Abstract
Using quantum tunneling of electrons into vibrating surface atoms, phonon oscillations can be observed on the atomic scale. Phonon interference patterns with unusually large signal amplitudes have been revealed by scanning tunneling microscopy in intercalated van der Waals heterostructures. Our results show that the effective radius of these phonon quasibound states, the realspace distribution of phonon standing wave amplitudes, the scattering phase shifts, and the nonlinear intermode coupling strongly depend on the presence of defectinduced scattering resonance. The observed coherence of these quasibound states most likely arises from phase and frequencysynchronized dynamics of all phonon modes, and indicates the formation of manybody condensate of optical phonons around resonant defects. We found that increasing the strength of the scattering resonance causes the increase of the condensate droplet radius without affecting the condensate fraction inside it. The condensate can be observed at room temperature.
Introduction
The concept of phonons describes quantum behavior and elementary excitations of sound waves in solids^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}. Phonons in layered twodimensional (2D) materials have been extensively studied over recent years^{1,2,3,4,5} including low frequency interlayer vibrations^{1}, flat optical bands with large lifetimes^{2,3,4}, and thickness dependent anharmonicities^{5}. The model system selected for this particular study is a defectsupported quasifreestanding 2D monolayer, whose phonon oscillations can be directly probed by scanning tunneling microscope (STM)^{20}, as shown schematically in Fig. 1a. Due to coherent scattering at defects, thermal phonons in such heterostructures can form spontaneous standing wave patterns resembling electronic Friedel oscillations in metals^{7,21}. A significant difference stems from the fact that while for electrons only the density function can be detected^{21}; for phonons both the standing wave amplitude A(r) and the time factor e^{iωt} are observable characteristics^{13}. In Fig. 1c we show the simulated interference wave packet for an interesting case of onedimensional dispersionless (ω_{k} = const) optical phonons scattering at massive point defect. The construction of this curve involves summation of all standing wave modes across the 1^{st} Brillouin zone (BZ):
where K = π/a is BZ edge wave vector, and a is interatomic distance. The timedynamics for all isofrequency modes in equation (1) is anticipated to be synchronized because of nonlinear intermode coupling^{22,23,24,25,26}. The curve exhibits 2aperiodic nodes and 2aperiodic oscillation maxima decaying by x^{−1} law. Due to coherent nature of a wave packet, described by equation (1), and due to superposition of multiple phonon modes, the periodicity of interference maxima corresponds to wavelength cutoff (λ = 2a) at the BZ edge, not ½ λ as it usually occurs (see Fig. 1d). In a more applicable to our study 2D case, separation between neighboring atomic rows a_{0} determines the BZ edge wave vector π/a_{0}; whereas equation (1), as we shall show in the Discussion part, describes a distribution of standing wave amplitude in orthogonal to these rows directions. In real 2D materials, due to formation of optical phonon bound states at defects^{15,16} and due to intermode phase and frequency synchronization^{22,23,24,25,26}, the requirement of zero phonon bandwidth (Δω =0) most likely can be replaced with a softer requirement . An interesting example of materials for experimental study of coherent optical phonon oscillations are semiconducting transition metal dichalcogenide (TMD) monolayers (ML)^{1,2,3,4,5} possessing an almost dispersionless homopolar optical branch ZO_{2} with estimated ∆ω/ω ≈ 3%^{2,3} and large (~300 oscillations) phonon lifetimes^{3}. Although in the absence of intermode coupling, the wave packets from Fig. 1c would only exist for ω/∆ω~30 oscillation periods, they may become essentially bound to defects when strong intermode coupling takes place. Apparently, for such transition to occur, the synchronization time has to be significantly shorter than both the optical phonon lifetime and the dephasing time ∆ω^{−1}. Synchronization of quasiparticles, and especially on the length scale ~λ, calls for analogy with BoseEinstein condensation^{27,28}, and the connection between these two effects was already shown in theory, on the examples of polaritons^{27} and magnons^{28}.
