state to the final state. The two-photon generation rate with incident light frequency *ω* can be represented in second-order perturbation theory with respect to the electron-hole interaction as [21]

*E*,

_{i}*E*,

_{f}*E*represent the energies of the initial, final and intermediate states of an electron, respectively.

_{v}*H*= (

^{int}*ev*/

_{F}*c*)

**A·σ**describes the electron-hole interaction and

**A**=

*A*

**e**is the vector potential of the light wave with the amplitude

*A*and the polarization vector

**e**, and

*γ*is the inverse of the lifetime in excited state. Based on the electron energy state we derived above, the expression of matrix elements for one-photon transition from initial state (

*m*

_{0},

*n*

_{0}) to intermediate state (

*m*

_{1},

*n*

_{1}) in

*K*valley can be written as

**e**

*,*

_{x}**e**

*are the Cartesian components of the polarization vector),*

_{y}*δ*is the Kronecker delta function. It is obvious that the matrix elements for one-photon transition are independent on

_{i,j}*R*. Similar matrix elements for transition from intermediate state to final state can be obtained as well. It is easily found that a two-photon transition can only occur from initial state (

*m*

_{0},

*n*

_{0}) via intermediate state (

*m*

_{1},

*n*

_{1}) to final state (

*m*

_{2},

*n*

_{2}) for which the quantum numbers of the electron satisfy the relations

*m*

_{1}-

*m*

_{0}= ± 1 and

*m*

_{2}-

*m*

_{1}= ± 1, ∆n = 0. These are the selection rules for a two-photon transition. For interband transitions from valence band to conduction band, both the states in valence band (

*v*) and in conduction band (

*c*) can be as the intermediate state. Therefore, each integrated 2PA process from the valence band to conduction band contains two possibilities, i.e.

*v*

_{1}→

*c*

_{1}→

*c*

_{2}, and

*v*

_{1}→

*v*

_{2}→

*c*

_{1}. Combining Eq. (7) with Eq. (8), the two-photon generation rate for interband transitions can be rewritten as

*T*(

*R*) function is left because there is only one transition possibility for intraband transitions. Obviously the two-photon generation rate depends upon the size by means of

*T*(

*R*) function. However, the current state-of-the-art nanotechnology gives very little scope to manufacture GQDs with a uniform size. Hence, one has to study the GQDs with an inhomogeneous size dispersion, which is characterized by a size-distribution function

*f*(

*R*). Considering size distribution, the 2PA coefficient

*β*for an ensemble of QDs is related to the average two-photon generation rate ${\overline{W}}^{(2)}$ bywhere

*N*is the GQDs concentration and

*I*is the incident radiation intensity $I={\epsilon}_{\omega}^{1/2}{\omega}^{2}{A}^{2}{(2\pi c)}^{-1}$, and

*ε*is the dielectric constant of the material at the light frequency. The size distribution results from the conditions of sample preparation. Usually, The Gaussian function is mostly used [22]. For an arbitrary function

_{ω}*f*(

*R*), the 2PA coefficient can be expressed in the SI system as follows,

## 5. Results and discussion

Following the analytical expressions derived above, we perform the calculations to predict the two-photon absorption spectra for GQDs with the shape of circular. Hereinafter calculations and discussions are carried out using the following parameters: *v _{F}* = 10

^{6}m/s,

*N*= 3 × 10

^{24}m

^{−3}, ℏ

*γ*= 1 meV,

*ε*= 3.

_{ω}The two-photon absorption coefficient is dependent on the energy states of electrons, so we display the energy spectrum first. Figure 2 shows the energy levels of a GQD with radius *R* = 2 nm in *K* valley as a function of angular momentum quantum number *m*. The arrows in green, blue and red stand for the interband transition, intra conduction band transition and intra valence band transition, respectively. It can be seen that the bandgap between conduction band and valence band is about 1.8 eV for this sized GQD. Several interesting properties can be observed from the energy spectrum. Firstly, the electron and hole states in Fig. 2 are related by the symmetry${E}_{m,n}^{e}=-{E}_{-(m+1),n}^{h}$. Also the energy levels in conduction band are of bilateral symmetry by${E}_{m,n}^{e}={E}_{-m,n}^{e}$, while in valence band the symmetry is${E}_{m,n}^{h}={E}_{-(m+2),n}^{h}$. This feature is coincident with our theoretical analysis in previous section. Secondly, the spaces between energy levels in either conduction band or valence band change slightly, which is quite different from conventional semiconductor QDs. The energy level spacing in conventional semiconductor QDs increases with the increasing of quantum numbers [21].

