Raman spectroscopy of graphene under ultrafast laser excitation
The equilibrium optical phonons of graphene are well characterized in terms of anharmonicity and electron-phonon interactions, however their non-equilibrium properties in the presence of hot charge carriers are still not fully explored. Here we study the Raman spectrum of graphene under ultrafast laser excitation with 3ps pulses, which trade off between impulsive stimulation and spectral resolution. We localize energy into hot carriers, generating non-equilibrium temperatures in the1700-3100K range, far exceeding that of the phonon bath, while simultaneously detecting the Raman response. The linewidth of both G and 2D peaks show an increase as function of the electronic temperature. We explain this as a result of the Dirac cones’ broadening and electron-phonon scattering in the highly excited transient regime, important for the emerging field of graphene-based photonics and optoelectronics.
The distribution of charge carriers has a pivotal role in determining fundamental features of condensed matter systems, such as mobility, electrical conductivity, spin-related effects, transport and optical properties. Understanding how these proprieties can be affected and, ultimately, manipulated by external perturbations is important for technological applications in diverse areas ranging from electronics to spintronics, optoelectronics and photonicsBonaccorso2010 (); Nanoscale2015 (); Koppens2014 ().
The current picture of ultrafast light interaction with single layer graphene (SLG) can be summarized as followsBrida2013 (). Absorbed photons create optically excited electron-hole (e-h) pairs. The subsequent relaxation towards thermal equilibrium occurs in three steps. Ultrafast electron-electron (e-e) scattering generates a hot Fermi-Dirac distribution within the first tens fsTomadin2013 (). The distribution then relaxes due to scattering with optical phonons (electron-phonon coupling), equilibrating within a few hundred fsLazzeri2005 (); Butscher2007 (). Finally, anharmonic decay into acoustic modes establishes thermodynamic equilibrium on the ps timescaleLui2010 (); Wu2012 (); Bonini2007 ().
Raman spectroscopy is one of the most used characterization techniques in carbon science and technologyFerrRob2004 (). The measurement of the Raman spectrum of grapheneFerrariPRL2006 () triggered a huge effort to understand phonons (ph), e-ph, magneto-ph, and e-e interactions in graphene, as well as the influence of the number and orientation of layers, electric or magnetic fields, strain, doping, disorder, quality and types of edges, and functional groupsFerrariNN2013 (); Malard2009 (); Froehlicher2015 (). The Raman spectra of SLG and few layer graphene (FLG) consist of two fundamentally different sets of peaks. Those, such as D, G, 2D, present also in SLG, and due to in-plane vibrationsFerrariPRL2006 (), and others, such as the shear (C) modesTan2012 () and the layer breathing modesSato2011 (); Lui2012 () due to relative motions of the planes themselves, either perpendicular or parallel to their normal. The G peak corresponds to the high frequency E phonon at . The D peak is associated to the ring breathing mode, and requires the presence of a defect for momentum conservationTuinstra1970 (); Thomsen2000 (); FerrariPRB2000 (). The 2D peak is the D overtone, it is always allowed as momentum conservation is satisfied in this case by two phonons with opposite wavevectors FerrariPRL2006 (). Both D and 2D are activated by a double resonance (DR) mechanism, and are dispersive in nature due to a Kohn Anomaly at KPiscanec2004 ().
Raman spectroscopy is usually performed under continuous wave (CW) excitation, therefore probing samples in thermodynamic equilibrium. The fast e-e and e-ph non-radiative recombination channels establish equilibrium conditions between charge carriers and lattice, preventing the study of the vibrational response in presence of an hot e-h population. Using an average power comparable to CW illumination (a few mW), ultrafast optical excitation can provide large fluences (Jm at MHz repetition rates) over sufficiently short timescales (0.1-10ps) to impulsively generate a strongly out-of-equilibrium distributions of hot e-h pairsLui2010 (); Yan2009 (); Breusing2011 (); Brida2013 (). The potential implications of coupled electron and phonon dynamics for optoelectronics were discussed for nanoelectronic devices based on CW excitationChae2010 (); PhysRevLett.104.227401 (); PhysRevB.93.075410 (); Mogulkoc201485 (); kim_bright_2015 (). However, understanding the impact of transient photoinduced carrier temperatures on the colder SLG phonon bath is important for mastering out of equilibrium e-ph scattering, critical for photonics applications driven by carrier relaxation, such as ultrafast laserssun2010 (), detectorsBonaccorso2010 (); Koppens2014 () and modulatorsLiu2011 (). E.g, SLG can be used as a much broader-band alternative to semiconductors saturable absorberssun2010 (), for mode-locking and Q switchingsun2010 (); Bonaccorso2010 ().
