Third order optical nonlinearity of graphene

We perform a perturbative calculation of the third order optical conductivities of doped graphene, using approximations valid around the Dirac points and neglecting effects due to scattering and electron–electron interactions. In this limit analytic formulas can be constructed for the conductivities. We discuss in detail the results for third harmonic generation, the Kerr effect and two-photon carrier injection, parametric frequency conversion, and two-color coherent current injection. We find a complicated dependence on the chemical potential and photon energies. The linear dispersion causes resonances over a wide range of photon energies, and it is possible to obtain large optical nonlinearities by tuning the chemical potential.

( ) x »+ , respectively. We thank Dr Ilya Shadrivov for pointing out an error in generating the curves of  (20) and other equations. The correct diagram is given as follows.
These corrections do not impact other discussions in the paper, and do not affect the conclusions.

Introduction
Due to the gapless linear dispersion of its low energy electronic excitations, graphene exhibits unique and remarkable optical properties: the linear optical response is characterized by a universal conductivity [1] σ =  e 4 0 2 at wavelengths from the mid-infrared to the visible, with a monolayer absorbance about ∼2.3% [2]. Thus while a single layer of graphene is highly transparent, considered as a monolayer it is a highly absorbing material, and saturated absorption is easy to achieve [3]. With its strong optical coupling, broadband absorption, and other novel material and electronic properties [4], graphene is a natural candidate for use in optically controlled devices in photonics and optoelectronics [5,6]. In moving towards any application, an important step is understanding the optical nonlinearities of graphene. The graphene structure has center-of-inversion symmetry, and while second-order nonlinear response can arise from interface effects [7], nonuniformity of optical field [8], or the presence of dc currents [9], the first nonvanishing nonlinear susceptibility arising in pristine graphene is the third order susceptibility [10], and that is the focus of this paper.
Third order nonlinearities in graphene have been experimentally investigated; however, the extracted effective bulk susceptibilities χ eff (3) values show discrepancies and strongly depend on measurement method, light frequency, and perhaps sample preparation. In their pioneering work on the nonlinear response of graphene, Hendry et al [11] presented an expression for χ eff (3) for four-wave mixing experiments, and then approximated their expression to find a value of about − − 10 m V 15 2 2 in the near-infrared, weakly dependent on frequency; they found agreement between experiment and their approximated expression by comparing their signal from graphene with that from a gold film. Using a sample with about 100 layers of graphene, Wu et al [12] found agreement with this approximated expression as well. However, in fact a direct calculation from the expression of Hendry et al gives a value only about − − 10 m V 19 2 2 3 . Yang et al [13] observed no detectable two-photon absorption in monolayer graphene by the Z-scan method at a wavelength of 780 nm, which limits χ ⎡ ⎣ ⎤ ⎦ Im eff (3) to less than − − 10 m V 18 2 2 . The same experimental technique was used by Zhang et al [14] to measure the nonlinear refractive index at 1.55 μm, and they found ∼ − − n 10 m W 1 obtained at a similar wavelength by Gu et al [6], which matches a theoretical prediction using Hendryʼs original expression. Kumar et al [15] found a value of about × − − 8 10 m V 17 2 2 by measuring third harmonic generation at a fundamental wavelength of 1720 nm, while Hong et al [16] found a value about two orders of magnitude smaller for third harmonic photon transitions at the M points in graphene. Sun et al [17] showed that two-color coherent injected currents can induce an observable terahertz radiation signal. Theoretically, besides the expression provided by Hendry et al, Rioux et al [18] investigated two-photon absorption and two-color coherent by using Fermiʼs Golden Rule for pristine graphene, and Jafari [19] calculated the nonlinear optical response in gapped graphene by adding a mass term. Zhang et al [20] used the density matrix method to study four wave mixing at the saturation regime in undoped graphene, and found an effective . We mention that the experimental comparison they made with the nonlinear response of gold could be questioned, since while their experiments on gold and graphene were done at 820 nm with 6 ps pulses, the reference value they used for the χ ( ) 3 of gold came from experiments done at 532 nm with 35 ps pulses; the effective nonlinear optical response of metals is well-known to be strongly dependent on wavelength [37] and pulse duration. with increasing light intensity. A giant optical nonlinearity [21][22][23] in the presence of a strong magnetic field has also been calculated from a density matrix formalism. Third and higher harmonic generations from intraband contributions have been investigated in detail in the THz regime [10,24,25]. Yet a general calculation of even third order nonlinearities at higher frequencies, including the optical regime where interband contributions can play a central role, is still lacking.
A full calculation of the nonlinear response, even in the perturbative limit, would require the inclusion of electron-electron and electron-phonon scattering in the construction of a selfenergy. Thermal effects caused by a high repetition rate of laser pulses, as used in Z-scan experiments, also cannot be ignored [26,27]. In this paper, we neglect these complications, restrict ourselves to frequencies such that we can use approximations relevant around the Dirac cone, and perform a simple, zero-temperature perturbative calculation at the independent particle level, but fully taking into account both the interband and intraband motion of the electrons as perturbed by an incident field. The advantage of such a simplified approach is that we can obtain analytic results for the third order conductivity σ ω ω ω ( ) , , dabc (3); 1 2 3 in doped graphene; one can then determine an effective bulk susceptibility [11] by ϵ 0 being the vacuum permittivity, and ≈ d 3.3 gr Å an effective thickness of graphene. These elementary but analytic results allow us to explore predictions for a number of third order nonlinear optical effects; they provide both an indication of what ranges of parameter space would be interesting to explore experimentally, and a benchmark for more sophisticated calculations.
For undoped graphene, we find that this model leads to a simple expression for the fully symmetrized conductivity where v F is the Fermi velocity. For doped graphene with a chemical potential μ, however, our formulas show divergences related to the resonances between any involved photon energy and the chemical potential gap μ 2 , with results quite different for different frequency combinations. Combined with the tunability of μ by an external gate voltage [28] or chemical doping [29] this should lead to novel approaches for controlling the nonlinear optical properties of graphene, and indeed to the possibility of graphene-based 'nonlinear optics on demand'.

