Appl. Phys. A (2006)
Applied Physics A
DOI: 10.1007/s00339-006-3711-0
Materials Science & Processing
Time-resolved two-color interferometric photoemission of image-potential states on Cu(100)
u ¨ j. gudde m. rohleder ¨ u. hofer
Fachbereich Physik und Zentrum für Materialwissenschaften, Philipps-Universität, 35032 Marburg, Germany
Received: 2 June 2006/Accepted: 9 August 2006 © Springer-Verlag 2006 ABSTRACT We describe an interferometric time-resolved photoemission technique that makes it possible to simultaneously observe the decay of optical induced polarizations and populations at surfaces in a two-color excitation scheme. In this scheme initially unoccupied electronic surface states are coherently excited by the interaction of laser pulses with frequency ωa and the two-photon polarization which is induced by laser pulses with frequency ωa /2. Interference is observed by changing the delay between both laser pulses using an actively stabilized two-color Mach–Zehnder interferometer. We demonstrate this technique for excitation of the n = 1 image-potential state on a Cu(100) surface. PACS 78.47.+p;
1
79.60.Bm; 73.20.-r; 82.53.Kp; 42.50.Md
Introduction
Time-resolved two-photon photoemission (2PPE) has become a very powerful tool for the investigation of electron dynamics at solid surfaces [1–5]. In the most basic excitation scheme, a first photon of energy hωa excites an electron from an occupied initial state |i below the Fermi energy E F to an intermediate state |n with energy E n . A second photon of energy hωb is used to lift the electron above the vacuum energy E vac into the final state | f . Usually the kinetic energy distribution of final states is measured for different parallel momenta k , i.e., with angle resolution, as a function of time delay ∆t between pump and probe pulses hωa and hωb . Via the relationship E n = hωa − E kin the temporal evolution of the population of excited electrons in the state |n is then deduced. After the pump pulse is over, the population in state |n is only affected by inelastic scattering of the excited electrons. In this way it is possible to determine inelastic lifetimes τn of electrons in surface states [2–12] or energy shifts correlated with molecular motion [13–16]. The strength of 2PPE for these types of experiments lies in the fact that both energy and parallel momentum of the excited electrons can be followed directly by the experiment. The use u Fax: +49 6421 28-24218, E-mail:
[email protected]
of different photon energies hωa = hωb for the pump and the probe pulse has two main advantages for the precise determination of the inelastic lifetime of the intermediate state. Firstly, it makes it possible to choose different intensities for the pump and probe beam, which results in a high signal at low background. Secondly, the resulting asymmetry and shift of the time-resolved signal relative to zero delay carry additional information about the lifetime. This helps one to extract inelastic lifetimes that are even much shorter than the temporal width of the laser pulses in the limit that the population of the intermediate state instantaneously follows the excitation by the pump pulses [4, 17]. However, a short excitation laser pulse does not immediately change the population of electronic states. The primary action of the incident electric field is the creation of a coherent superposition of initial and intermediate state, i.e., the electric field induces a polarization. In second-order in the applied electric field, this then leads to a change of the population. For this reason, 2PPE as other ultrafast optical probes is principally capable of accessing also coherent electron dynamics, i.e., it can follow the loss (dephasing) of the optically induced coherence between electronic states due to inelastic and elastic interaction with the environment. Here, the total dephasing rate of the polarization between state |i and |n can be written as 1/Tin = 1/2τn + 1/Tin∗ , where 1/Tin∗ is the "pure" dephasing rate due to elastic scattering, which can be decomposed into the contributions of initial and intermediate state by 1/Tin∗ = 1/Ti∗ + 1/Tn∗ . Whereas optical surface spectroscopies based on secondharmonic generation and sum-frequency generation are increasingly exploiting such coherent effects [18–26], only a few studies have addressed coherence phenomena using 2PPE. Perhaps the most well-known example is quantum-beat spectroscopy of image-potential states [27–32]. In this case, several close lying states are excited within the bandwidth of a short laser pulse. The resulting beating of the 2PPE signal as function of time delay provides not only an accurate measure of the energy separation of the states. The comparison of the decay of the oscillations with the overall decay of the 2PPE signal also directly reflects quasielastic scattering processes of the electrons in the intermediate states with rate 1/Tn∗ , which results in a loss of phase correlation between their wavefunctions [28–30].
