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Experimental observation of anharmonic effects in coherent phonon dynamics in diamond
Accepted Manuscript Experimental observation of anharmonic effects in coherent phonon dynamics in diamond
M. Zukerstein, M. Kozák, F. Trojánek, P. Malý PII: DOI: Reference:
S0925-9635(18)30694-0 https://doi.org/10.1016/j.diamond.2018.10.018 DIAMAT 7235
To appear in:
Diamond & Related Materials
Received date: Revised date: Accepted date:
8 October 2018 24 October 2018 24 October 2018
Please cite this article as: M. Zukerstein, M. Kozák, F. Trojánek, P. Malý , Experimental observation of anharmonic effects in coherent phonon dynamics in diamond. Diamat (2018), https://doi.org/10.1016/j.diamond.2018.10.018
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Experimental observation of anharmonic effects in coherent phonon dynamics in diamond M. Zukerstein,1,* M. Kozák,1 F. Trojánek,1 and P. Malý1 1 *
Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 12116 Prague 2, Czech Republic Corresponding author: [email protected]
Abstract
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We present a study of coherent optical phonon dynamics in monocrystalline and polycrystalline diamond using transient transmission spectroscopy with few-cycle laser pulses. Besides the oscillations at the fundamental phonon frequency of 40 THz we observe a signal at second harmonic frequency and four weak frequency sidebands (25.0 THz, 32.5 THz, 47.5 THz and 55.0 THz). Potential origin of these sidebands is discussed along with the dependence of the observed signal on experimental conditions (pump power, sample thickness, etc.). Phonon dephasing time in polycrystalline diamond T2,poly = 6.0 ± 0.4 ps is found to be shorter than in monocrystalline diamond T2,mono = 7.0 ± 0.4 ps.
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Keywords
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Single crystal diamond; polycrystalline diamond; phonons; anharmonic effects; ultrafast spectroscopy; few-cycle laser pulses.
Introduction
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Ultrafast switching and information processing at frequencies > 10 THz may become accessible via the manipulation of coherent excitations in solids with light. Time-resolved optical spectroscopy provides information about various ultrafast processes in solids. Relevant time scales (fs – ps), which can be studied, cover electronic relaxation and recombination as well as dynamics of lattice vibrations. These can be generated in the form of incoherent population or as a coherent phonon wave.
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Coherent phonon population is excited either by the fast relaxation of carriers excited by above -band gap absorption [1,2,3] or via stimulated Raman scattering (SRS) [1,4]. The first principle of coherent phonon generation is through resonant carrier excitation, which, due to the coupling between the lattice and the electronic excitations, leads to a local transient change of lattice con stants and an impulse excitation of coherent phonons. The second principle is generation of coherent phonons by SRS, which requires two photons whose frequencies differ by the vibrational energy of the coherent optical oscillations. This can be reached by application of a single laser beam whose spectral width is much bigger than the energy of generated phonons and the pulse duration is shorter than one lattice vibration period [1] or by two laser beams whose frequencies differ by the frequency of the coherent optical oscillations. These pulses may then be longer in the time domain than the period of the generated phonons because the total intensity is modulated in time at the difference frequency of the two laser fields. This corresponds to the train of pulses separated by the period of the phonon. Additionally, phonon population can be coherently controlled using a sequence of two pump pulses [5]. Lattice vibrations can be observed via transient transmission (reflection) signal using the pump-and-probe method. Once a pump pulse generates the coherent phonons, they subsequently modulate the electrical susceptibility, and hence the refractive index, in the material. These slight oscillatory changes can be measured by another pulse (probe) whose transmissivity (reflectivity) is modulated. As shown in [6], the spectrally resolved modulation of reflectivity at optical frequency ω is given by: 𝛥𝑅 (𝜔 ) = 2𝜔0 𝛼√𝛺⁄2ℏ 𝑄0 [{𝜒 (𝜔 + 𝛺) − 𝜒 (𝜔 )} 𝐸0 (𝜔 + 𝛺) − {𝜒 (𝜔 ) − 𝜒 (𝜔 − 𝛺)}𝐸 (𝜔 − 𝛺)] 𝐸0 (𝜔 ) 𝑠𝑖𝑛 𝛺𝜏𝑝 , (1)
ACCEPTED MANUSCRIPT where α is the dimensionless electron-phonon coupling constant, ω0 is the central frequency of the probe pulse, Ω is the phonon frequency, Q0 is the amplitude of oscillation, χ(ω) is the electric susceptibility of the crystal, E0 (ω) ≡ exp(–iωτp )E(ω) is a real quantity in the Fourier transform limit and τp is the delay between pump and probe pulses. From the equation (1) it can be easily shown that the components on the frequencies ω + Ω and ω Ω oscillate out of phase so the spectrally integrated signal is almost cancelled out [6]. The nonzero signal can only be caused by the curvature of the dispersion curve [6]. For this reason, the spectrally resolved detection leads to a strong increase of the oscillatory signal, both in transmission and reflection geometries.
