LETTER pubs.acs.org/NanoLett
Giant Two-Photon Absorption in Bilayer Graphene Hongzhi Yang,† Xiaobo Feng,†,‡ Qian Wang,† Han Huang,† Wei Chen,†,§ Andrew T. S. Wee,† and Wei Ji*,† †
Department of Physics, National University of Singapore, Singapore 117542 School of Physics and Electronic Information Technology, Yunnan Normal University, Kunming, China 650092 § Department of Chemistry, National University of Singapore, Singapore 117543 ‡
bS Supporting Information ABSTRACT: We present a quantum perturbation theory on two-photon absorption (2PA) in monolayer and bilayer graphene which is Bernalstacked. The theory shows that 2PA is significantly greater in bilayer graphene than monolayer graphene in the visible and infrared spectrum (up to 3 μm) with a resonant 2PA coefficient of up to ∼0.2 cm/W located at half of the bandgap energy, γ1 = 0.4 eV. In the visible and terahertz region, 2PA exhibits a light frequency dependence of ω3 in bilayer graphene, while it is proportional to ω4 for monolayer graphene at all photon energies. Within the same order of magnitude, the 2PA theory is in agreement with our Z-scan measurements on high-quality epitaxial bilayer graphene deposited on SiC substrate at light wavelength of 780 and 1100 nm. KEYWORDS: Bilayer graphene, two-photon absorption, second-order perturbation thoery, Z-scan, ballistic photocurrent
G
raphene exhibits unique one-photon absorption (1PA) features that have attracted a great deal of research interest.13 Recently, two-photon absorption (2PA) in graphene and graphite has also received attention.4,5 Quantum interference between 1PA and 2PA has been demonstrated4 to provide a solution to the coherent control and noncontact generation of ballistic photocurrent in multilayer graphene, which have applications in quantum technology. Furthermore, 2PA in graphene is one of the fundamental yet relevant processes if graphene is considered as a material candidate for next-generation photonic and optoelectronic devices. Despite such potentials and importance, up to date there is no report on the characteristics and magnitudes of 2PA in graphene. Here, we present a quantum perturbation theory and experimental observation of frequency-degenerate 2PA in graphene. Our findings show that 2PA is significantly different and greater in Bernal-stacked (or AB-stacked) bilayer graphene than in monolayer graphene. Such significant difference provides a practically efficient and explicit way of indentifying the stacking orders of bilayer graphene. Our theory is limited to both undoped monolayer and bilayer graphene since their band structures and 1PA have been wellunderstood.13,615 As illustrated in Figure 1, 2PA is an interband transition wherein two photons, each of energy pω, are absorbed simultaneously, thereby creating an electron in the conduction band and a hole in the valence band. In monolayer graphene, the electronic dispersion of the conduction (π*) or valence (π) band can be written as E((k) = (pυF|k|, due to the FermiDirac-like nature of the two-dimensional atomic layer.15,16 The electron wave function ψ(r) close to the K point, obeys the two-dimensional Dirac equation as follows15 iυF σ 3 rψðrÞ ¼ EψðrÞ r 2011 American Chemical Society
ð1Þ
In the momentum space, for momentum around the K point, the wave function has the form of15 ! 1 eiθk =2 ψ( ðkÞ ¼ pffiffiffi ð2Þ iθ =2 2 (e k where the ( signs correspond to the eigenenergies for the π* and π band, respectively, and θk is given by θk = arg(kx þ iky). By using the second-order perturbation theory of quantum mechanics, the second-order interband transition probability rate per unit area (W2) can be written as17 2 Z 2π Æψþ ðkÞjHjψi ðk0ÞæÆψi ðk0ÞjHjψ ðkÞæ W2 ¼ δ½Eþ ðkÞ p i Ei ðk0Þ E ðkÞ pω
∑
d2 k E ðkÞ 2pω ð2πÞ2
ð3Þ
where ψ and ψþ are the electronic wave function for the initial and final state of the transition, respectively; ψi is the all-possible intermediate state; and H is the interaction Hamiltonian: H = (e/c)A 3 υFσ, where A = Ae is the vector potential of the light wave with the amplitude A and the polarization vector e. The 2 2 amplitude A is related to the light irradiance by I = ε1/2 ω ω A 1 (2πc) . The density-of-state argument is implicitly included in the integration of k in eq 3.18 By including all possible intermediate states in the two bands, we find an analytical expression for the 2PA Received: February 19, 2011 Revised: May 15, 2011 Published: June 08, 2011 2622
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Figure 1. The 2PA spectra of monolayer and bilayer grapehene. Inset shows four possible transitions in bilayer graphene.