Here we characterize the optical phonons of SLG at high electronic temperatures by performing Raman spectroscopy under pulsed excitation. We use a 3ps pulse to achieve a trade off between the narrow excitation bandwidth required for spectral resolution (10cm, being [Hz] the laser frequency and c the speed of light, a condition met under CW excitation) and a pulse duration, , sufficiently short (10ps, achieved using ultrafast laser sources) to generate an highly excited carrier distribution over the equilibrium phonon population, being those two quantities Fourier conjugatespapoulis_1962 () (cm). This allows us to determine the dependence of both phonon frequency and dephasing time on the hot carriers temperature, which we explain by a broadening of the Dirac cones.
Fig.1a plots a sequence of AntiStokes (AS) Raman spectra of SLG following ultrafast excitation at 1.58eV, as a function of excitation power . The broad background stems from hot photoluminescence (PL) due to the inhibition of a full non-radiative recombination under high excitation densitiesfreitag_2010 (); Lui2010 (); Stohr2010 (); PhysRevLett.104.227401 (). This process, absent under CW excitation in pristine SLGGokus2009 (), is due to ultrafast photogeneration of charge carriers in the conduction band, congesting the e-ph decay pathway, which becomes progressively less efficient with increasing fluence. This non equilibrium PL recalls the grey body emission and can be in first approximation described by Planck’s lawfreitag_2010 (); Lui2010 (); PhysRevLett.104.227401 (); kim_bright_2015 ():
where is the emissivity, defined as the dimensionless ratio of the thermal radiation of the material to the radiation from an ideal black surface at the same temperature as given by the Stefan-Boltzmann lawcallen_1985 (), is the emission time and is the frequency-dependent, dimensionless responsivity of our detection chainprinceton (); princeton2 (). Refs.kim_bright_2015 (); freitag_2010 (); Lui2010 () reported that, although Eq.1 does not perfectly reproduce the entire grey body emission, the good agreement on aeV energy window is sufficient to extract . By fitting the backgrounds of the Raman spectra with Eq.1 (solid lines in Fig.1a) we obtain as a function of . Fig.1b shows that can reach up to 3100K under our pulsed excitation conditions.
An upper estimate for the lattice temperature, , can be derived assuming a full thermalization of the optical energy between vibrational and electronic degrees of freedom, i.e. evaluating the local equilibrium temperature, , by a specific heat argument (see Methods). We get K at the maximum excitation power, mW. This is well below the corresponding , indicating an out-of-equilibrium distribution of charge carriers. Thus, over our 3ps observation timescale, is well below .
Fig.1c plots the AS and S G peaks, together with fits by Lorentzians (blue lines) convoluted with the laser bandwidth (cm) and spectrometer resolution (cm), which determine the instrumental response function, IRF (see Methods). The S data have a larger noise due to a more critical background subtraction, which also requires a wider accessible spectral range (see Methods). For this reason, we will focus on the AS spectral region, with an higher spectrometer resolution (1.2 cm), Fig.2. We obtain a full width at half maximum of the G peak, FWHM(G)cm, larger than the CW one (cm). Similarly, we get FWHM(2D)50-60cm over our range, instead of FWHM(2D)cm as measured on the same sample under CW excitation. To understand the origin of such large FWHM(G) and FWHM(2D) in pulsed excitation, we first consider the excitation power dependence of the SLG Raman response in the mW range (the lower bound is defined by the detection capability of our setup). This shows that the position of the G peak, Pos(G), is significantly blueshifted (as reported for graphite in Ref.Yan2009 ()), while the position of the 2D peak, Pos(2D), is close to that measured under CW excitation, while both FWHM(G) and FWHM(2D) increase with . Performing the same experiment on Si proves that the observed peaks broadening is not limited by our IRF (see Methods). Moreover, even the low resolution S data of the G band, collected in the range 1.8-7.0mW (a selection is shown in Fig.1c), display a broadening ( cm/K) and upshift ( cm/K), which is compatible with that of the high resolution AS measurements (Fig. 3d-e), cm/K and cm/K, respectively.