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In the case of weak driving force acting on the atoms in a lattice, the interatomic potential can be treated as harmonic. As a consequence, after an impulsive excitation, the lattice vibrates at a single frequency given by the strength of the chemical bonds between neighboring atoms. However, if the excursion of an atom from its equilibrium position is large enough, anharmonic terms of the potential lead to an anharmonicity of the coherent vibrations and refractive index modulation. This manifests itself in a comb of harmonic frequencies of the fundamental phonon frequency, which can be observed in the transient reflectivity signal [7,8]. The origin of the Raman frequency comb can be different. It can be generated through anharmonic crystal lattice response [7,8] or through cascaded high-order Raman scattering. Study of strongly driven atomic vibrations and monitoring the anharmonicity of the generated infrared photons has recently been used for reconstruction of the interatomic potential [9]. In this paper we investigate properties of coherent phonons in diamond generated by few-cycle laser pulses. In the measured transient transmission signal we observe oscillations at optical phonon diamond frequency of 40 THz, its second harmonics and four additional sidebands.
Experimental
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The studied samples are commercially available diamonds (Element Six) prepared by chemical vapor deposition (CVD). The first sample is monocrystalline diamond of type IIa with dimensions 4.5 × 4.5 × 0.5 mm. It has a very low concentration of point defects and impurities (< 5 ppb of nitrogen, < 1 ppb of boron, < 0.03 ppb of nitrogen-vacancy centers). Its front face has crystallographic orientation <100> with the roughness better than 5 nm. In some experiments, a thinned monocrystalline samples with the same properties was used (thickness of 5.8 μm). The sample was thinned down by electrochemical etching by Diamond Materials Company in Freiburg, Germany. The other sample is polycrystalline CVD diamond with a diameter of 10 mm and thickness of 0.5 mm. The surface is polished with roughness better than 15 nm. The average diameter of the randomly oriented grains of polycrystalline diamond sample is approximately 10 % of the film thickness and the grains are elongated in the direction perpendicular to the surface. High concentration of defects results in relatively high strain and strong scattering of visible light.
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The pump and probe technique was used to measure the dynamics of coherent optical phonons in diamond. The time delay between pump and probe pulses obtained by amplitude splitting of the output of a few-cycle Ti:sapphire laser (Rainbow (Newport)) was controlled by an optical delay line providing temporal resolution of 0.1 fs. The pump beam was modulated by an optical chopper at fchopper = 500 Hz and the signal of the probe transient transmissivity was detected by a lock-in phase-sensitive amplifier operating at the frequency fchopper. The transmitted (reflected) probe was measured by a silicon photodiode. Sp ectrally resolved detection was provided by the monochromator or bandpass filters in the probe beam situated between the sample and the photodiode. The pulse durations of both the pump and the probe pulses were measured by the Spectral Phase Interferometry for Direct Electric Field Reconstruction (SPIDER) method to be τFWHM = 7 fs. In the collinear geometry, both laser beams were focused on the sample by off-axis parabolic mirror with focal length of 15 mm to the 1/e2 radius of w0 = 2.6 μm. The maximum pulse energy was Epump = 33 mJ/cm2 for the pump and Eprobe = 2 mJ/cm2 for the probe pulses, respectively. Linear polarization of the pump beam was parallel to the <110> crystallographic axis and perpendicular to the probe polarization, in order to minimize the contribution of scattered pump light to the detection. The transient reflectivity measurements were performed in the non collinear geometry with two spherical mirrors with focal lengths of 150 mm. However, in this geometry the combination of reflectivity and transmissivity due to the reflection of the probe beam from the back surface of the sample is measured.
ACCEPTED MANUSCRIPT Results and discussion
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In the inset of Fig. 1(a) we show the measured differential transmissivity for the monocrystalline diamond in the spectral window of 795 ‒ 805 nm. The Fourier transform of the signal shown in Fig. 1(a) reveals the fundamental lattice vibration frequency of νph = 39.95 ± 0.05 THz. This frequency is independent of the pump peak energy in the range 0 ‒ 33 mJ/cm2 , of spectral window of detection and of the type of diamond (the same frequency is observed for both the monocrystalline and polycrystalline samples). The corresponding wavenumber 1333 ± 2 cm-1 , agrees well with incoherent Raman scattering data [10] and time-resolved CARS [11]. The mechanism of coherent phonon generation in diamond can be explained in time domain [12]. When the polarization of the excitation light is parallel to <100> crystallographic direction, the force acting on an atom in the crystal lattice is linearly proportional to the oscillating light field. After the interaction with a short pulse, the total momentum change of the atom is zero. However, for polarization along <110> direction, the force has a component, which is quadratic with the incident light field (linear with the intensity), which causes acceleration of the atom in the propagation direction of the driving light [12]. Moreover, the force acting on the two neighboring atoms from the unit cell has opposite sign leading to generation of high -frequency coherent optical phonons. This excitation mechanism was verified by rotating the sample by 45° around the axis perpendicular to the front surface, where the oscillatory signal completely disappears, which agrees well with the Γ25 ’ symmetry of the Raman tensor of optical phonons in diamond [4].