In order to investigate the size dependence of energy levels in GQDs, results for the energy levels as a function of the dot radius are shown in Fig. 3 with *m* = 0. The spectrum shows a 1/*R* dependence which can be linked to the relation $E={x}_{m}^{n}\hslash {v}_{F}/R$ we derived in section 2. The larger the radius is, the lower the energy level for electron is. Especially for the QDs with small radius (< 2 nm), the electron energy drops significantly with the increase of radius. Actually, if the radius trends to infinite, the ground states of electron and hole approach to 0. Then the band gap vanishes. It happens to be the situation of monolayer graphene.

By applying the TPA model presented in the previous section, the TPA spectra associated with interband transitions, intra conduction band transitions as well as intra valence band transitions for 2nm-average-sized GQDs with a Gaussian size distribution (FWHM = 20%) are shown in Figs. 4(a) and 4(c). All the transitions satisfied selection rules for −10≤*m*≤10 and 1≤*n*≤10 are included in our calculation. It is found that the TPA coefficient associated with intra band transitions is 8 orders of magnitude greater than that of interband transitions. It can be explained by the tiny changed energy level spaces in conduction or valence band, which can be seen obviously in Fig. 2. This means a large number of two photon transition resonance can occur easily, while for interband transition, the resonance occurs much more difficulty because of the bandgap between conduction band and valence band. In addition, there is a difference of absorption peak value position between intraband transitions and interband transitions. We can see that the absorption peak for interband transition is at 0.88 eV (~1411 nm), while for intra conduction band is at 0.37 eV (~3357 nm) and 0.39eV (~3185 nm) for intra valence band. It is predicted that for GQDs with a fixed size, if the incident light wavelength is varied from a lower to a higher one, two photon resonance associated with interband transition occurs with lower wavelength. The positions of absorption peaks in the TPA spectra are determined by the energy denominators in the term T($\Re $) in Eq. (11). In order to explain why the peak locates here, Figs. 4(b) and 4(d) demonstrate the low-energy spectra of the form function *F*(*R*) which is nonzero when TPA resonance occurs. We can find that the allowed TPA transitions are squeezed to the particular photon energy, which is corresponding to the absorption peak. The minor differences of TPA spectra between intra conduction band and intra valence band result from the non absolute symmetry of energy states in conduction band and valence band.

Figures 5(a)-5(c) demonstrates the simulated photon energy-dependent TPA spectra for GQDs with three different radii: *R* = 1.8, 2.0 and 2.2 nm, associated with interband transitions, intra conduction band transitions and intra valence band transitions, respectively. It is found that with the increasing of the GQD’s radius, there is a red shift for the absorption peak. This is owing to the fact that as the consequence of quantum size effect, energy differences become smaller when *R* increases, which can be seen in Fig. 3. Also, with the increase of *R*, the magnitude of TPA coefficient *β* increases too. This also results from the quantum size effect. 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 [21].

## 4. Conclusion

In conclusion, by solving the Dirac-Weyl equation analytically under infinite-mass boundary condition, we have obtained the energy spectrum of a circular GQD. There is a bandgap between electron and hole energy levels. The energy states of electrons and holes are of symmetry in some extent. On the basis of energy levels and electronic states, the analytical expressions for the size-dependent two-photon absorption coefficient associated with interband as well as intraband transitions have been deduced taking the size distribution into account. The TPA spectra reveal that the absorption peak resulted from the intraband transition is much higher than that from interband transition, and the intraband TPA resonance occurs with lower incident photon energy. With the increase of GQD’s radius, there is a red shift for the absorption peak and the magnitudes of the TPA coefficient *β* increases too. These theoretical analyses are of great importance to applications based on two-photon fluorescence imaging as well as academic interest. In our future work, we may conduct the theoretical researches on TPA in GQDs under zigzag and armchair boundary condition where the electronic states are of great differences.

## Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (Grant No. 11304275, No. 11164034), Yunnan Provincial Department of Education (2013Z013), Yunnan Province Natural Science Foundation (Grant No.2010DC053), the Key Applied Basic Research Program of Science Technology Commission Foundation of Yunnan Province (Grant No. 2013FA035), and Innovative Talents of Science and Technology Plan Projects of Yunnan Province (Grant No. 2012HA007, No. 2014HB0010).