coefficient (β = 4W2pω/I2) as follows β
monolayer
4π2 υF e2 ¼ 3 4 c εω ω p
!2 ð4Þ
As for AB-stacked bilayer graphene, it is differentiated from monolayer graphene because bilayer possesses two more bands. In the continuum limit, by expanding momentum close to the K point in the Brillouin Zone (BZ), the Hamiltonian in the momentum space can be written as a matrix as follows15 0 1 0 3γ3 ak V υF kp B C B υF kp V γ1 0 C C ð5Þ HK ¼ B B 0 V υF kp C γ1 @ A 0 υF k p V 3γ3 ak where γ1 is the hopping energy between atom A1 and atom A2, γ3 is the hopping energy between atom B1 and atom B2 in the graphite nomenclature; and V is half the shift in electrochemical potential between the two layers. If V = 0, γ3 = 0, and υFkp , γ1 (= 0.4 eV), eq 5 gives rise to four parabolic bands, as shown in the inset of Figure 1. Two bands (E1 ≈ (p2υ2Fk2/γ1) touch each other at k = 0, while other two bands (E2 ≈ (γ1 ( p2υ2Fk2/γ1) have a minimum energy of (γ1 (or a bandgap energy of 2γ1) at k = 0. The electron effective mass can be derived as m = γ1/(2υ2F). There are four possible transitions, as shown in the inset of Figure 1. All the four transitions are included in the calculation of 2PA coefficient if pω > γ1, which leads to an analytical expression as follows " 2 pω 1 1 2 2 bilayer monolayer þ ¼ 8β 16p ω β γ1 pω γ1 þ pω 2 2 1 1 2 þ þ ðpω γ1 Þ þ pω pω þ γ1 pω γ1 2 1 1 þ þ 16ðpω γ1 Þ2 pω pω γ1 2 2 1 1 þ þ ð2pω γ1 Þ2 þ ð6Þ pω pω þ γ1 pω γ1 where the four terms in the square bracket correspond to Transitions 14, respectively, in the inset of Figure 1. There are
four possibilities of interband transitions for incident photon energy pω > γ1: from the lower valence band (E2) to the conduction band (E1þ or E2þ), and from the upper valence band (E1) to the conduction band (E1þ or E2þ). As they all satisfy the law of energy conservation, they all contribute to the 2PA coefficient. If pω < γ1/2, only Transition 1 is possible because three other transitions do not satisfy the law of energy conservation. As such, the first term in eq 6 remains and the other terms disappear. If γ1/2 < pω < γ1, Transitions 1, 2, and 4 meet the law of energy conservation but Transition 3 does not. In this case, eq 6 includes the three terms and the third term is omitted. Owing to two more bands originated from the interlayer interaction in bilayer, which considerably enhances two-photon transition probability, eq 6 shows that 2PA in bilayer is significantly different from monolayer with the following characteristics: (i) there is a resonant feature (or singularity) centered at pω = γ1; (ii) 2PA in monolayer has a ω4 dependence for all photon frequencies, while 2PA in bilayer becomes proportional to ω3 in the off-resonant regions of pω . γ1 or pω , γ1/2; and (iii) in the computation of eq 6, there is a difficulty at pω = γ1 as it is a singularity. To avoid this problem, a common approach is to introduce a phenomenological term called the line broaden factor, Γ, which is inversely proportional to the dephasing time. As such, the denominator in eq 3 becomes Ei(k) Eþ 1 (k) pω iΓ, and the 2PA coefficient is given by 2 β
bilayer
monolayer pω4
¼ 8β
γ1
1 1 þ 4p ω pω iΓ pω þ iΓ 2
2
2 1 1 þ þ γ1 þ pω iΓ γ1 þ pω þ iΓ 1 1 1 2 þ þ þ ð2pω γ1 Þ pω iΓ pω iΓ pω þ γ1 þ iΓ 2 1 1 1 2 þ þ þ 4ðpω γ1 Þ pω iΓ pω þ iΓ pω γ1 iΓ 2 1 1 1 2 þ þ þ ð2pω γ1 Þ pω iΓ pω γ1 iΓ pω γ1 þ iΓ 2 # 1 1 1 þ þ þ ð7Þ pω iΓ pω þ γ1 þ iΓ pω γ1 iΓ