We note that phonons temperature estimates based on the AS/S intensity ratioSchomacker1986 (); FilhoPRB2001 () (corrected for the wavelength dependent grating reflectivity and CCD efficiency) are hampered in graphene by two concurring effects. First, SLG’s resonant response to any optical wavelength gives a non trivial wavelength dependent Raman excitation profile which modifies the Raman intensities with respect to the non-resonant case. Consequently, the AS/S ratio is no longer straightforwardly related to the thermal occupation goldstein2016raman (). Second, in graphene one S created phonon may be subsequently annihilated by a correlated AS event. Although a complete theoretical description for this phenomenon is still laking, in practice, it results into an extra pumping in the AS side which does not allow to relate in the standard way AS/S ratio and phonon temperature via the thermal occupation factor ParraMurilloPRB2016 (). Accordingly, the AS/S ratio approaching one at the largest excitation power in Fig.1c (black circles) does not necessary imply a large increase of the G phonon temperature.
Fig.3 plots Pos(2D), FWHM(2D), Pos(G), FWHM(G) as a function of , estimated from the hot-PL. A comparison with CW measurements (633nm) at room temperature (RT) is also shown (blue diamonds). Under thermodynamic equilibrium, the temperature dependence of the Raman spectrum of SLG is dominated by anharmonicity, which is responsible for mode softening, leading to a redshift of the Raman peaksApostolov (); Basko2008 (); Bonini2007 (). This differs from our experiments (Figs.4a-d), in which Pos(G) has an opposite trend (blue shift), and Pos(2D) is nearly -independent, in agreement with Density Functional Perturbation Theory (DFPT) calculations, giving Pos(2D)5cm in the range K (see Methods). This indicates the lack of significant anharmonic effects and suggests a dominant role of e-ph interaction on FWHM(G) and Pos(G), in the presence of a cold phonon bath at constant decoupled from the (large) .
Here is a dimensionless constant, is the Fermi velocity, is the upper cutoff of the linear dispersion , is the carbon atom mass, eV the bare phonon energy, is a positive arbitrary small number (meV), eV is proportional to the e-ph coupling (EPC) Piscanec2004 (); Lazzeri2005 (); LazzeriPRL06 (); Ferrari200747 (), , are the energy integration variables and is the Fermi-Dirac distribution with the Fermi energy. Although our samples are doped, significantly decreases as a function of Chae2010 (). Hence, we assume in the following calculations. The two indexes denote the e and h branches, and is the corresponding spectral function, which describes the electronic dispersion.
so that the shift Pos(G) and FWHM(G) can be written as:
where is the Planck constant. FWHM(G) can be further simplified since the evaluation of leads to in Eq.2, so that we get:
In the limit of vanishing broadening of the quasiparticle state, the SLG gapless linear dispersion is represented by the following spectral functionAndo2006 ():
where cmBonini2007 (). This value, with the additional2cm contribution arising from anharmonic effectsBonini2007 (), is in agreement with the CW measurement at K (see diamond in Fig.3e) corresponding to fluencesJ/m. Eq.7 also shows that, as increases, the conduction band becomes increasingly populated, making progressively less efficient the phonon decay channel related to e-h formation and leading to an increase of the phonon decay time (Fig.4b). This leads to a decrease of FWHM(G) for increasing (black solid line in Fig.3e), which is in contrast with the experimentally observed increase (blue circles in Fig.3e).
A more realistic description may be obtained by accounting for the effect of on the energy broadening () of the linear dispersion , along with the smearing of the Fermi function. can be expressed, to a first approximation, as the sum of three termsVenezuela2011 ():
where , and are the e-e, e-ph and defect contributions to . The only term that significantly depends on is , while the others depend on the energy Pisana2007 (); Pinczuk2007 (); Neumann2015 (); Bonini2007 (); Venezuela2011 (); BaskPRB2009 ().
where cm/eV Vidano1981 (); FerrariNN2013 (), i.e. the ratio between the phonon and Fermi velocity, defined as the slope of the phononic (electronic) dispersion at the ph (e) momentum corresponding to a given excitation laser energy FerrariNN2013 (). Since the DR process responsible for the 2D peak involves the creation of e-h pairs at energy , the variation of FWHM(2D) allows us to estimate the variation of at eV. Then, and , both proportional to (), will give an additional constant contribution to FWHM(2D), but not to its variation with . Our data support the predictedSchutt2011 () linear increase of with , with a dimensionless experimental slope , Fig.3c.
In order to compute FWHM(G) from Eq.2, we note that the terms and are negligible at the relevant low energy eV . Hence .
The Dirac cone broadening can now be introduced by accounting for in the spectral function of Eq.6:
accordingly, all the processes where the energy difference is less than (which guarantees the overlap between the spectral functions of the quasiparticles) will now contribute in Eq.2. Amongst them, those transitions within the same (valence or conduction) band, as shown in Fig.4d.