## References and links

**1. **Y. Zhu, S. Murali, W. Cai, X. Li, J. W. Suk, J. R. Potts, and R. S. Ruoff, “Graphene and graphene oxide: synthesis, properties, and applications,” Adv. Mater. **22**(35), 3906–3924 (2010). [CrossRef] [PubMed]

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

**3. **A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. **6**(3), 183–191 (2007). [CrossRef] [PubMed]

**4. **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]

**5. **K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature **438**(7065), 197–200 (2005). [CrossRef] [PubMed]

**6. **H. Hiura, “Tailoring graphite layers by scanning tunneling microscopy,” Appl. Surf. Sci. **222**(1–4), 374–381 (2004). [CrossRef]

**7. **L. Tang, R. Ji, X. Cao, J. Lin, H. Jiang, X. Li, K. S. Teng, C. M. Luk, S. Zeng, J. Hao, and S. P. Lau, “Deep ultraviolet photoluminescence of water-soluble self-passivated graphene quantum dots,” ACS Nano **6**(6), 5102–5110 (2012). [CrossRef] [PubMed]

**8. **Y. Zhang, C. Wu, X. Zhou, X. Wu, Y. Yang, H. Wu, S. Guo, and J. Zhang, “Graphene quantum dots/gold electrode and its application in living cell H_{2}O_{2} detection,” Nanoscale **5**(5), 1816–1819 (2013). [CrossRef] [PubMed]

**9. **H. Sun, L. Wu, W. Wei, and X. Qu, “Recent advances in graphene quantum dots for sensing,” Mater. Today **16**(11), 433–442 (2013). [CrossRef]

**10. **G. S. He, K. T. Yong, Q. Zheng, Y. Sahoo, A. Baev, A. I. Ryasnyanskiy, and P. N. Prasad, “Multi-photon excitation properties of CdSe quantum dots solutions and optical limiting behavior in infrared range,” Opt. Express **15**(20), 12818–12833 (2007). [CrossRef] [PubMed]

**11. **M. Zarenia, A. Chaves, G. A. Farias, and F. M. Peeters, “Energy levels of triangular and hexagonal graphene quantum dots: a comparative study between the tight-binding and the Dirac equation approach,” Phys. Rev. B Condens. Matter **84**(24), 245403 (2011). [CrossRef]

**12. **S. Schnez, K. Ensslin, M. Sigrist, and T. Ihn, “Analytic model of the energy spectrum of a graphene quantum dot in a perpendicular magnetic field,” Phys. Rev. B Condens. Matter **78**(19), 195427 (2008). [CrossRef]

**13. **D. R. da Costa, A. Chaves, M. Zarenia, J. M. Pereira Jr, G. A. Farias, and F. M. Peeters, “Geometry and edge effects on the energy levels of graphene quantum rings: a comparison between tight-binding and simplified Dirac models,” Phys. Rev. B Condens. Matter **89**(7), 075418 (2014). [CrossRef]

**14. **J. Peng, W. Gao, B. K. Gupta, Z. Liu, R. Romero-Aburto, L. Ge, L. Song, L. B. Alemany, X. Zhan, G. Gao, S. A. Vithayathil, B. A. Kaipparettu, A. A. Marti, T. Hayashi, J. J. Zhu, and P. M. Ajayan, “Graphene quantum dots derived from carbon fibers,” Nano Lett. **12**(2), 844–849 (2012). [CrossRef] [PubMed]

**15. **R. P. Choudhary, S. Shukla, K. Vaibhav, P. B. Pawar, and S. Saxena, “Optical properties of few layered graphene quantum dots,” Mater. Res. Express **2**(9), 095024 (2015). [CrossRef]

**16. **M. Grujić, M. Zarenia, M. Tadić, and F. M. Peeters, “Interband optical absorption in a circular graphene quantum dot,” Phys. Scr. **T149**, 014056 (2012). [CrossRef]

**17. **H. D. Ha, M. H. Jang, F. Liu, Y. H. Cho, and T. S. Seo, “Upconversion photoluminescent metal ion sensors via two photon absorption in graphene oxide quantum dots,” Carbon **81**(1), 367–375 (2015). [CrossRef]

**18. **M. Grujić and M. Tadić, “Electronic states and optical transitions in a graphene quantum dot in a normal magnetic field,” Serbian J. Electrical Eng. **8**(1), 53–62 (2011). [CrossRef]

**19. **D. S. L. Abergel, V. M. Apalkov, and T. Chakraborty, “Interplay between valley polarization and electron-electron interaction in a graphene ring,” Phys. Rev. B Condens. Matter **78**(19), 193405 (2008). [CrossRef]

**20. **P. Recher, B. Trauzettel, A. Rycerz, Y. M. Blanter, C. W. J. Beenakker, and A. F. Morpurgo, “Aharonov-Bohm effect and broken valley-degeneracy in graphene rings,” Phys. Rev. B Condens. Matter **76**(23), 235404 (2007). [CrossRef]

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

**22. **W. Y. Wu, J. N. Schulman, T. Y. Hsu, and U. Efron, “Effect of size nonuniformity on the absorption spectrum of a semiconductor quantum dot system,” Appl. Phys. Lett. **51**(10), 710–712 (1987). [CrossRef]