By using the dielectric constant εω = 9 (see ref 19) and Γ = 66 or 6.6 meV, eq 4 and eq 7 plotted in Figure 1 demonstrate that the 2PA is 2 orders of magnitude greater in bilayer than in monolayer in the near-infrared region (8001100 nm). It also shows that 2PA is nearly independent of Γ in this spectral region if the values of Γ are between 6 and 66 meV, which correspond to dephasing times between 50 and 5 fs, respectively. To verify our theory, we carried out optical experiments on high-quality epitaxial bilayer graphene. The sample used was epitaxial bilayer graphene on the C-face of 4HSiC wafers grown at temperatures ∼1100 C in ultrahigh vacuum condition with pressure of 2 109 Torr or better.2022 The sample size was 3 mm 10 mm with high homogeneity. Raman spectroscopy of the sample showed a single-resonant G peak at 1600 cm1, a double-resonant D peak at 2728 cm1, and a relatively low intensity double-resonant D peak near 1371 cm1, close to the reported values.23,24 It is known that the D peak is not allowed in perfect graphene layers but is observed in the presence of 2623
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Nano Letters disorder due to SiC substrate effects.25 The Raman signal of the D peak, therefore, indicates the presence of disorder in the sample. The ratio of the magnitude of G to D peak (at 1371 cm1) is ∼9, implying a very low concentration of disorder. The number of graphene layers was determined to be two with both scanning tunneling microscope (STM) and Raman characterizations in a similar way to ref 26. More details can be found in Supporting Information. The nonlinear light absorption of the above sample was detected by using both transient absorption (or frequencydegenerate, time-resolved pumpprobe) measurement and Z-scan technique27 at 780 nm at room temperature. The laser pulses (780 nm, 400 fs, 1 kHz) were produced by a mode-locked Ti:Sapphire laser (Quantronix, IMRA), which seeded a Ti: Sapphire regenerative amplifier. The 400 fs full width at e1 of maximum (FWe1H) enables us to perform modeling on the properties of quasi-equilibrium under which the FermiDirac distribution is valid for the description of photoexcited hot carriers, as shown later. In the transient absorption measurement, see the setup in Figure 2a, the pump and probe pulses were orthogonally polarized with the incident angle between them less than three degrees. The energy ratio of the probe to pump pulse was less than 5%. The setup was calibrated by using a wide-gap semiconductor CdS as a standard sample, see Supporting Information. The measured signals were fit with a two-exponential-component decay, the fast lifetime accounts for the laser pulse duration and the slow lifetime indicates the carrier recombination time (∼1.2 ps for the bilayer graphene sample). In the Z-scans, the laser pulses were focused with the minimum beam waist of 25 ( 2 μm. The Z-scan setup was calibrated by using another standard sample, ZnSe. As displayed in Figure 3a, the transmitted laser pulse energies were monitored by moving the sample along the propagation direction (or z-axis) of the laser pulses. Additional Z-scans at 1100 nm were measured with an optical parametric amplifier (Coherent TOPAZ) pumped by another regenerative amplifier (Coherent Legend, 780 nm, 100 fs, 1 kHz). For all the Z-scans reported here, the maximum onaxis irradiance (I00) of the laser pulses was 150 GW/cm2 or lower at the focus. No laser-induced damage was observed at 150 GW/ cm2 or less. As the sample was scanned and the incident laser pulse energies were kept at a constant level, the sample experienced different laser irradiances at different z-positions, giving rise to a different transmission if the sample absorbed light nonlinearly, see examples in Figure 3b,c. More data are available in Supporting Information. The measured nonlinear transmittance (Z-scan or ΔT/T0 at zero delay) was analyzed by using the nonlinear propagation equation, dI/dz = [R0/(1þI/Is) þ βI]I, which was solved numerically and then integrated over space and time to obtain the transmitted laser pulse energy. In the numerical simulation for the best fit, the R0 parameter used matches to the theoretical 1PA coefficient.811 Figures 2 and 3 indicate that their signals are dominated by saturable absorption due to Pauli