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In Fig. 1(b) we display the time decay of the oscillatory transient reflectivity signal in the monocrystalline diamond in comparison with that in the polycrystalline diamond. Phonon dephasing times of these two samples T2,mono and T2,poly are highlighted with black arrows. Data from these measurements were fitted by a sine function with exponentially decaying amplitude ΔR/R = Aexp(-τ/T2,mono/2,poly )sin(2πνph τ + φ0 ). The measured dephasing time in monocrystalline diamond is T2,mono = 7.0 ± 0.4 ps, while in polycrystalline diamond it is only T2,poly = 6.0 ± 0.4 ps. Faster phonon dephasing in latter case can be explained by higher density of defects in the crystal lattice of polycrystalline sample. The resulting relaxation time T2,poly is then given by the expression 1/T2,poly = 1/T1 + 1/Tpure dep., where T1 is the decay time caused by the decrease in the population of phonons, and Tpure dep. is the decay caused by the effect of dephasing [11]. As in [4,6] we assume that in the monocrystalline diamond, the phonons are perfectly coherent all the time, and the effect of Tpure dep. can be neglected. Fig. 1(b) also shows noticeably smaller oscillation amplitude of the polycrystalline diamond. The average oscillation amplitude obtained from long time scale measurements at 737 nm for the monocrystalline diamond is Amono = (1.5 ± 1.1) ×
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10-4 , whereas the polycrystalline diamond has this amplitude approximately 4 times smaller Apoly = (3.9 ± 1.2) × 10-5 . Smaller amplitude and signal isotropy were expected in the polycrystalline diamond due to random orientation of single-crystalline grains. The observed dephasing times are much longer than those reported for micro- and nanocrystalline diamond films with grain sizes ranging from 10 ‒ 500 nm where the phonon decay times between 0.7 ‒ 1.72 ps were measured [11]. This discrepancy is in line with expected much larger amount of lattice defects in diamond nanocrystals.
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In addition to the main peak at 40 THz, in the Fourier Transform we observe sidebands distributed symmetrically around this main peak (Fig. 1(a)). The frequencies and relative intensities (the ratio between sidebands and the peak at the fundamental frequency) of these sidebands do not change when varying the pump intensity and the orientation of the sample. Such sidebands are observed in the transient transmission measurements in diamond for the first time to the best of our knowledge. Their origin is unclear with two possible hypotheses. The first is generation of coherent superposition of two acoustic phonons with opposite momenta by anharmonic decay of optical phonons [13–16]. If the frequency of these two phonons would be different, it may result in frequency mixing with the fundamental vibration mode. We expect the process, in which two phonons are involved, to be nonlinear with the excitation power. We cannot confirm this hypothesis as the nonlinear dependence on excitation power was not measurable due to high signal noise. The second possible explanation involves acoustic phonons generated at the surface of the sample, where the symmetry of the crystal is broken. Excitation of surface acoustic modes was theoretically predicted in [17]. To verify the role of the surface, we measured the transient transmission signal with the same experimental conditions at two monocrystalline samples with different thicknesses (500 μm vs 5.8 μm). No difference was observed between the relative ratios of the sidebands and the fundamental frequency peak. Therefore the role of surface-excited acoustic phonons is not likely. These peaks were not observed in previous measurements, in which the transient reflection anisotropy was detected [4,6].
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Fig. 1. (a) Fourier transform spectrum of coherent phonon oscillations with fundamental frequency at νph = 39.95 THz and second harmonic frequency at νSH = 79.90 THz. Sidebands symmetrically around this main peak could be explained as TA and LA phonons. The time window used for FT spectrum calculation was 1.5 – 12.5 ps. Inset shows the oscillating differential transient transmissivity signal. (b) The decay of differential reflectivity signal for monocrystalline (red line) and polycrystalline (green line) diamond. Black arrows illustrate the dephasing time of monocrystalline (τmono = 7.0 ± 0.4 ps) and polycrystalline diamond (τpoly = 6.0 ± 0.4 ps). The amplitude of polycrystalline diamond is approximately 4 times smaller. The spike-like signals at ≈8 ps are caused by optical interference of the probe pulse with the twice-reflected pump pulse from the back and front surface of the sample and do not influence the interpretation of the data. The plotted data are offset for clarity.