The broadened interband contributions still follow, approximately, Eq.7 (see Fig.4e). However, the Dirac cone broadening gives additional channels for G phonon annihilation by carrier excitation. In particular, intra-band transitions within the Dirac cone are now progressively enabled for increasing , as sketched in Fig.4d. In Fig.4e the corresponding contributions to FWHM(G) are shown. Calculations in the weak-coupling limitSchutt2011 () suggest that should be suppressed as , due to phase-space restriction of the Dirac-cone dispersion. Our results, however, indicate that this effect should appear at an energy scale smaller than , as the theory captures the main experimental trends, just based on a -independent .
Critically, the G peak broadening has a different origin from the equilibrium caseBaladin2008 (). The absence of anharmonicity would imply a FWHM(G) decrease with temperature due to the e-ph mechanism. However, the Dirac cone broadening reverses this trend into a linewidth broadening above K producing, in turn, a dephasing time reduction, corresponding to the experimentally observed FWHM(G) increase. The blueshift of the G peak with temperature is captured by the standard e-ph interaction, beyond possible calibration accuracy. Importantly, the Dirac cone broadening does not significantly affect Pos(G).
In conclusion, we measured the Raman spectrum of SLG with impulsive excitation, in the presence of a distribution of hot charge carriers. Our excitation bandwidth enables us to combine frequency resolution, required to observe the Raman spectra, with short pulse duration, needed to create a significant population of hot carriers. We show that, under these strongly non-equilibrium conditions, the Raman spectrum of graphene cannot be understood based on the standard low fluence picture, and we provide the experimental demonstration of a broadening of the electronic linear dispersion induced by the highly excited carriers. Our results shed light on a novel regime of non-equilibrium Raman response, whereby the e-ph interaction is enhanced. This has implications for the understanding of transient charge carrier mobility under photoexcitation, important to study SLG-based optoelectronic and photonic devicesPhysRevB.93.075410 (); Mogulkoc201485 (), such as broadband light emitterskim_bright_2015 (), transistors and optical gain mediaengel_lightmatter_2012 ().
Sample preparation and CW Raman characterization
SLG is grown on a 35m Cu foil, following the process described in Refs.57,58. The substrate is heated to 1000C and annealed in hydrogen (H, 20 sccm) for 30 minutes. Then, 5 sccm of methane (CH) is let into the chamber for the following 30 minutes so that the growth can take placeBaeNN2010 (); LiS2009 (). The sample is then cooled back to RT in vacuum (1 mTorr) and unloaded from the chamber. The sample is characterized by CW Raman spectroscopy using a Renishaw inVia Spectrometer equipped with a 100x objective. The Raman spectrum measured at 514 nm is shown in Fig.5 (red curve). This is obtained by removing the non-flat background Cu PLLagaAPL2013 (). The absence of a significant D peak implies negligible defectsFerrariNN2013 (); FerrariPRL2006 (); CancNL2011 (); FerrariPRB2000 (). The 2D peak is a single sharp Lorentzian with FWHM(2D)23cm, a signature of SLGFerrariPRL2006 (). Pos(G) is1587cm, with FWHM(G)14cm. Pos(2D) is2705cm, while the 2D to G peak area ratio is 4.3. SLG is then transferred on glass by a wet methodbonaccorso2012 (). Poly-methyl methacrylate (PMMA) is spin coated on the substrate, which is then placed in a solution of ammonium persulfate (APS) and deionized water. Cu is etchedBaeNN2010 (); bonaccorso2012 (), the PMMA membrane with attached SLG is then moved to a beaker with deionized water to remove APS residuals. The membrane is subsequently lifted with the target substrate. After drying, PMMA is removed in acetone leaving SLG on glass. The SLG quality is also monitored after transfer. The Raman spectrum of the substrate shows features in the D and G peak range, convoluted with the spectrum of SLG on glass (blue curve in Fig.5). A point-to-point subtraction is needed to reveal the SLG features. After transfer, the D peak is still negligible, demonstrating that no significant additional defects are induced by the transfer process, and the fitted Raman parameters indicate p doping250meVBaskPRB2009 (); DasNN2008 ().