Model
We describe the electronic states in the π and π * bands of graphene by a tight binding model employing carbon p 2 z orbitals; we denote by ϕ r ( ) centered at the origin. The Bloch states can be written as Moving to a continuous range of crystal wavevectors k, we write the Bloch functions as n m for any n m , . From the normalization of the where the integral is over all space and we take k and ′ k to be in the first Brillouin zone, the r u ( ) k s are normalized according to where the indicated integral is over a unit cell with area  cell in the xy plane and over all z. The Berry connections [30][31][32], are then found to be , ands the index of the band that is not the s band. At the Dirac points, χ k is singular.
Under the approximation that the optical field is treated as uniform, the light-matter interaction can in principle be described either through the use of a scalar potential or a vector potential associated with the electric field ( ) E t . Both treatments have disadvantages. The first involves introducing the position operator r, which does not have well-behaved matrix elements between the periodic Bloch functions ψ r ( ) k s . Starting with the work of Blount [30], a number of strategies and techniques have been developed for working with the effective matrix elements of the position operator that arise; these matrix elements are linked to the Berry connections [32]. Such problems do not arise if the vector potential is used to represent the electric field, but the inevitable band truncation in numerical calculations can result in false divergences [32,33]. This problem does not plague calculations based on the position operator. And while these divergences can be eliminated using proper sum rules that involve all the bands, since only two bands are included in our tight binding model we use the approach based on introducing the position operator, and take the interaction Hamiltonian to be where we take the electronic charge to be = − e e. In the independent particle approximation we adopt in this paper, the proper treatment of the position operator then leads to the description of the system by the semiconductor Bloch equations [32] . and ξ k denotes the matrix with elements ξ k s s 1 2 . The areal current density is then calculated by where the integral is over the Brillouin zone and the v k are the matrix elements of the velocity given by The direct perturbative solution of equation (8) has been discussed in detail before [32], in fact for a 3D crystal, using an approach in which interband and intraband contributions are separated as the perturbation expansion is developed. Here we introduce a simpler strategy by shifting to a moving frame that essentially follows the intraband motion of the carriers; we put where ( ) A t is the vector potential used to describe the electric field, In a rough sense we are using the vector potential to treat the intraband motion, and the scalar potential to treat the interband motion. Equation (8) The a th Cartesian component of the areal current density is then written as We wish to generate a perturbative expansion of the solution of (10), which will then allow us to write an expansion of ( ) J t in terms of powers of the electric field, Because graphene has center-of-inversion symmetry the second order term ( ) J t (2) will be identically zero in the dipole approximation.
We begin by writing (2); 1 2 , . Hereafter we keep the dependence of the terms on frequencies implicit, always linking the frequencies ω ω ω 1 2 3 to the Cartesian components a b c. The matrix function ω ( ) of (11) according to (12) we have where the linear conductivity σ ω We also find where as in equation (1)   (1); . For the third order conductivity there are in all eight nonzero components, among which three are independent; we have  , and they satisfy the symmetry relations given above.

Approximations around dirac points
The expressions given above are exact for the two-band model. However, for studying optical transitions around the Dirac points , it is usual to approximate the electronic dispersion relation as linear, i.e. ε ≈ , the velocity matrix elements as ≈ , and the interband Berry connection as ξ ≈ˆ× + z k k 2 K k ss 2 . These approximations follow immediately from equations (5), (6), and (9), and when they are made an analytic result is found for the linear optical conductivity, is a dimensionless complex function of a real variable x. Here θ ( ) y is the Heaviside step function, equal to 0 for < y 0 and 1 for > y 0. Compared to a full band structure calculation based on our tight-binding model, this analytic result gives an error less than 15% for ω <  3 eV. We also find an analytic result for the third order optical conductivity, (3); (3);

6
(1) The linear conductivity is the same as found by, e.g. Gu et al [6], and reduces to the universal conductivity σ 0 when taking the Fermi energy μ → 0. We plot σ xx (1); in figure 1(a). Due to the gapless linear dispersion, the third order conductivities contain many possible divergences as the photon energies involved-including all the ω i and their combinations-go to zero or to the doping induced gap ( μ 2 ).
In limit of zero doping, μ → 0 and . These terms all exhibit power-law divergences as photon energies vanish. For doped graphene, the chemical potential is only involved in the function ( ) G x , and a logarithmic divergence appears for suitable chemical potentials. This could be utilized as a flexible method of controlling the third order nonlinearities, by tuning the doping through adjusting a gate voltage or applying chemical doping. To present a summary of the results we scale all photon energies by the chemical potential and rewrite the third order conductivity as . This value is at the high end of the range of electric field amplitudes used in experiments.