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The anharmonicity of the lattice potential can lead to the generation of higher harmonic frequencies of the fundamental phonon frequency – the generation of Raman frequency comb [7,8]. However, for the interaction in the bulk of diamond crystal, the symmetry does not allow even terms in the Taylor series of the atomic potential. This means that only odd terms of harmonic frequencies are expected in the frequency spectrum of atomic oscillations. Our observation of the signal at the second harmonic frequency νSH = 80 THz of the fundamental optical phonon is in contradiction to these expectations. Therefore its origin can be assigned to the cascade d Raman scattering process [18], in which the wave at a certain frequency from the bandwidth of the laser pulses twice inelastically scatters via stimulated Raman process. This leads to the oscillations of the phase of the scattered wave at twice the phonon frequency. In the narrow spectral windo w of detection, the scattered wave interferes with a part of the original laser spectrum leading to intensity modulation at νSH . High-order χ(3)nonlinear optical interaction in CVD diamond like multi-octave Raman frequency comb generation was studied in [18], where the cascaded stimulated Raman scattering was observed in frequency domain. In this work we perform the cascaded SRS by measuring the oscillating signal in time domain. To reveal the origin of the frequency sidebands and the second harmonics of t he fundamental optical phonon in the transient transmission signal, we process the data using continuous wavelet transform (CWT) [19]. This technique performs discrete Fourier transform from the signal in a narrow time window. By shifting this window, the frequency spectrum is obtained as a function of time delay. The results of CWT are shown in Fig. 2(a). For our system, we analyzed the damping of two frequency components (40 and 80 THz) corresponding to the fundamental phonon frequency and its second harmonic frequency after the 7 fs laser pulse excitation. The profiles created from the CWT by integrating the time-dependent spectral amplitude in the spectral window of the width of 6 THz for the fundamental phonon frequency and its second harmonic frequency including the exponential decay fits are shown in Fig. 2(b). Due to the cascaded process, in which two subsequent coherent scattering events are required, the amplitude of the scattered wave should depend quadratically on the amplitude of the phonon wave. Therefore we expect twice faster decay of the spectral component at the second harmonic frequency. The component at the fundamental phonon frequency decays exponentially with the decay time T2,mono = 7.0 ps, which corresponds to the phonon dephasing time. This value agrees well with published data measured in monocrystalline diamond [2,6,20]. The second harmonic frequency component has a decay time T2,mono,SH = 3.8 ps, which is approximately half of the measured dephasing time of the optical phonon.
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The sidebands (not shown) have the same damping as the fundamental phonon frequency at 40 THz. This observation supports the hypothesis that these sidebands are due to the presence of coherent acoustic phonon waves excited by short laser pulses in the crys tal. Frequency domain signal containing a central frequency and two symmetric sidebands corresponds to the amplitude-modulated oscillations at the fundamental frequency in time domain. The amplitude modulation probably arises from the presence of acoustic phonons. However, the momentum of optical phonons generated by stimulated Raman scattering is close to zero. To fulfil the energy and momentum conservation laws, an interaction with single acoustic phonon is not possible due to phonon dispersion relations in diamond [21]. Therefore the most probable explanation is an interaction with two acoustic phonons with opposite momenta. If the sum of their energies is equal to the energy of the optical phonon ( 160 meV), coherent scattering of the laser pulse might be possible. The generation mechanism of the coherent superposition of two acoustic phonons is, however, not clear. The decay of optical phonons to two acoustic phonon modes of equal and opposite momentum which belong to the same acoustic branch was suggested in [15]. The other decay channels were taken into account in [22], where the relative contributions between different channels were compared. In semiconductors such as Si and Ge is shown that the greatest probability (95 %) involves decay into one TA and one LA mode, on the other hand decay into two acoustic phonons of the same branch is very unlikely. The situation is quite different in a diamond, where the decay into two acoustic phonons of the opposite momenta of the same branch is enhanced by the high two phonon density of states at the zone center. The relative contributions of decay via Klemens channel into TA + TA and LA + LA are 31 % and 15 %, respectively, while the decay to two acoustic phonons from different branches is with 34 % less probable. However, this decay channel is most probably the origin of the coherent acoustic phonons, which cause the observed frequency sidebands.
Fig. 2. (a) Continuum wavelet transform of transient transmissivity signal. (b) The damping of two frequency components at νph = 40 THz (black squares) and νSH = 80 THz (blue circles). Dephasing times of the fundamental phonon frequency τmono = 7.0 ps and the second harmonic frequency τmono,SH = 3.8 ps were obtained from a fitting the profiles by exponential decay.
Conclusion In summary, we used femtosecond transient transmission measurements to study the dynamics of coherent optical phonons in crystalline and polycrystalline diamond. The measured dephasing times of coherent optical phonons in monocrystalline and polycrystalline diamond are T2,mono = 7.0 ± 0.4 ps and T2,poly = 6.0 ± 0.4 ps. Phonon dephasing time in the polycrystalline diamond is faster due to higher density of lattice defects. Frequency sidebands around the frequency corresponding to the optical phonon were observed in the Fourier transform of the transient transmission signal. Their origin remains unclear with the possible explanation by the presence of coherent acoustic phonons generated by the anharmonic decay of coherent optical phonons . We observed a second harmonic of the fundamental ph onon frequency at 80 THz at high excitation fluencies. This
ACCEPTED MANUSCRIPT double frequency component was analyzed by CWT and found to have damping approximately twice as fast (T2,mono,SH = 3.8 ps) as the phonon oscillation at 40 THz. Higher orders of Raman frequency comb are not observable in our experiment because of the limited bandwidth of the laser spectrum. Centrosymmetry of the diamond crystal leads us to a conclusion that the observed second harmonic frequency can be assigned to the cascade Raman scattering process.
Funding
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This work was supported by the Czech Science Foundation (project GA18-0486Y) and by Charles University (Center of nano- and bio-photonics, UNCE/SCI/010 and SVV-2018-260445).
