Phonon-pump XUV-photoemission-probe in graphene: evidence for non-adiabatic heating of Dirac carriers by lattice deformation
Isabella Gierz, Matteo Mitrano, Hubertus Bromberger, Cephise Cacho, Richard Chapman, Emma Springate, Stefan Link, Ulrich Starke, Burkhard Sachs, Martin Eckstein, Tim O. Wehling, Mikhail I. Katsnelson, Alexander Lichtenstein, Andrea Cavalleri
PPhonon-pump XUV-photoemission-probe in graphene: evidencefor non-adiabatic heating of Dirac carriers by lattice deformation
Isabella Gierz, ∗ Matteo Mitrano, Hubertus Bromberger, CephiseCacho, Richard Chapman, Emma Springate, Stefan Link, UlrichStarke, Burkhard Sachs, Martin Eckstein, Tim O. Wehling, MikhailI. Katsnelson, Alexander Lichtenstein, and Andrea Cavalleri
1, 7 Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany Central Laser Facility, STFC Rutherford Appleton Laboratory, Harwell, United Kingdom Max Planck Institute for Solid State Research, Stuttgart, Germany I. Institut f¨ur Theoretische Physik, Universit¨at Hamburg, Germany Institut f¨ur Theoretische Physik, Universit¨at Bremen, Bremen, Germany Institute for Molecules and Materials,Radboud University Nijmegen, Nijmegen, The Netherlands Department of Physics, Clarendon Laboratory,University of Oxford, Oxford, United Kingdom (Dated: November 8, 2018)
Abstract
We modulate the atomic structure of bilayer graphene by driving its lattice at resonance with thein-plane E lattice vibration at 6.3 µ m. Using time- and angle-resolved photoemission spectroscopy(tr-ARPES) with extreme ultra-violet (XUV) pulses, we measure the response of the Dirac electronsnear the K-point. We observe that lattice modulation causes anomalous carrier dynamics, withthe Dirac electrons reaching lower peak temperatures and relaxing at faster rate compared to whenthe excitation is applied away from the phonon resonance or in monolayer samples. Frozen phononcalculations predict dramatic band structure changes when the E vibration is driven, which weuse to explain the anomalous dynamics observed in the experiment. a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov IG. 1: (a) Sketch of the tr-ARPES experiment. The pump pulse (red) at normal incidenceresonantly excites the in-plane E lattice vibration in bilayer graphene. A collinear extreme ultra-violet probe pulse (blue) ejects photoelectrons that pass through a hemispherical analyzer andimpinge on a two-dimensional detector. (b) Equilibrium band structure for hydrogen-intercalatedmonolayer (ML) and bilayer (BL) graphene measured with HeII α radiation for a cut through theK-point perpendicular to the ΓK-direction (see inset). Optical excitation of Dirac carriers in graphene has to date been known to occur throughtwo mechanisms. For photon energies higher than twice the chemical potential ( (cid:126) ω pump > | µ e | ), direct interband excitation takes place [1, 2], resulting in population inversion forsufficiently high fluences [3–5]. For doped samples and lower photon energies ( (cid:126) ω pump < | µ e | ), the Dirac carriers are heated by metallic free carrier absorption [4, 6], where the peakelectronic temperature is determined by the pump fluence [4].Here, we introduce a new mechanism, active when an infrared optical field is made res-onant with a vibrational mode that modulates the band structure. We show that anoma-lous heating of the Dirac carrier distribution occurs when the E phonon mode of bilayergraphene [7–13] is driven to large amplitudes with a coherent mid-infrared field. This mode isparticularly interesting as it exhibits a pronounced Fano profile and large oscillator strength2n the conductivity spectrum, originating from an anomalously strong coupling to electronicinterband transitions [15, 16, 39]. Further, the E motion of the two triangular latticeunits (Fig. 1a) is expected to periodically open and close a band gap at the K-point [17], aprospect of general interest for the physics of graphene. Finally, the frequency of the modeis fast compared to the electron-phonon scattering time, resulting in a breakdown of the adi-abatic Born-Oppenheimer approximation as proposed in the context of Raman scatteringexperiments for monolayer graphene [18].Quasi-freestanding epitaxial graphene mono- and bilayers, grown on the silicon-terminated face of silicon carbide (SiC), are used in this work [19] (for further details see[20]). The equilibrium ARPES spectra for these samples, measured at room temperaturewith HeII α radiation at (cid:126) ω = 40 . π -band dispersion of graphene. Both samplesare lightly hole-doped due to charge transfer from the substrate with hole concentrations of n = 6 × cm − and n = 4 × cm − in monolayer and bilayer graphene, respectively.Mid-infrared pulses were used to excite these samples, either resonantly with the E mode in bilayer graphene ( λ pump = 6 . µ m) or at other wavelengths between 4 µ m and 9 µ m.The resulting non-equilibrium Dirac carrier distributions were tracked with tr-ARPES as afunction of pump-probe time delay using