Before and after the pulsed laser experiment, equilibrium CW measurements are performed at room temperature using a micro-Raman setup (LabRAM Infinity). A different energy and momentum of the D phonon is involved, for a given excitation wavelength, in the S or AS processes, due to the phonon dispersion in the DR mechanismbaranov_1987 (); thomsen_2000 (). Hence, in order to measure the same D phonon in S and AS, different laser excitations () must be used according to FerrariNN2013 (); tan2002 (); cancado2002 (). Given our pulsed laser wavelength (783nm), the corresponding CW excitation would be649.5nm. Hence, we use a 632.8nm He-Ne source, accounting for the small residual wavelength mismatch by scaling the phonon frequency as FerrariNN2013 ()
Pulsed Raman measurements
The ps-Raman apparatus is based on a mode-locked Er:fiber amplified laser atnm, producing 90fs pulses at a repetition rate RR=40MHz. Using second-harmonic generation in a 1cm Periodically Poled Lithium Niobate crystalMarangoni2007 (), we obtain 3ps pulses at 783nm with acm bandwidth. The beam is focused on SLG through a slightly underfilled 20X objective (NA), resulting in a focal diameter m. Back-scattered light is collected by the same objective, separated with a dichroic filter from the incident beam and sent to a spectrometer (with a resolution of 0.028 nm/pixel corresponding to 1.2cm). The overall IRF, therefore, is dominated by the additional contribution induced by the finite excitation pulse bandwidth. Hence, in order to extract the FWHM of the Raman peaks, our data are fitted convolving a Lorentzian with the spectral profile of the laser excitation.
When using ultrafast pulses, a non-linear PL is seen in SLGLui2010 (). Such an effect is particularly intense for the S spectral rangePhysRevB.82.081408 (); Stohr2010 (). The S signal in Fig.1c is obtained as the difference spectrum of two measurements with excitation frequencies slightly offset by130cm, resulting in PL suppression. The background subtraction requires in this case a wider spectral range, at the expenses of spectrometer resolution which is reduced to 0.13 nm/pixel corresponding to6cm, as additional contribution to the IRF. Although this procedure allows to isolate the S Raman peaks, the resulting noise level is worse than for AS. For this reason we mostly focus on the AS features.
To verify that the observed peaks broadening is not limited by our IRF, we perform the same experiment on a Si substrate (6a). For this we retrieve, after deconvolution of the IRF, the same Raman linewidth measured in the CW excitation regime (Fig.6a). The FWHM of the Si optical phonon is independent of , in contrast with the well-defined dependence on observed in SLG, Fig.6b.
Estimate of the local equilibrium temperature
Photoexcitation of SLG induces an excess of energy in the form of heat Q per unit area, that can be expressed as:
where % is the SLG absorption, approximated to the undoped caseNair2008 (), m is the waist of focused beam and MHz is the repetition rate of the excitation laser. The induced can be derived based on two assumptions: (i) in our ps time scale the energy absorbed in the focal region does not diffuse laterally, (ii) the energy is equally distributed on each degree of freedom (electrons, optical and acoustic ph). Then, can be described as:
where is the SLG T-dependent specific heat. In the K range, can be described asPop2012 (): , where J/(Km) and J/(Km). Therefore, considering Eqs.11,12, for mW, we get K, well below the corresponding , indicating an out-of-equilibrium condition (). Any contributions from the substrate and taking into account for the heat profile would contribute in reducing even further estimation.
Estimate of Pos(2D) as a function of
We perform calculations within the Local Density Approximation in DFPTGironcoli1995 (); Giannozzi2009 (). We use the experimental lattice parameter 2.46Åwang_crystal_2012 () and plane waves (45Ry cutoff), within a norm-conserving pseudopotential approachGiannozzi2009 (). The electronic levels are occupied with a finite fictitious with a Fermi Dirac distribution, and we sample a Brillouin Zone with a 160x160x1 mesh. This does not take into account anharmonic effects, assuming K. Fig.7 shows a weak (cm) in the range K. In equilibrium, would induce a non-negligible anharmonicityNef2014 (), which would lead to a Pos(2D) softening: cmK. The weak dependence (blue circles in Fig.7) rules out a dominant anharmonicity contribution and, consequently, . The minor disagreement with DFPT suggests a slightly larger than RT, but definitely smaller than .
We acknowledge funding from the Graphene Flagship, ERC Grant Hetero2D, EPSRC Grants EP/K01711X/1, EP/K017144/1, EP/N010345/1, EP/L016087/1 and MAECI under the Italia-India collaborative project SuperTop-PGR04879.
TS led the research project, conceived with GC, FM and ACF. CFe, AV and TS designed and built the pulsed Raman setup. CFe and AV performed the out of equilibrium Raman experiments, with contribution from MM. CFa, PP, DDF, US, AKO and DY performed the equilibrium CW Raman experiment. LB and FM developed the modelling and carried out the numerical simulations, with contribution from AV DDF, US, AKO and DY prepared and characterised the sample. CFe, AV, LB, GC, FM, ACF and TS interpreted the data and the simulations and wrote the manuscript.
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