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Graphical abstract
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Frequency sidebands in coherent lattice vibrations in diamond Transient transmission signal at second harmonic of phonon frequency Phonon dephasing time varies with the diamond crystallinity
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M. Zukerstein, M. Kozák, F. Trojánek, P. Malý PII: DOI: Reference:
S0925-9635(18)30694-0 https://doi.org/10.1016/j.diamond.2018.10.018 DIAMAT 7235
To appear in:
Diamond & Related Materials
Received date: Revised date: Accepted date:
8 October 2018 24 October 2018 24 October 2018
Please cite this article as: M. Zukerstein, M. Kozák, F. Trojánek, P. Malý , Experimental observation of anharmonic effects in coherent phonon dynamics in diamond. Diamat (2018), https://doi.org/10.1016/j.diamond.2018.10.018
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Experimental observation of anharmonic effects in coherent phonon dynamics in diamond M. Zukerstein,1,* M. Kozák,1 F. Trojánek,1 and P. Malý1 1 *
Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 12116 Prague 2, Czech Republic Corresponding author: [email protected]
Abstract
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We present a study of coherent optical phonon dynamics in monocrystalline and polycrystalline diamond using transient transmission spectroscopy with few-cycle laser pulses. Besides the oscillations at the fundamental phonon frequency of 40 THz we observe a signal at second harmonic frequency and four weak frequency sidebands (25.0 THz, 32.5 THz, 47.5 THz and 55.0 THz). Potential origin of these sidebands is discussed along with the dependence of the observed signal on experimental conditions (pump power, sample thickness, etc.). Phonon dephasing time in polycrystalline diamond T2,poly = 6.0 ± 0.4 ps is found to be shorter than in monocrystalline diamond T2,mono = 7.0 ± 0.4 ps.
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Single crystal diamond; polycrystalline diamond; phonons; anharmonic effects; ultrafast spectroscopy; few-cycle laser pulses.
Introduction
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Ultrafast switching and information processing at frequencies > 10 THz may become accessible via the manipulation of coherent excitations in solids with light. Time-resolved optical spectroscopy provides information about various ultrafast processes in solids. Relevant time scales (fs – ps), which can be studied, cover electronic relaxation and recombination as well as dynamics of lattice vibrations. These can be generated in the form of incoherent population or as a coherent phonon wave.
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Coherent phonon population is excited either by the fast relaxation of carriers excited by above -band gap absorption [1,2,3] or via stimulated Raman scattering (SRS) [1,4]. The first principle of coherent phonon generation is through resonant carrier excitation, which, due to the coupling between the lattice and the electronic excitations, leads to a local transient change of lattice con stants and an impulse excitation of coherent phonons. The second principle is generation of coherent phonons by SRS, which requires two photons whose frequencies differ by the vibrational energy of the coherent optical oscillations. This can be reached by application of a single laser beam whose spectral width is much bigger than the energy of generated phonons and the pulse duration is shorter than one lattice vibration period [1] or by two laser beams whose frequencies differ by the frequency of the coherent optical oscillations. These pulses may then be longer in the time domain than the period of the generated phonons because the total intensity is modulated in time at the difference frequency of the two laser fields. This corresponds to the train of pulses separated by the period of the phonon. Additionally, phonon population can be coherently controlled using a sequence of two pump pulses [5]. Lattice vibrations can be observed via transient transmission (reflection) signal using the pump-and-probe method. Once a pump pulse generates the coherent phonons, they subsequently modulate the electrical susceptibility, and hence the refractive index, in the material. These slight oscillatory changes can be measured by another pulse (probe) whose transmissivity (reflectivity) is modulated. As shown in [6], the spectrally resolved modulation of reflectivity at optical frequency ω is given by: 𝛥𝑅 (𝜔 ) = 2𝜔0 𝛼√𝛺⁄2ℏ 𝑄0 [{𝜒 (𝜔 + 𝛺) − 𝜒 (𝜔 )} 𝐸0 (𝜔 + 𝛺) − {𝜒 (𝜔 ) − 𝜒 (𝜔 − 𝛺)}𝐸 (𝜔 − 𝛺)] 𝐸0 (𝜔 ) 𝑠𝑖𝑛 𝛺𝜏𝑝 , (1)
ACCEPTED MANUSCRIPT where α is the dimensionless electron-phonon coupling constant, ω0 is the central frequency of the probe pulse, Ω is the phonon frequency, Q0 is the amplitude of oscillation, χ(ω) is the electric susceptibility of the crystal, E0 (ω) ≡ exp(–iωτp )E(ω) is a real quantity in the Fourier transform limit and τp is the delay between pump and probe pulses. From the equation (1) it can be easily shown that the components on the frequencies ω + Ω and ω Ω oscillate out of phase so the spectrally integrated signal is almost cancelled out [6]. The nonzero signal can only be caused by the curvature of the dispersion curve [6]. For this reason, the spectrally resolved detection leads to a strong increase of the oscillatory signal, both in transmission and reflection geometries.