blocking resulting from 1PA, which is in agreement with previous reports.2831 At relatively lower excitation (I00 < 10 GW/cm2), the 2PA is too small to outperform the saturation of 1PA. However, the 2PA manifests itself at higher excitation, see the bending of ΔT/T0 (zero delay) in Figure 2c and the dip on the Z-scan (z = 0) in Figure 3c. It should be pointed out that in the pumpprobe experiment, the minimum beam waist of the pump pulses was adjusted to ∼50 μm, in order to ensure the total overlapping between the pump and probe pulses. As such, the maximum excitation level (or I00) was reduced to ∼50 GW/cm2.
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Figure 2. (a) Schematic setup for transient absorption measurement on the bilayer graphene. (b) Experimental data (black) and theoretical fits R 2 2 τ/τ1 with ¥ þ A2eτ/τ2)e(tτt0) /τp dτ, where A1, A2, and t0 are 0 (A1e constants, and τp ≈ 200 fs (HWe1 M for pulse duration). The red curve is the biexponential fit with A1 6¼ 0, while the blue curve is the monoexponential fit with A1 = 0. (c) Experimental data (black) for transient absorption at zero delay. The error bars are calculated from five series of repeated measurements at each intensity, taking into account of estimated error (∼5%) in the measurement of laser pulse energy. The curves are the theoretical fits with details described in the text.
Though the excitation level could be up to 150 GW/cm2 in the Z-scans, our analysis is limited to the range from a few to 70 GW/ cm2, avoiding other high-order nonlinear mechanisms (such as three-photon absorption). Even within this limited range of excitation, it is found that the data cannot be fit with a saturation model alone, as discussed in the following . 2624
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Figure 4. Schematic diagrams for photodynamics. (a) Carrier distribution of n-doped bilayer graphene prior to laser excitation; (b) carrier excitation by 1PA (including both interband and intraband transitions) and 2PA; (c) quasi-equilibrium after carriercarrier scattering; and (d) equilibrium after carrier recombination through carrier-phonon scattering.
Figure 3. Schematic setup and Z-scans on the bilayer graphene on the substrate (blue symbols) and the substrate alone (red symbols) at 780 and 1100 nm. The upper Z-scans are vertically shifted for clear presentation. The on-axis maximum power density at focus for each Z-scan is shown. The theoretical fits (red solid line) to the Z-scan data are calculated from the nonlinear propagation equation, dI/dz = [R0/(1þI/Is) þ βI]I, where R0 is the linear absorption coefficient, Is is the saturation intensity, and β is the 2PA coefficient. More details on modeling can be found in the text.
The 2PA contribution to the overall nonlinear signal can be uncovered by analyzing the data with the total absorption coefficient, R0/(1 þ I/Is) þ βI, where the first term is a phenomenological expression for absorption saturation (with R0 being the small-signal 1PA coefficient and Is being the saturation irradiance); and the second term quantifies 2PA. The best fits to all the measurements, obtained at two different light wavelengths, various excitation irradiance (I00 = a few to 70 GW/cm2) and different positions on the sample, lead to that Is = 6 ( 2, 1.5 ( 0.5 GW/cm2; and β = 10 ( 2, 20 ( 4 cm/MW at wavelengths of 780 and 1100 nm, respectively. If β = 0 cm/MW, the above simulation becomes a simple saturation modeling. The dashed curves in Figure 2c show the failure of the saturation model [R = R0/(1 þ I/Is)]. We have also conducted similar experiments on a monolayer graphene sample (see Supporting Information) and found that its 2PA was too insignificant to be detected, consistent with eq 4. To gain more insights, we adopt the following photodynamic model to fit both Z-scan and pumpprobe data at 780 nm.
