## 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* = 3*M*_{0} ± 1 (*M*_{0} is an integer), the GQD exhibits a semiconductor while for *M* = 3*M*_{0} 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 10^{11} 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* = 3*M*_{0} ± 1 (*M*_{0} 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* = 3*M*_{0}, 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 *L _{ZZ}* =

*Ma*,${L}_{AC}=(N+\frac{1}{3})\frac{\sqrt{3}}{2}a$, 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/3

*a*, 0) and K' = (4π/3

*a*, 0) obeys the following Dirac-like equation in momentum space [19]

*v*≈10

_{F}^{6}m/s is the Fermi velocity at the Dirac points [5]. And ${k}_{x(y)}=-i{\partial}_{x(y)}$ is an operator to measure the momentum deviation from K or K' point. The total wavefunction $\Psi (r)$ of the sublattice A and B are written by the four components of the spinor wavefunction in Eq. (1) as

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

*θ*= arctan(

_{k}*k*/

_{y}*k*), and

_{x}*C*is the normalization constant determined by the normalization condition,

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

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*k*and

_{x}*k*around K point need to satisfy the following quantized condition

_{y}*m*and

_{k}*n*are integers, which give the quantized electron states. So it is convenient to denote the electron state by the symbol (

_{k}*m*,

*n*), where

*m*and

*n*stands for the orders of

*k*and

_{x}*k*. For example, for

_{y}*M*= 12, the lowest

*k*can be zero when

_{x}*m*16. So we sign (1,

_{k}=*n*) for the states for

*m*16.

_{k}=In most experiments, TPA is measured in terms of the absorption cross-section σ_{2} 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 (*m*_{0}, *n*_{0}) to intermediate state (*m*_{1}, *n*_{1}) in K valley for electron-photon interaction *H* = (*ev _{F}*/

*c*)

**A·σ**is deduced as

The transition matrix element for rectangular GQDs is in proportion to the size of GQDs *L _{AC}* and

*L*, 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 ∆

_{ZZ}*m*= 0. This is the selection rule for a two-photon absorption transition in rectangular graphene quantum dots.

_{k}## 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 ℏ*γ* = 60meV (corresponding to relaxation time 10fs), and dielectric constant at the light frequency *ε _{ω}* = 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 *L*_{ZZ} with a fixed *L*_{AC} or vary the size *L*_{AC} with a fixed *L*_{ZZ}. 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, *L*_{AC}, i.e. *N* = 5, 10, 20, while zigzag-edged size *L*_{ZZ} with *M* fixed at 10, the quantized *k _{x}* keeps the same. That is to say, in this case the varying of energy sates only results from

*k*. This can be explained by Eq. (8), which tell us

_{y}*k*is only a function of

_{x}*L*

_{ZZ}or

*M*, and

*k*depends on both

_{y}*M*and

*N.*So in Figs. 2(d), 2(b) and 2(e), even though

*N*is fixed,

*k*and

_{x}*k*both contribute to the increasement of density of state.

_{y}In Fig. 3, we plot the curves of three energy states (1, 1), (1, 2) and (1, 3) of rectangular GQDs with varied *L*_{ZZ} or *M* and a fixed *L*_{AC} for *N* = 10. Something special appears when *M* = 3*M*_{0} with *M*_{0} an integer. First of all, when *M* = 3*M*_{0}, the rectangular GQD is metallic since *k _{x}* and

*k*can be both zero then the lowest energy state (1, 1) is zero. While for

_{y}*M*= 3

*M*

_{0}± 1 it is always a semiconductor since

*k*and

_{x}*k*cannot be both zero. Secondly, the states (1,

_{y}*n*) remain unchanged provided

*M*= 3

*M*

_{0}with fixed

*N*even though

*M*is changed. Deriving from Eq. (8), when

*k*= 0 we can deduce

_{x}*k*= 0 or

_{y}*k*= (l + 1/2)π/

_{y}*L*with

_{AC}*l*an integer. So, the energy difference between states (1, 1) and (1, 2) is ℏ

*v*π/(2

_{F}*L*) while the energy differences between other neighboring states are ℏ

_{AC}*v*π/

_{F}*L*. These equidistant energy levels provide the possibility of resonance for the two-photon absorption transitions.

_{AC}Figure 4 shows the size-dependent TPA spectra of rectangular GQDs with non-3*M*_{0} 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} m^{4}s/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.

We also demonstrate the TPA spectra for 3*M*_{0}-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 *k _{x}* = 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

*L*, i.e.

_{AC}*N*, is fixed, the energy level spaces between the (1,

*n*) states are confirmed to be ℏ

*v*π/(2

_{F}*L*) or ℏ

_{AC}*v*π/

_{F}*L*. So the position of absorption peak will be estimated at ℏω = ℏ

_{AC}*v*π/(2

_{F}*L*) and ℏ

_{AC}*v*π/

_{F}*L*around. If we alter the size of armchair-edged dimension

_{AC}*L*, i.e.

_{AC}*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.

## 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* = 3*M*_{0} ± 1 (*M*_{0} 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* = 3*M*_{0}, 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).

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