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In the case of weak driving force acting on the atoms in a lattice, the interatomic potential can be treated as harmonic. As a consequence, after an impulsive excitation, the lattice vibrates at a single frequency given by the strength of the chemical bonds between neighboring atoms. However, if the excursion of an atom from its equilibrium position is large enough, anharmonic terms of the potential lead to an anharmonicity of the coherent vibrations and refractive index modulation. This manifests itself in a comb of harmonic frequencies of the fundamental phonon frequency, which can be observed in the transient reflectivity signal [7,8]. The origin of the Raman frequency comb can be different. It can be generated through anharmonic crystal lattice response [7,8] or through cascaded high-order Raman scattering. Study of strongly driven atomic vibrations and monitoring the anharmonicity of the generated infrared photons has recently been used for reconstruction of the interatomic potential [9]. In this paper we investigate properties of coherent phonons in diamond generated by few-cycle laser pulses. In the measured transient transmission signal we observe oscillations at optical phonon diamond frequency of 40 THz, its second harmonics and four additional sidebands.
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The studied samples are commercially available diamonds (Element Six) prepared by chemical vapor deposition (CVD). The first sample is monocrystalline diamond of type IIa with dimensions 4.5 × 4.5 × 0.5 mm. It has a very low concentration of point defects and impurities (< 5 ppb of nitrogen, < 1 ppb of boron, < 0.03 ppb of nitrogen-vacancy centers). Its front face has crystallographic orientation <100> with the roughness better than 5 nm. In some experiments, a thinned monocrystalline samples with the same properties was used (thickness of 5.8 μm). The sample was thinned down by electrochemical etching by Diamond Materials Company in Freiburg, Germany. The other sample is polycrystalline CVD diamond with a diameter of 10 mm and thickness of 0.5 mm. The surface is polished with roughness better than 15 nm. The average diameter of the randomly oriented grains of polycrystalline diamond sample is approximately 10 % of the film thickness and the grains are elongated in the direction perpendicular to the surface. High concentration of defects results in relatively high strain and strong scattering of visible light.
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The pump and probe technique was used to measure the dynamics of coherent optical phonons in diamond. The time delay between pump and probe pulses obtained by amplitude splitting of the output of a few-cycle Ti:sapphire laser (Rainbow (Newport)) was controlled by an optical delay line providing temporal resolution of 0.1 fs. The pump beam was modulated by an optical chopper at fchopper = 500 Hz and the signal of the probe transient transmissivity was detected by a lock-in phase-sensitive amplifier operating at the frequency fchopper. The transmitted (reflected) probe was measured by a silicon photodiode. Sp ectrally resolved detection was provided by the monochromator or bandpass filters in the probe beam situated between the sample and the photodiode. The pulse durations of both the pump and the probe pulses were measured by the Spectral Phase Interferometry for Direct Electric Field Reconstruction (SPIDER) method to be τFWHM = 7 fs. In the collinear geometry, both laser beams were focused on the sample by off-axis parabolic mirror with focal length of 15 mm to the 1/e2 radius of w0 = 2.6 μm. The maximum pulse energy was Epump = 33 mJ/cm2 for the pump and Eprobe = 2 mJ/cm2 for the probe pulses, respectively. Linear polarization of the pump beam was parallel to the <110> crystallographic axis and perpendicular to the probe polarization, in order to minimize the contribution of scattered pump light to the detection. The transient reflectivity measurements were performed in the non collinear geometry with two spherical mirrors with focal lengths of 150 mm. However, in this geometry the combination of reflectivity and transmissivity due to the reflection of the probe beam from the back surface of the sample is measured.
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In the inset of Fig. 1(a) we show the measured differential transmissivity for the monocrystalline diamond in the spectral window of 795 ‒ 805 nm. The Fourier transform of the signal shown in Fig. 1(a) reveals the fundamental lattice vibration frequency of νph = 39.95 ± 0.05 THz. This frequency is independent of the pump peak energy in the range 0 ‒ 33 mJ/cm2 , of spectral window of detection and of the type of diamond (the same frequency is observed for both the monocrystalline and polycrystalline samples). The corresponding wavenumber 1333 ± 2 cm-1 , agrees well with incoherent Raman scattering data [10] and time-resolved CARS [11]. The mechanism of coherent phonon generation in diamond can be explained in time domain [12]. When the polarization of the excitation light is parallel to <100> crystallographic direction, the force acting on an atom in the crystal lattice is linearly proportional to the oscillating light field. After the interaction with a short pulse, the total momentum change of the atom is zero. However, for polarization along <110> direction, the force has a component, which is quadratic with the incident light field (linear with the intensity), which causes acceleration of the atom in the propagation direction of the driving light [12]. Moreover, the force acting on the two neighboring atoms from the unit cell has opposite sign leading to generation of high -frequency coherent optical phonons. This excitation mechanism was verified by rotating the sample by 45° around the axis perpendicular to the front surface, where the oscillatory signal completely disappears, which agrees well with the Γ25 ’ symmetry of the Raman tensor of optical phonons in diamond [4].