Applied Physics A – Materials Science & Processing
As long as h/Tn∗ is large compared to the experimental energy resolution given by the detector resolution and the bandwidth of the probe pulses, it can also be determined in the frequency domain by measuring the linewidth of the photoemission peak for large delays between pump and probe pulses if inhomogeneous broadening can be neglected and the laser pulse form is known [33]. Another photoemission scheme to observe dephasing in the time domain is interferometric time-resolved 2PPE introduced by Petek an co-workers [34–36]. In these experiments the probe pulse is a time-delayed replica of the pump pulse (hωb = hωa ). Therefore the intermediate state can be populated not only by one of the laser pulses, but also by the interaction of laser field E1(ωa ) with the polarization that is induced by laser field E2(ωa ) and vice versa. Quantum interference between these different pathways makes it possible to control the population of the intermediate state by changing the relative phase between the two laser fields [34]. This has been realized by changing the delay between the two laser pulses, which was kept interferometrically stable. The observed interferometric resolved photoemission intensity, i.e., the population of the final state | f , oscillates as a function of delay with ωa and 2ωa due to the induced coherence between |i and |n, on the one hand, and between |i and | f , on the other hand [3, 37]. This makes it possible to determine not only the inelastic lifetime from the phase-averaged photoemission signal, but also the dephasing rates 1/Tin and 1/Ti f from the ωa and 2ωa component of the oscillating signal. The detailed analysis of this single-color excitation scheme, however, requires well-characterized and extremely short laser pulses. The associated large bandwidth may lead to quantum beats due to excitation of several intermediate states, which are unwanted here, since they complicate the extraction of the ωa and 2ωa components.
In this contribution we report on an extension of the 2PPE technique in which an initially unoccupied surface state is coherently excited by two laser pulses with frequency ωa and ωa /2. Such simultaneous coupling of two electronic states by a one- and two-photon excitation becomes possible in dipole approximation in non-centrosymmetric materials [38] or at surfaces where the inversion symmetry is broken in general. Then the intermediate state can be populated not only by the interaction of the laser field E1 (ωa ) with the first-order optical polarization P (1) (ωa ) = χ (1) E1(ωa ) that is induced by the same field, but also by interaction of E1(ωa ) with the secondorder polarization P (2) (ωa ) = χ (2) : E2(ωa /2)E2(ωa /2) that is induced by a second laser field E2(ωa /2) with frequency ωa /2 (cp. Fig. 1). Here, χ (1) and χ (2) are the linear and second-order non-linear dielectric susceptibilities of the system, where χ (2) = 0 only in a non-centrosymmetric environment. The two-color excitation pathway allows the control of the population of the intermediate state by changing the relative phase between laser field E1 (ωa ) and E2 (ωa /2). This makes it possible to perform interference resolved photoemission experiments in analogy to single-color interference resolved 2PPE, but with the advantages of a time-resolved two-color excitation and photoemission scheme. For this purpose we have developed a two-color actively stabilized Mach–Zehnder interferometer which allows a continuous phase-locked scanning of the delay between two laser pulses of frequency ωa and ωa /2. We demonstrate this technique for the excitation of the n = 1 image-potential state at the Γ¯ -point on the Cu(100) surface (cp. Fig. 1) and show how the incoherent dynamics of the image-state population and the coherent dynamics of the optically induced polarization between the initial bulk state and the image-potential state can be simultaneously observed. 2
FIGURE 1 Schematic energy diagram for a coherent two-color excitation of electrons from a occupied initial bulk state |i into an image-potential state |n and subsequent photoemission into an unbound final state | f . In a noncentrosymmetric environment, like a surface, the states |i and |n can be coupled even in dipole approximation by a one- as well as by a two-photon excitation. Inelastic scattering of the excited electrons leads to a decay of the population of the intermediate state |n with rate Γn . Elastic scattering of the excited electrons and the holes as well as inelastic population decay contribute to the total dephasing rate Γin of the optically induced coherent polarization between states |i and |n
Experiment
The experiments were performed at room temperature in an ultrahigh vacuum chamber with a base pressure of 6 × 10−11 mbar. The Cu(100) sample was prepared by standard sputtering and annealing procedures. Surface cleanliness and order were verified by X-ray photoelectron spectroscopy (XPS), low energy electron diffraction (LEED), and by line width measurements with 2PPE [28]. Photoelectrons emitted from the sample along the surface normal were detected by a hemispherical electron energy analyzer with an energy resolution of 10 meV. The optical setup consists of a Ti:sapphire femtosecond laser system (Coherent RegA) operating at 800 nm with a repetition rate of 100 kHz that generates 60-fs laser pulses with a pulse energy of 6 µJ. The output of the amplifier is used to pump a traveling-wave optical parametric amplifier (OPA) that operates in the visible range (470 – 730 nm). For the present experiments the OPA was tuned to produce 570-nm pulses with a pulse energy of 400 nJ. The dispersion of these visible laser pulses was compensated by a prism compressor consisting of a pair of LaFN28 Brewster prisms. Laser pulses in the UV, which served as pump pulses with frequency ωa , were generated by frequency doubling of the visible pulses (frequency ωa /2) in a 100-µm thick type I phase-matched BBO crystal.