Upon photoexcitation by ultrashort optical pulses, hot carriers (electrons in the conduction bands and holes in the valence bands) are created by one-photon absorption (1PA) (including both interband and intraband) and two-photon absorption (2PA), see Figure 4b. Subsequently, these nonequilibrium carrier distributions broaden and reach quasi-equilibrium through carriercarrier scattering, as shown in Figure 4c. These hot-carrier thermalization processes take an ultrashort time between 10 and 150 fs. The quasi-Fermi Dirac-like equilibrium with an elevated temperature (∼1000 K, much higher than the lattice temperature) may last up to a few picoseconds, when the carrier recombination takes place, until the room-temperature equilibrium is restored as displayed by Figure 4d. The carrier recombination (τ2) may involve the following two mechanisms: namely (i) carrier-phonon scattering through collisions with the lattices in graphene (and/or substrate), and (ii) carrierphonon scattering with disorders in graphene. The carrier densities excited by two-photon and one-photon interband transitions, N2P and N1P, are given respectively by (dΔN2P)/(dt) = (βI2)/(2pω) (ΔN2P)/(τ2P) and (dΔN1P)/ (dt) = (RinterI)/(pω) (ΔN1P)/(τ1), where Rinter is the interband 1PA coefficient, and τ2P and τ1 are the carriercarrier scattering lifetime from the 2PA- and 1PA-involved states to the quasiequilibrium in Figure 4c. They are shorter than the pulse duration (400 fs). Thus, the total density of carriers excited by interband transitions is approximated by Rinter τ1 I βτ2p I 2 þ ð8Þ pω 2pω Then, Fermi energies, Ecf and Evf , for the quasi-equilibrium electron and hole distributions, are calculated through32,33 Z d2 k ΔN þ N0 ¼ 4 fc ðE Ef c Þ ð9Þ ð2πÞ2 ΔN
and
Z ΔP ¼ ΔN ¼ 4
d2 k ½1 fv ðE Ef v Þ ð2πÞ2
ð10Þ
where N0 is the carrier concentration due to doping or thermal excitation at room temperature. Such carriers are also excited to higher energy states through intraband 1PA, but intraband 1PA does not generate excess carriers within the upper (or lower) 2625
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Figure 5. Normalized total absorption coefficient as a function of I00 (or photoexcited carrier density). The solid triangles are the experimental data by using R/R0 = 1/(1 þ I00/Is) þ βI00/R0 with both Is- and βvalues being determined from both transient absorption measurements and Z-scans at 780 nm. The curves are the modeling described in the text.
bands. Within the upper (or lower) bands, however, it promotes carriers (which result from doping, thermal excitation at room temperature, or light excitation by interband transitions) to higher energy states, at which they subsequently merge with the 2PA-excited hot carriers. The total absorption coefficient is given by R ¼ Rinter þ intra þ βI
ð11Þ
where Rinterþintra includes both interband and intraband 1PA. Depending on both Fermi energies (at room temperature) and elevated temperatures, it can be evaluated by using eq 2 and eq 3 in ref 33 both of them are expanded to take the four bands into account. The numerical solutions to eqs 811 are plotted as a function of light excitation (I00) in Figure 5. The reliable calculation critically relies on the precise values of the parameters in eqs 811. There have been many attempts to determine the accurate values of carriercarrier scattering lifetimes and hot-carrier temperature. With a time resolution of 10 fs, Breusing et al.34 measured τ1 to be 30 fs by measuring the transient transmission of freestanding thin graphite films. They also simulated their measurements to reveal that the change in the temperature of hot electrons, Te. Initially, the temperature can reach beyond 1000 K and then cool down quickly within a characteristic time of τ1 to be in the range of 700 K ∼ 900 K for the quasi-equilibrium, similar to the situation in Figure 4c. By assuming that Te = Th = 900 K, β = 10 cm/MW, τ2P = 10 ( 4 fs, τ1 = 130 ( 30 fs, and N0 = 5 1011 cm2, (a very low density, corresponding to the Fermi energy of ∼0.0 eV at room temperature under dark condition), we numerically compute the total absorption coefficient with the four bands being taken into account. Compared to the measurements, good agreement can be reached as shown in Figure 5. The interband 1PA gives rise to absorption saturation with a saturation irradiance Is. The intraband 1PA mitigates absorption saturation as shown by the dotted lines in Figure 5. If one ignores 2PA by letting β = 0 cm/MW, Figure 5 shows a large discrepancy between the modeling (dotted lines) and the experimental data in the high-excitation domain (I00 > 10 GW/cm2), thereby reconfirming that the 2PA term (βI) contribution should not be overlooked.