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In Fig. 1(b) we display the time decay of the oscillatory transient reflectivity signal in the monocrystalline diamond in comparison with that in the polycrystalline diamond. Phonon dephasing times of these two samples T2,mono and T2,poly are highlighted with black arrows. Data from these measurements were fitted by a sine function with exponentially decaying amplitude ΔR/R = Aexp(-τ/T2,mono/2,poly )sin(2πνph τ + φ0 ). The measured dephasing time in monocrystalline diamond is T2,mono = 7.0 ± 0.4 ps, while in polycrystalline diamond it is only T2,poly = 6.0 ± 0.4 ps. Faster phonon dephasing in latter case can be explained by higher density of defects in the crystal lattice of polycrystalline sample. The resulting relaxation time T2,poly is then given by the expression 1/T2,poly = 1/T1 + 1/Tpure dep., where T1 is the decay time caused by the decrease in the population of phonons, and Tpure dep. is the decay caused by the effect of dephasing [11]. As in [4,6] we assume that in the monocrystalline diamond, the phonons are perfectly coherent all the time, and the effect of Tpure dep. can be neglected. Fig. 1(b) also shows noticeably smaller oscillation amplitude of the polycrystalline diamond. The average oscillation amplitude obtained from long time scale measurements at 737 nm for the monocrystalline diamond is Amono = (1.5 ± 1.1) ×
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10-4 , whereas the polycrystalline diamond has this amplitude approximately 4 times smaller Apoly = (3.9 ± 1.2) × 10-5 . Smaller amplitude and signal isotropy were expected in the polycrystalline diamond due to random orientation of single-crystalline grains. The observed dephasing times are much longer than those reported for micro- and nanocrystalline diamond films with grain sizes ranging from 10 ‒ 500 nm where the phonon decay times between 0.7 ‒ 1.72 ps were measured [11]. This discrepancy is in line with expected much larger amount of lattice defects in diamond nanocrystals.
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In addition to the main peak at 40 THz, in the Fourier Transform we observe sidebands distributed symmetrically around this main peak (Fig. 1(a)). The frequencies and relative intensities (the ratio between sidebands and the peak at the fundamental frequency) of these sidebands do not change when varying the pump intensity and the orientation of the sample. Such sidebands are observed in the transient transmission measurements in diamond for the first time to the best of our knowledge. Their origin is unclear with two possible hypotheses. The first is generation of coherent superposition of two acoustic phonons with opposite momenta by anharmonic decay of optical phonons [13–16]. If the frequency of these two phonons would be different, it may result in frequency mixing with the fundamental vibration mode. We expect the process, in which two phonons are involved, to be nonlinear with the excitation power. We cannot confirm this hypothesis as the nonlinear dependence on excitation power was not measurable due to high signal noise. The second possible explanation involves acoustic phonons generated at the surface of the sample, where the symmetry of the crystal is broken. Excitation of surface acoustic modes was theoretically predicted in [17]. To verify the role of the surface, we measured the transient transmission signal with the same experimental conditions at two monocrystalline samples with different thicknesses (500 μm vs 5.8 μm). No difference was observed between the relative ratios of the sidebands and the fundamental frequency peak. Therefore the role of surface-excited acoustic phonons is not likely. These peaks were not observed in previous measurements, in which the transient reflection anisotropy was detected [4,6].
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Fig. 1. (a) Fourier transform spectrum of coherent phonon oscillations with fundamental frequency at νph = 39.95 THz and second harmonic frequency at νSH = 79.90 THz. Sidebands symmetrically around this main peak could be explained as TA and LA phonons. The time window used for FT spectrum calculation was 1.5 – 12.5 ps. Inset shows the oscillating differential transient transmissivity signal. (b) The decay of differential reflectivity signal for monocrystalline (red line) and polycrystalline (green line) diamond. Black arrows illustrate the dephasing time of monocrystalline (τmono = 7.0 ± 0.4 ps) and polycrystalline diamond (τpoly = 6.0 ± 0.4 ps). The amplitude of polycrystalline diamond is approximately 4 times smaller. The spike-like signals at ≈8 ps are caused by optical interference of the probe pulse with the twice-reflected pump pulse from the back and front surface of the sample and do not influence the interpretation of the data. The plotted data are offset for clarity.