GÜDDE et al.
Time-resolved two-color interferometric photoemission of image-potential states on Cu(100)
identical with the beam-splitting substrate BS in order to compensate the group-delay of UV and visible laser pulses. For the same reason a further fused-silica substrate (DS), identical with the entrance window of the UHV-chamber, is placed in the path of the control beam. Rotation of PS changes the difference of the effective path lengths and therefore the relative phase between UV and visible pulses due to their different dispersion within the window. The phase shift of a laser field with frequency ω passing a substrate of thickness d and index of refraction n(ω) depends nonlinearly on the angle of incidence α and is given by Φ(ω) = −
FIGURE 2 Optical setup for the generation of phase-locked laser pulses at frequency ωa and ωa /2. See text for details
The time delay and relative phase between the visible and UV laser pulses could be continuously tuned with high reproducibility by employing an actively stabilized two-color Mach–Zehnder interferometer depicted in Fig. 2. The interferometer consists of two dichroitic beam splitters (DBS) that split and recombine the two collinear laser beams and a piezodriven 75-µm translation stage (PZT) which varies the path length of the visible beam. The polarization of the visible pulses can be changed by means of a half-wave plate placed in the ‘visible’ arm of the interferometer. A small part of the two-color output of the interferometer is split off by an uncoated plan-parallel fused-silica substrate (BS) for feedback control. The two collinear laser beams in the control arm are focused by an Al mirror ( R = 20 cm) into a second 100-µm thick BBO crystal. The interference of the frequency doubled visible and the UV part of the control beam is detected by a photomultiplier (PM). The remaining visible beam is blocked by a color filter (CF). A phase-locking feedback loop has been realized by applying a small modulation voltage with a frequency of 160 Hz at the piezo-stage that modulates the phase between visible and UV laser pulses. The resulting intensity modulation of the interference signal is detected by a lock-in amplifier. The output of the latter serves as errorsignal input of a proportional-integral controller which drives the piezo-stage. A continuously tunable phase-shift between the visible and UV laser pulses has been realized by placing a plan-parallel fused-silica substrate (PS) with a thickness of 6.5 mm in the control beam, which can be rotated by a computer-controlled rotation stage. The substrate PS is
ωdn(ω)/c0 , [1 − sin2 (α)/n 2 (ω)]1/2
(1)
where c0 is the speed of light in vacuum. The feedback control compensates the relative phase-shift ∆Φ = Φ(ωa ) − 2Φ(ω2 /2) in the control arm by changing the path length of the visible beam in the interferometer and thereby the relative phase between UV and visible pulses in the main output of the interferometer, which has been used in the experiment. Figure 3 shows the position of the piezo-driven translation stage as a function of angle of incidence α of the laser beams on the phase-shifting substrate PS during a phase-locked rotation of PS. Rotation of PS from 10 to 40 degrees leads to a continuous and smooth change of the pathlength between UV and visible pulses of 11 µm which corresponds to a phaseshift of about 77π at a wavelength of 285 nm. The data shown in Fig. 3 represent phase-locked operation of the interferometer over a period of 2.4 h without phase-jumps, which would lead to sudden changes of the piezo position of at least λ/2 = 142.5 nm. In the present setup, phase-locking is only possible for delays in the order of the pulse widths due to the difference of phase and group delay between UV and visible pulses within the phase-shifting substrate. However, the feedback control can be straightforwardly extended in order to phase-stabilize pulses also for delays that are larger than the pulse width by
FIGURE 3 Position of the piezo-driven translation stage as a function of tilt angle α of the phase-shifting fused silica substrate in the control beam path of the two-color interferometer during a phase-locked scan. The position of the translation stage was measured by an internal capacitive encoder which has a resolution of 10 nm. The translation stage is controlled by the phase-locked loop which is not limited by the encoder resolution
Applied Physics A – Materials Science & Processing
detecting the interference of the laser pulses in the frequency domain [35, 39]. 3
Results and discussion