Results
The reported here STM measurements were performed on WSe_{2} layers grown by metalorganic chemical vapor deposition (MOCVD) method on top of graphene (see Methods). The STM studies revealed phonon standing wave patterns only on quasifreestanding 1 ML islands, like a triangular shaped island presented in Fig. 2a. On STM crosssections a quasifreestanding geometry of WSe_{2} nanostructures manifests as additional 2.5 Å elevation of the first atomic layer. As we show in Fig. 2c, measured by STM heights are 6.4 Å for some of 1 ML islands (dashed line), whereas for other 1 ML islands measured heights are 8.9 Å (solid line). For WSe_{2} pyramids^{29}, also present at the surfaces of our samples (see Fig. 2b and d), STM crosssections typically reveal 6.5 Å layer heights, which is close to bulk value; although in some instances the bottom layers in pyramids can also be elevated to 8.9 Å, as shown in Fig. 2e. The additional 2.5 Å elevation of the first atomic layer was observed in more than 50% of nanostructures in our samples. As we discuss in Fig. 2f, these observations indicate that a significant portion of WSe_{2} nanostructures in our samples possesses quasifreestanding geometry^{4,30} being supported by intercalation defects produced during sample growth.
The room temperature STM measurements on quasifreestanding 1 ML WSe_{2} islands are presented in Fig. 3. The STM image in Fig. 3a shows the 90 × 75 Å^{2} fragment of the surface of such island. The image was obtained at 2 V tunneling bias and 35 pA tunneling current. The interference rings in this image can be clearly observed. On the right side of the image, one can see a singlering pattern (later in the text: typeA pattern) with a diameter of ≈8.2 Å. On the left side of the image, we observe several multiring (typeB) patterns. For this type, the first ring also has a diameter of ≈8.2 Å; the second ring has a diameter of ≈19.4 Å; and the fragments of the third interference ring can also be seen in some cases. The STM crosssections of typeB and typeA patterns are shown in Fig. 3c,d. The crosssections are oriented perpendicular to atomic rows, and their horizontal axes are normalized on a_{0} (a_{0} = 2.8 Å^{31}, see the right inset in Fig. 3e). The central minima have 0.2 Å depths, and a typical height of the first interference maxima is 0.1 Å. The absence of surface adatoms or other visible defects in the centers of these patterns indicates that the interference is induced by subsurface defects. The ring diameters correspond to 3a_{0} and 7a_{0} in exact accordance with Fig. 1a prediction for intercalation defects. As it follows from the comparison of Fig. 1a and c, because intercalating atoms are attached at highcoordination interatomic sites, the “reflection points” for phonons become ±½ a_{0} shifted, increasing, thus, the interference ring diameters from Fig. 1c predicted {2, 6} to experimentally observed {3, 7}. As we show in Fig. 3b, the ring diameters do not depend on the bias voltage, and the separation between interference maxima always corresponds to 2a_{0}. Please note: a closer look at typeB patterns reveals visible angular segmentation that will be later discussed in more detail.
In Fig. 3e we show a larger scale view of phonon standing wave patterns. The size of this STM image is 260 × 260 Å^{2}. To enhance the visibility of interference patterns the image was differentiated along the vertical (scan) axis. From this image, we conclude that typeB patterns appear in 85% of all observed cases. From the total number of interference patterns in Fig. 3e, knowing the total number of WSe_{2} unit cells, we find that the density of intercalation defects significantly contributing to phonon scattering is ≈0.3%. It seems very likely that the two types of interference patterns on STM images are associated with two different kinds of boundary conditions, decay laws, and scattering regimes^{32} imposed by intercalation defects. The observed difference could be induced by two types of intercalating molecules with different masses, for example CO vs. H_{2}O molecules, or it could be caused by two preferential attachment sites for intercalation defects^{33,34} (see left inset of Fig. 3e). The right inset of Fig. 3e demonstrates another type of observed patterns, the typeC pattern possessing only a broad central minimum and originating most likely from weakly scattering defects and/or interfacial charges. The typeC patterns are unnoticeable on gradientcontrast Fig. 3e. The total density of intercalation defects is apparently more than 0.3%.