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The anharmonicity of the lattice potential can lead to the generation of higher harmonic frequencies of the fundamental phonon frequency – the generation of Raman frequency comb [7,8]. However, for the interaction in the bulk of diamond crystal, the symmetry does not allow even terms in the Taylor series of the atomic potential. This means that only odd terms of harmonic frequencies are expected in the frequency spectrum of atomic oscillations. Our observation of the signal at the second harmonic frequency νSH = 80 THz of the fundamental optical phonon is in contradiction to these expectations. Therefore its origin can be assigned to the cascade d Raman scattering process [18], in which the wave at a certain frequency from the bandwidth of the laser pulses twice inelastically scatters via stimulated Raman process. This leads to the oscillations of the phase of the scattered wave at twice the phonon frequency. In the narrow spectral windo w of detection, the scattered wave interferes with a part of the original laser spectrum leading to intensity modulation at νSH . High-order χ(3)nonlinear optical interaction in CVD diamond like multi-octave Raman frequency comb generation was studied in [18], where the cascaded stimulated Raman scattering was observed in frequency domain. In this work we perform the cascaded SRS by measuring the oscillating signal in time domain. To reveal the origin of the frequency sidebands and the second harmonics of t he fundamental optical phonon in the transient transmission signal, we process the data using continuous wavelet transform (CWT) [19]. This technique performs discrete Fourier transform from the signal in a narrow time window. By shifting this window, the frequency spectrum is obtained as a function of time delay. The results of CWT are shown in Fig. 2(a). For our system, we analyzed the damping of two frequency components (40 and 80 THz) corresponding to the fundamental phonon frequency and its second harmonic frequency after the 7 fs laser pulse excitation. The profiles created from the CWT by integrating the time-dependent spectral amplitude in the spectral window of the width of 6 THz for the fundamental phonon frequency and its second harmonic frequency including the exponential decay fits are shown in Fig. 2(b). Due to the cascaded process, in which two subsequent coherent scattering events are required, the amplitude of the scattered wave should depend quadratically on the amplitude of the phonon wave. Therefore we expect twice faster decay of the spectral component at the second harmonic frequency. The component at the fundamental phonon frequency decays exponentially with the decay time T2,mono = 7.0 ps, which corresponds to the phonon dephasing time. This value agrees well with published data measured in monocrystalline diamond [2,6,20]. The second harmonic frequency component has a decay time T2,mono,SH = 3.8 ps, which is approximately half of the measured dephasing time of the optical phonon.
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The sidebands (not shown) have the same damping as the fundamental phonon frequency at 40 THz. This observation supports the hypothesis that these sidebands are due to the presence of coherent acoustic phonon waves excited by short laser pulses in the crys tal. Frequency domain signal containing a central frequency and two symmetric sidebands corresponds to the amplitude-modulated oscillations at the fundamental frequency in time domain. The amplitude modulation probably arises from the presence of acoustic phonons. However, the momentum of optical phonons generated by stimulated Raman scattering is close to zero. To fulfil the energy and momentum conservation laws, an interaction with single acoustic phonon is not possible due to phonon dispersion relations in diamond [21]. Therefore the most probable explanation is an interaction with two acoustic phonons with opposite momenta. If the sum of their energies is equal to the energy of the optical phonon ( 160 meV), coherent scattering of the laser pulse might be possible. The generation mechanism of the coherent superposition of two acoustic phonons is, however, not clear. The decay of optical phonons to two acoustic phonon modes of equal and opposite momentum which belong to the same acoustic branch was suggested in [15]. The other decay channels were taken into account in [22], where the relative contributions between different channels were compared. In semiconductors such as Si and Ge is shown that the greatest probability (95 %) involves decay into one TA and one LA mode, on the other hand decay into two acoustic phonons of the same branch is very unlikely. The situation is quite different in a diamond, where the decay into two acoustic phonons of the opposite momenta of the same branch is enhanced by the high two phonon density of states at the zone center. The relative contributions of decay via Klemens channel into TA + TA and LA + LA are 31 % and 15 %, respectively, while the decay to two acoustic phonons from different branches is with 34 % less probable. However, this decay channel is most probably the origin of the coherent acoustic phonons, which cause the observed frequency sidebands.
Fig. 2. (a) Continuum wavelet transform of transient transmissivity signal. (b) The damping of two frequency components at νph = 40 THz (black squares) and νSH = 80 THz (blue circles). Dephasing times of the fundamental phonon frequency τmono = 7.0 ps and the second harmonic frequency τmono,SH = 3.8 ps were obtained from a fitting the profiles by exponential decay.
Conclusion In summary, we used femtosecond transient transmission measurements to study the dynamics of coherent optical phonons in crystalline and polycrystalline diamond. The measured dephasing times of coherent optical phonons in monocrystalline and polycrystalline diamond are T2,mono = 7.0 ± 0.4 ps and T2,poly = 6.0 ± 0.4 ps. Phonon dephasing time in the polycrystalline diamond is faster due to higher density of lattice defects. Frequency sidebands around the frequency corresponding to the optical phonon were observed in the Fourier transform of the transient transmission signal. Their origin remains unclear with the possible explanation by the presence of coherent acoustic phonons generated by the anharmonic decay of coherent optical phonons . We observed a second harmonic of the fundamental ph onon frequency at 80 THz at high excitation fluencies. This
ACCEPTED MANUSCRIPT double frequency component was analyzed by CWT and found to have damping approximately twice as fast (T2,mono,SH = 3.8 ps) as the phonon oscillation at 40 THz. Higher orders of Raman frequency comb are not observable in our experiment because of the limited bandwidth of the laser spectrum. Centrosymmetry of the diamond crystal leads us to a conclusion that the observed second harmonic frequency can be assigned to the cascade Raman scattering process.
Funding
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This work was supported by the Czech Science Foundation (project GA18-0486Y) and by Charles University (Center of nano- and bio-photonics, UNCE/SCI/010 and SVV-2018-260445).
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Graphical abstract
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Frequency sidebands in coherent lattice vibrations in diamond Transient transmission signal at second harmonic of phonon frequency Phonon dephasing time varies with the diamond crystallinity
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