The coherent two-color excitation of the population of the n = 1 image-potential state as depicted in Fig. 1 is demonstrated in Fig. 4 where the photoemission yield is plotted as a function of relative phase ∆Φ = Φ(ωa ) − 2Φ(ω2 /2) between the UV and visible laser pulses. The solid curve is a cos(∆Φ)-fit to the data with the known angle of incidence, fixed thickness of the phase-shifting substrate PS and index of refraction for the frequencies ωa and ω2 /2 taken from the Sellmeier equation of fused silica [40]. Thus, ∆Φ is fixed, except for a constant Φ0 which is chosen here by demanding that the photoemission yield should have a maximum for ∆Φ = 0. The variation of the photoemission yield from the image-potential state with cos(∆Φ) shows that the excitation pathway has a strong contribution of a coherent two-color process where one photon of frequency ωa and two photons of frequency ωa /2 are participating. One of these pathways is the interaction of the UV laser field E1(ωa ) with the two-photon polarization P (2) (ωa ) = χ (2) : E2 (ωa /2)E2(ωa /2) that is generated by the visible laser field E2(ωa /2). Further equivalent pathways result from consideration of all permutations of these three photons in the interaction sequence, which all contribute to a cos(∆Φ) dependence. The strength of the interference contrast results from the competition between the coherent two-color interaction and the exclusive excitation by the laser field E1(ωa ), which is responsible for the incoherent background. While the population for the first excitation pathway is proportional to E 1 (ωa )[E 2 (ωa /2)]2 , it depends quadratically on the field strength E 1 (ωa ) for the latter. Thus, the interference contrast can be maximized by increasing the intensity ratio between the visible and UV laser pulses on the cost of total signal strength. The data depicted in Fig. 4 show an interference contrast of about 20% and have been taken with a ratio between the laser power of visible and UV pulses of about 400 : 1. By further decreasing the UV power the interference contrast
FIGURE 4 Photoemission yield as a function of relative phase ∆Φ = Φ(ωa ) − 2Φ(ω2 /2) between the two phase-locked laser pulses exciting the n = 1 image-potential state. The solid curve is a fit of a cos(∆Φ) dependence
could be increased up to 50% as shown in Fig. 5 where the interference contrast and the phase-averaged signal is plotted as a function of UV intensity IUV while the intensity of the visible laser pulses √ was kept constant. The interference contrast follows a 1/ IUV ∝ 1/E 1 (ωa ) dependence while the incoherent 2PPE background increases linearly with IUV as depicted by the solid lines. Both findings are in perfect agreement with the expected intensity dependencies of the two excitation pathways. The potential of the coherent two-color excitation scheme for a simultaneous but independent determination of the inelastic population lifetime and the dephasing rate of the coherent polarization between the initial and intermediate state is demonstrated in Fig. 6 where Fig. 6a shows the photoemission yield for excitation of the n = 1 image-potential state as a function of delay between UV and visible pulses. Negative delays correspond to an excitation sequence with preceding visible pulses. The feedback control for the active phase-stabilization was switched off during this scan, but even the passive stability of the interferometer was sufficient to observe a well modulated photoemission yield with an interference contrast of up to 20%. The phase-averaged signal represents the population dynamics of electrons in the n = 1 state as can be observed in a usual two-color 2PPE experiment. Its asymmetric shape reflects the finite population lifetime of electrons in the intermediate state. The rapidly modulated contribution of the photoemission yield results from the coherent interaction of the laser field E2(ωa /2) with the optical polarization induced by the laser field E1(ωa ) and represents the dynamics of the coherence between a initially occupied bulk state and the n = 1 image-potential state. The different dynamics of population and polarization are clearly visible on first sight due to the different asymmetric shape of the phase-averaged and the strongly modulated contribution to the photoemission yield. This is emphasized by the two blow-ups of the increasing and decreasing tail of the transient data (Fig. 6b and c), which show comparable averaged photoemission yields, but a modulation amplitude
FIGURE 5 Interference contrast of the 2PPE yield as a function of the UV laser intensity IUV ∝ E a (ω)2 (data points) for coherent two-pulse excitation of the n = 1 image-potential state on Cu(100). The curve depicts a 1/E a (ω) dependence, where E a (ω) is the electric field strength of the UV laser pulses. The inset shows the (linear) intensity dependence of the incoherent (phaseaveraged) 2PPE signal
GÜDDE et al.