extreme ultra-violet (XUV) femtosecond pulses at31 eV photon energy. The experiments were performed at 30 K base temperature.In Fig. 2 we present snapshots of the carrier distributions (upper panel) of bilayergraphene, along with the pump-induced changes of the photocurrent (lower panel) for dif-ferent pump-probe delays after excitation of the E lattice vibration at λ pump = 6 . µ m.These measurements were taken along the ΓK-direction, where only two of the four π -bandsare visible due to photoemission matrix element effects [21]. Zero time delay was set to thepeak of the pump-probe signal.Consistent with previous measurements performed in the free carrier absorption regime[4], we observe a broadening of the Fermi edge due to an increase in electronic tempera-ture, without signature of population inversion. The transient distributions of Fig. 2 wereintegrated over momentum, and plotted as a function of pump-probe delay (Fig. 3a). Ateach time delay, the momentum-integrated photocurrent was fitted by a Fermi-Dirac distri-bution, to obtain the width of the Fermi cut-off, related to the electronic temperature, T e .3 IG. 2: Snapshots of the electronic structure along the ΓK-direction (see inset) of bilayer graphenefor different pump-probe delays (upper panel) together with the corresponding pump-probe signal(lower panel). The excitation wavelength was 6.3 µ m in resonance with the in-plane E latticevibration. The fluence was F = 0.26 mJ/cm . The time evolution of the Fermi width (Fig. 3b) is found to follow a double exponentialdecay, generally ascribed to the emission of optical ( τ ) and acoustic phonons ( τ ) [22–34].Experiments for different excitation wavelengths were performed at a constant pump flu-ence of F = 0.26 mJ/cm (see [20]). The wavelength dependence of the peak electronic tem-perature, max T e , and the fast relaxation time, τ , are plotted in Fig. 3c and d, respectively.Both quantities exhibit a strong anomaly at the phonon resonance, with a reduced peak tem-perature and faster relaxation time ( τ ). The second relaxation time ( τ = 2 . ± . τ values for monolayer graphene (black data points) which are foundto be independent of pump wavelength. 4 IG. 3: (a) Momentum-integrated photocurrent from Fig. 2 as a function of energy and pump-probe delay in the vicinity of the Fermi level. (b) Width of the Fermi-Dirac distribution as afunction of pump-probe delay (data points). The continuous line is a fit including an error func-tion for the rising edge and the sum of two exponentials to describe the decay. (c) Dependenceof the peak electronic temperature on excitation wavelength for constant excitation fluence of0.26 mJ/cm . (d) Dependence of the fast relaxation time τ on excitation wavelength for constantexcitation fluence of 0.26 mJ/cm . The corresponding data for monolayer graphene for fluencesbetween 0.26 mJ/cm and 0.8 mJ/cm is shown in black for direct comparison. Continuous linesin (c) and (d) are guides to the eye. Dashed lines in (c) and (d) show the Fano line shape of thephonon in the real part of the optical conductivity ∆ σ from [39]. The data presented above points then to important differences between excitation on andoff resonance with the in-plane lattice vibration, which is interpreted in the following.In the case of free carrier absorption [4], the excitation is a result of periodic accelerationand deceleration of the carriers along the direction of light polarization, k y (ΓM-direction,green arrows in Fig. 4a). Carrier heating occurs in this case because the Dirac carriersscatter during this oscillatory motion.When excitation is resonant with the E phonon, in addition to this time dependentpolarization, one expects a modulation of the band structure, as the motion of the atomsmakes the hopping between sites time dependent. Qualitatively, one can think of the motion5 IG. 4: (a) Sketch of the excitation due to electric field acceleration (green) and band structureshift (blue). The field polarization lies along k y (ΓM). The displacement of the atoms in realspace leads to a corresponding displacement of the Dirac cone in momentum space along k x (ΓK).The combined effect results in a complicated sloshing motion of the electrons that depends on therelative phase and amplitudes of the electric field acceleration and the band structure displacement,respectively. (b) Density functional theory calculations of the electronic structure around the K-point in the vicinity of the Fermi level for different lattice distortions of 0%, 2% and 3% of theequilibrium lattice constant along the E mode coordinate. of the atoms in real space directly translating into an oscillatory displacement of the Diraccone itself [18]. Note that this motion of the Dirac cone occurs along k x (ΓK-direction, bluearrows Fig. 4a), that is, perpendicular to the free carrier absorption occurring along k y .Crucially, since the atomic motion is fast compared to the electron-phonon scattering time,the electrons do not follow the motion of the atoms promptly and experience a complexmotion that breaks the adiabatic Born-Oppenheimer approximation [18].6o understand the physics at hand more quantitatively, we calculate the changes in theband structure for a static lattice distortion along the normal mode coordinate. For apump field of approximately 1 MV/cm, as employed in our experiments, we estimate atomicdisplacements of ∼ ∼