Third harmonic generation
For third harmonic generation we set ω ω ω ω = = = . If we take μ → 0, using (1) we find the effective bulk susceptibility is χ = × xxyy is plotted in figure 1(b). The divergence at w = 2 is very weak compared to those at = w 2 3 and w = 1, due to cancellations in T(x). Setting ω =  0.72 eV, as in the experiment by Kumar et al [15], our results are χ =  (3) has the same importance as the real part, and the contribution from k points far away from Dirac points may be not ignorable. (ii) Even around the Dirac points, the widely used linear dispersion approximation, which we have adopted here, may not be adequate for an accurate calculation of optical nonlinearities. Likely more important are the following reasons: (iii) for a very low doping level, the strong light intensity used in THG measurements may cause saturated absorption, especially if excited by high-repetition-rate laser pulses, this would lead to a nonlinearity of a totally different type, beyond the perturbation approach we apply here. (iv) Thermal effects may lead to a higher effective nonlinearity than the intrinsic one calculated here; their study is beyond the scope of the present work.

Kerr effects and two-photon absorption
We now look at the frequency components ω ω ω ω = − = − = − The divergent terms are related to the optical transition shown at the left hand side in figure 2 (b) and determined by and δ δ π δ δ = +   figure 1(c). The two-photon absorption rate is written as This result agrees with calculations based on Fermiʼs Golden Rule [18] where only transitions at the right hand side of figure 2 (b) are considered and give ξ ω ∝ −  ( ) dabc 2 5 for μ ω μ < <  2 . In bulk materials the Kerr effect can be described by a nonlinear index of refraction n 2 , leading to an index of refraction given by = + n n n I 0 2 where I is the light intensity and n 0 is the index at very low intensities. With the very naïve model of graphene behaving as a thin layer of macroscopic material with effective response coefficients, we would have [34]; in these formulas we assume that the light is linearly level were much lower the experiment could have fallen in the divergent regime identified above, where we would not expect the perturbation approach adopted here would be sufficient.

Parametric frequency conversion
With a strong pump at frequency ω p , a signal photon at frequency ω s can be converted to an idler 2 . Here only the first term contributes to the injection rates. This corresponds to the interference of one-photon absorption at ω 2 with two photon absorption at ω (see the transitions at the right hand side of figure 2 (c)), and corresponds to the usual current injection observed in semiconductors where ω  2 is greater than the band gap but ω  is not. (ii) ω μ >  2 . In this regime one-photon absorption occurs at the fundamental frequency, and the second order contribution to current injection at the same position in the Brillouin zone leads to a divergent result. However, a new contribution to the coherent current injection appears and is finite; it arises from a new process of interfering pathways involve ω and ω ω − 2 transitions [36] (at the left hand side of figure 2 (c)). For experiments in semiconductors, where the fundamental energy ω  is typically less than the band gap, this regime is usually not explored. But note that in graphene the onset of this new channel leads to obvious amplitude changes in η, which could be easily detected in experiments by measuring the sudden change of the injection currents amplitude with scanning the fundamental frequencies across μ 2 .

Conclusion
We investigated the linear and third order nonlinear optical conductivity of doped graphene, at zero temperature in the tight binding model, using the semiconductor Bloch equations. Analytic results were obtained around the Dirac points by using the widely accepted linear dispersion approximation. The third order optical conductivities exhibit a complicated dependence on the photon frequencies and chemical potential. We discussed in detail third harmonic generation, Kerr effects and two-photon absorption, parametric frequency conversion, and two-color coherent current injection. A nonvanishing chemical potential mimics many of the features that result from the presence of a band gap in normal semiconductors, and divergences can result when the energy of the effective gap is matched to any of the photon energies involved. The different third order processes considered exhibit a wide range of behavior, with each process having its own signature features. The important role played by the chemical potential allows for the generation of desired nonlinearities by electrically tuning or chemical doping; for low doping levels, the easily saturated absorption may lead to a nonlinear response totally different than those calculated here in the perturbative regime. Both systematic experiments and full band structure calculations, including finite temperature effects, thermal effects, scattering processes, and saturation, are clearly required to identify the true nature of the nonlinear response. But the perturbative calculations presented here identify a host of frequency regimes where very interesting behavior can be expected, even if more sophisticated calculations are required to elucidate it.