Within the same order of magnitude, the measured β values are in agreement with 32 and 88 cm/MW predicted by eq 7 for the two wavelengths, respectively. The agreement on the order of magnitude implies that the four parabolic bands should play an essential role in the 2PA process. The discrepancy is expected due to the following two reasons. First, our theory considers the γ1-terms in eq 5 only and ignores other complexities by letting V = 0, γ3 = 0, which account for band perturbations arising from trigonal warping and electrohole asymmetry in “skew” interlayer coupling.11,12,15 Second, recent experimental evidence35,36 indicates the coexistence of both AB stacking and decoupled layers by azimuthal rotation in bilayer graphene on the C-face of SiC substrate, consistent with our STM studies, see Figure S3 in Supporting Information. This is also attributed to our observation of smaller 2PA coefficients. Interestingly, the 2PA of bilayer graphene reaches a maximum (∼0.2 cm/W if Γ = 6.6 meV) at pω = γ1 (= 0.4 eV, or 3.1 μm), resulting from resonance with half of the bandgap between the E2 bands. This β value is at least 5 orders of magnitude greater than that for many narrow-gap semiconductors (such as InSb, MgCdTe, etc.) in the infrared region.37 It is important to note that unlike 2PA coefficients in the nonresonant region (λ < 2000 nm), resonant β-values are sensitive to the broaden factor, Γ. This factor is related to the time for a dephasing process in which coherence in graphene caused by perturbation decays over time. Such decoherence could be caused by incoherent processes such as carriercarrier scattering. From eq 7 or Figure 1, the maximum 2PA coefficient should be ∼ 0.002 cm/W if Γ = 66 meV, which are still 3 orders of magnitude larger than the abovesaid narrow-gap semiconductors. The above-discussed giant 2PA should facilitate many quantum technologies, for example, the coherent control of ballistic photocurrent generated by quantum interference between 1PA and 2PA pathway in bilayer graphene. Besides, the generation of ballistic photocurrent by 1.55 μm photons through 1PA and 3.1 μm photons through 2PA is desirable because 1.55 μm wavelength is close to one of the telecom bands. Furthermore, such giant 2PA is predicted and found in Bernal-stacked bilayer graphene. If bilayer graphene is stacked in other ways whereby the interlayer interaction is insignificant, that is, γ1 = 0, its 2PA value is expected to be as twice as the βmonolayer value, which is 2 orders of magnitude less than the AB-stacked sample. Hence, the difference in 2PA measurements provides a practically efficient and explicit way of indentifying the stacking orders of bilayer graphene.
’ ASSOCIATED CONTENT
bS
Supporting Information. Details on the characterization of the sample including micro-Raman spectra, STM images, and nonlinear optical measurements. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail address:
[email protected].
’ ACKNOWLEDGMENT We are grateful to the National University of Singapore, Natural Science Foundation of China (Grant 11064017) and 2626
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Nano Letters Yunnan Provincial Department of Education (2010Y001) for financial support. We thank Zexiang Shen and Yingying Wang for conducting some of the Raman measurements, Guichuan Xing for femtosecond laser facilities, and Bing Gu for fruitful discussions on the Z-scan analyses.
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