2% of the equilibrium lattice constant (see [20]). In Fig. 4b, wereport calculated band structures for different lattice distortions along the E normal modecoordinate. Dramatic changes are induced by the vibration, with a momentum-splitting ofthe π -bands at the K-point and a huge shift of the σ -bands at the Γ-point towards the Fermi-level (see [20]). The oscillatory motion of the Dirac cone known from the monolayer (Fig.4a) affects the upper and lower layer of the bilayer in opposite directions. Together with theinterlayer coupling this leads to a splitting of the π -bands along ΓKM that increases withincreasing distortion (Fig. 4b). The non-equilibrium occupancy of the transient electronicstates is determined by the complex non-adiabatic motion sketched in Fig. 4a.To describe the experimental results we consider how the dynamical changes in the den-sity of states (see Fig. 4 in [20]) are expected to affect the electronic temperature. In order toapproximately estimate the change in peak electronic temperature due to the E lattice dis-tortion, we assume that during the lattice distortion both the electron number N(DOS, µ e ,T e )and the entropy of the system S(DOS, µ e ,T e ) are conserved (for further details see [20]). Thechemical potential and the peak electronic temperature in the absence of a lattice distortionare known from off-resonance tr-ARPES data ( µ ∼ −
200 meV, T ∼ µ ∼ −
140 meV and T ∼ µ m and 9 µ m, both on- and off-resonance with the in-plane E mode in bilayergraphene, and probed the response of the electronic structure with tr-ARPES. We findthat both the peak electronic temperature as well as the relaxation rate are significantlyperturbed when the excitation is made resonant with the E mode, an effect that is ab-sent in monolayer graphene, in which light cannot couple to the in-plane lattice vibration.We explain the data by discussing the dynamical band structure changes combined witha non-adiabatic temporal response, a genuinely new type of carrier excitation for the solidstate. Similar concepts may be extended beyond graphene, for example to the transition7etal dichalcogenides, opening up new avenues for electronic structure control with light.The complex circular motion of the electrons throughout momentum space, which involvesultrafast manipulation of the electronic structure of the system over very few femtoseconds,may be interesting for applications in optoelectronics at very high bit rates. ACKNOWLEDGMENTS
We thank J¨org Harms for assisting with the figures and Axel K¨ohler for hydrogen-etchingand argon-annealing of the samples. Access to the Artemis facility at the Rutherford Apple-ton Laboratory was funded by STFC. A.L. acknowledge financial support from the GermanScience Foundation (DFG, SFB 925) and the EU-Flagship Graphene. ∗ Electronic address: [email protected][1] J. C. Johannsen, S. Ulstrup, F. Cilento, A. Crepaldi, M. Zacchigna, C. Cacho, I. C. E. Turcu,E. Springate, F. Fromm, C. Raidel, T. Seyller, F. Parmigiani, M. Grioni, and P. Hofmann,
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Phys. Rev. B , 075126 (2014) UPPLEMENTAL MATERIALSample preparation
Prior to graphene growth, the SiC substrate was hydrogen-etched to remove scratchesfrom mechanical polishing, resulting in atomically flat terraces. The substrate was graphi-tized in argon atmosphere, and the resulting carbon layer(s) was (were) decoupled from thesubstrate by hydrogen intercalation. After characterization with angle-resolved photoemis-sion spectroscopy, the samples were transported under ambient conditions to the ARTEMISfacility, where they were reinserted into ultra-high vacuum and cleaned by annealing at200 ◦ C. Tr-ARPES experiments
The setup consists of a Ti:Sapphire laser system (780 nm, 1 kHz, 30 fs, 15 mJ), where partof the light is converted to MIR frequencies using an optical parametric amplifier (OPA)with difference frequency generation (DFG), resulting in wavelength-tunable pump pulseswith a spectral width of ∼ (cid:126) ω probe = 31 eV. The extreme ultra-violet (XUV) probeejects photoelectrons from the sample whose energies and momenta are determined by ahemispherical analyzer (SPECS, Phoibos 150), giving direct access to the occupied part ofthe electronic band structure. Frozen phonon calculations
First-principles density functional theory calculations were performed using the Vienna ab initio simulations package (VASP) [35] with projector augmented (PAW) plane waves[36, 37]. The local density approximation (LDA) was employed to the exchange-correlationpotential, and the Brillouin zone was sampled by a 45 × × a = 2 .