Time-resolved two-color interferometric photoemission of image-potential states on Cu(100)
FIGURE 6 (a) Photoemission yield as a function of delay time between the UV (frequency ωa ) and visible (frequency ωa /2) laser pulses in steps of 0.2 fs. (b,c) depict two blow-ups of (a) for positive and negative delay respectively. (d) The phase-averaged signal (cyan points) represents the incoherent population dynamics of the n = 1 image-potential state, which is plotted together with a fit using a rate-equation model (solid cyan line). The envelope of the fast oscillating contribution plotted in red represents the coherent dynamics of the optical polarization which is induced by the laser field E a (ωa )
which is a factor of two larger for the increasing tail (negative delays). The incoherent part of the photoemission yield has been separated from the data by low-pass filtering and is shown as cyan points in Fig. 6 This data is shown together with a fit of the population dynamics using a rate-equation model. It yields an inelastic lifetime of 31 ± 4 fs for the n = 1 state, which is slightly smaller than the reported lifetime of τ1 = 40 fs [27, 41, 42]. The observed reduction of the lifetime may be caused by a residual chirp of the laser pulses. The present experiment on Cu(100) has been set up primarily for the demonstration of the two-color excitation technique, where we made no attempt to fully compensate the dispersion simultaneously for the UV and visible laser pulses at the sample position. This requires a dispersion control within the Mach–Zehnder interferometer, which has not yet been realized. The envelope of the fast oscillation part of the photoemission yield, which is shown as red points in Fig. 6d, has been
extracted by Fourier-filtering in the following way: First the original data were filtered by a wide band-pass with a center frequency which corresponds to the frequency of the fast oscillating contribution of the data in order to get rid of the incoherent background. The absolute magnitude of this result was then low-pass filtered, which gives the magnitude of the envelope of the fast oscillating contribution. Clearly the polarization shows much faster dynamics compared to the population. If zero delay is known the dephasing rates of the coherent polarizations between the initial, intermediate and final state can be determined by modeling the data using the multi-level Bloch equation formalism [3, 33, 41]. The clear advantage of a two-color interferometric excitation scheme is the asymmetric shape of the transient data for final dephasing times, which is accompanied by a temporal shift of the curve maximum relative to zero delay. Both represent, in analogy to an incoherent two-color 2PPE experiment, an additional independent information, which makes it possible to determine dephasing times even for laser pulse durations that are long compared to the dephasing times. In the present experiment on the Cu(100) surface, however, zero delay is not well defined since it has not been determined independently as could be done, for example, by observation of non-resonant two-photon photoemission from the occupied Shockley surface state on the (111) surfaces of noble metals. Therefore we will not give a quantitative result for the dephasing time of the optical polarization for excitation of the n = 1 imagepotential state here. However, the qualitative observation of a dephasing time which is much shorter than the population lifetime can be in any case attributed to the contribution of the initial states to the total dephasing time Tin since the contribution of the electron in the n = 1 image-potential state ∗ (pure dephasing time Tn=1 due to elastic electron scattering) has been estimated from linewidth analysis to be larger than 130 fs [4, 41, 43]. For the evaluation of the initial-state contribution to the dephasing of the optical polarization, one has to consider that in contrast to the intermediate state which is here represented by a single image-potential state, the initial state is in fact a continuum of bulk states within the bandwidth of the laser pulses. Therefore dephasing can not only occur due to scattering of the photo-hole, but also due to destructive interference of a system of coupled polarizations in the case of homogeneous broadening. This can lead to the observation of an extremely rapid dephasing even if there is no scattering at all [44]. 4
Summary
In summary, we have shown that the intrinsic optical non-linearity of a surface can be used to apply a coherent two-color excitation scheme for the population of initially unoccupied surface states. We have demonstrated its extension to an interferometric time-resolved photoemission scheme for the n = 1 image-potential state on Cu(100) where we have simultaneously observe the inelastic decay of the population as well as the dephasing of the optical polarization with the advantages of a two-color setup. The combination of this technique in the time-domain with linewidth measurements can be applied to determine inelastic and elastic lifetimes of surface states as well as lifetimes of the photo-holes.
Applied Physics A – Materials Science & Processing ACKNOWLEDGEMENTS We thank W. Berthold for stimulating discussions and acknowledge funding by the Deutsche Forschungsgemeinschaft through HO 2295/3 and by the Center for Optodynamics, Marburg.
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