445 ˚A and an interlayerspacing of 3.29 ˚A. 11 stimate displacement of carbon atoms
The electric field amplitude E is related to the fluence F via E = (cid:114) Fc(cid:15) ∆ t , where c is the speed of light, (cid:15) is the vacuum permittivity, and ∆ t is the pulse duration. Thefluence in the present experiment was 0.26 mJ/cm . This corresponds to a field amplitude of E = 1 MV/cm for a pulse duration of ∆ t = 200 fs. The real part of the optical conductivityat resonance with the E phonon mode at ω = 1594 cm − for a hole concentration of n = 4 × cm as determined by static ARPES is σ ( ω ) ∼
250 Ω − cm − [39]. From there,we calculate the polarization P = σ ( ω ) ω E = 6 × − C cm − . The polarization arises due to a light-induced dipole moment P = d × n × Z eff , where n is the number of dipoles per unit volume. The effective charge Z eff = 0 . e is taken from [39].For bilayer graphene, there are two dipoles per unit cell with volume V = 17 .
66 ˚A . Thisresults in d = P/ ( n × Z eff ) = 5 pm which corresponds to 2% of the in-plane lattice constant. Tr-ARPES data for different pump wavelengths in monolayer and bilayer graphene
Figure 5 shows the pump-induced changes of the photocurrent at t = 0 fs for both bilayer(upper panel) and monolayer graphene (lower panel) for different pump wavelengths. Thepump-probe signal is determined by a change in the electronic temperature and a transientbroadening of the electronic structure. Figure 6 shows the corresponding evolution of theFermi width together with fits including an error function for the rising edge and the sumof two exponentials for the decay. The relevant fit parameters are shown in Fig. 3 of themain manuscript. In Fig. 7 we compare the pump wavelength and electronic temperaturedependence of the fast relaxation time τ in monolayer and bilayer graphene. The absence ofany wavelength or temperature dependence in the monolayer, where the light cannot coupleto the in-plane lattice vibration, indicates that the observed wavelength dependence in thebilayer has to be attributed to coherent phonon oscillations. In particular, Fig. 7 clearly12 IG. 5: Peak pump-probe signal for bilayer (top row) and monolayer graphene (bottom row) fordifferent pump wavelengths. Unless indicated otherwise, the sample temperature was 30 K and thepump fluence 0.26 mJ/cm . shows that we can exclude a heating effect [40] as possible origin of the observed pumpwavelength dependence. Electronic temperature minimum
Here, we calculate the change in electronic temperature due to a lattice distortion alongthe E phonon coordinate. One can see from Fig. 7 that a typical electron-lattice relaxationprocess takes τ > . ν = 48 THz. Thus ντ >
1, indicatingthat, at early times, the electronic subsystem can be considered to be isolated. Therefore,both the electron number N and the entropy of the system S are conserved during the latticedistortion. This results in the following system of coupled equations N (DOS , µ , T ) = N (DOS , µ , T ) ,S (DOS , µ , T ) = S (DOS , µ , T ) , IG. 6: Width of the Fermi cut-off for bilayer (left) and monolayer graphene (right) for differentpump wavelengths. Continuous lines represent fits including an error function for the rising edgeand the sum of two exponentials for the decay. Unless indicated otherwise, the sample temperaturewas 30 K and the pump fluence was 0.26 mJ/cm . IG. 7: Comparison of the fast relaxation time τ in monolayer (blue) and bilayer graphene (red)as a function of pump wavelength (left) and peak electronic temperature (right). Continuous redand blue lines are guides to the eye. where the particle number and the entropy can be obtained from N = (cid:90) DOS( E ) f ( E ) dE,S = (cid:90) DOS( E ) [ f ( E ) ln f ( E ) + (1 − f ( E )) ln (1 − f ( E ))] dE. Subscript 1 (0) refers to the (un)distorted lattice, f ( E ) is the Fermi-Dirac distribution.The above equation is only valid if the electronic system follows a thermal distribution.This is not necessarily the case for the non-adiabatic excitation mechanism discussed inthe main text. The experimental time resolution, however, is too slow to resolve this non-adiabatic dynamics, so that the measured photocurrent can be nicely fitted by a Fermi-Diracdistribution at all times. Thus, in order to compare the frozen phonon calculations to theexperimental data, we assume that the electronic system can be assigned a temperature T . The calculated band structure and density of states for different lattice distortions areshown in Fig. 8. With µ , T , and DOS , known from tr-ARPES data and frozen phononcalculations, respectively, we can easily calculate µ and T for a given lattice distortion.Using µ = −
200 meV and T = 3600 K, we find µ = −
140 meV and T = 2745 K for anatomic displacement of 3% of the lattice constant, in good agreement with our data.15 IG. 8: Calculated electronic structure (left) and density of states (right) for different latticedistortions along the E mode coordinate.mode coordinate.