The interference patterns were only observed on elevated monolayer islands. In order to illustrate this, in Fig. 4 we show a comparison of 2D STM Fourier maps corresponding to elevated (Fig. 4a) and nonelevated (Fig. 4b) monolayer islands. The six peaks from the hexagonal crystal lattice in Figs. 4ab represent the reciprocal lattice vectors ±2π/a_{0}. In Fig. 4a, the 1^{st} BZ is constructed from the intersection of six Bragg planes. The hexagonal disk in Fig. 4a represents the area of kspace that contributes to interference. Essentially, Fig. 4a confirms that the interference patterns are produced by superposition of all optical phonon modes from the 1^{st} BZ. The concept of BZ plays important role in condensed matter physics, and STM imaging optical phonon quasibound states represents an interesting method of its direct observation. Because for optical phonons an interference pattern represents a spatial distribution of a standing wave amplitude, each kpoint inside the 1^{st} BZ corresponds to the same kpoint of the Fourier map, not 2k as it occurs for electrons^{35}. The absence of interferencerelated features in Fig. 4b confirms that the standing waves are produced by phonon scattering at subsurface defects. The interference signals in Fig. 4a primarily originate from typeB patterns, because only for this type the pattern periodicity can be defined. The hexagonal symmetry of the Fourier image in Fig. 4a indicates that phonon wave packets may also possess hexagonal realspace symmetry. Indeed, the analysis of Fig. 3a,b shows that for typeB patterns a visible angular segmentation of the interference amplitude takes place. The outer interference rings on these images reveal visible 60° segmentation relevant to hexagonal symmetry. Such angular segmentation indicates that realspace distribution of phonon amplitudes for typeB patterns represents a superposition of three quasionedimensional oscillations directed perpendicular to atomic rows. The inner interference rings show visible 180° segmentation, indicating the symmetry breaking. Schematic view of these segmentations is shown in the lower inset of Fig. 4a. The experimentally observed ring segmentation can also be seen in the upperright inset of Fig. 4a obtained after image contrast enhancement. For typeA patterns, a similar symmetry breaking effect can also be observed upon significant readjustment of the image contrast. The possibility of complete or partial suppression of certain interference maxima represents a unique feature of phonon standing waves described by equation (1). Such suppression, for example, may occur due to formation of coherent and incoherent spatial domains predicted by phasesynchronization theories^{22,23}. The locations of the suppressed interference maxima are most likely determined by boundary conditions imposed by defect and are anticipated to be reproducible for a given type of interference pattern, in excellent agreement with our experimental results. At lower temperatures, the pattern contrast decreases, most likely due to decrease of phonon amplitudes and less efficient synchronization (see Supplementary Figure 3), and at 100 K the interference cannot be observed (see the upperleft inset of Fig. 4a).
Discussion
All our experimental findings can be successfully explained by formation of coherent phonon quasibound states around intercalation defects. The most likely source of these STM signals is nearly dispersionless outofplane optical phonon branch ZO_{2} , connected to Raman peak at 250 cm^{−1}, whose atomic motion represents mirrorsymmetric oscillations of Se atoms, as illustrated in the upper inset of Fig. 5a. For this phonon branch, the simulated dependences of topographic STM signals (h) on phonon amplitudes (A) are presented in Fig. 5a for symmetric (solid curve) and asymmetric (dashed curve) harmonic atomic motions. The construction of these curves takes into account highfrequency oscillations of a vacuum gap (see Fig. 1a) and exponential distance dependence of tunneling current with decay length L = 0.4 Å established for combination of studied islands and STM tip. For simulation of a dashed curve in Fig. 5a, a factor two outward vs. inward motion asymmetry has been assumed (see lower inset of Fig. 5a). Although optical phonon oscillations are too fast to be detected by STM in realtime, the increase of average tunneling current takes place due to these oscillations forcing the STM tip to retract from the surface. We found that for symmetric harmonic oscillations h(A) ∝ A^{2}/L, whereas for asymmetric oscillations h(A) ∝ A. Typical room temperature phonon amplitude (oscillation amplitude for each Se atom), estimated for ZO_{2} branch using quantum harmonic oscillator model, is (5 pm) from which about ${\scriptstyle \frac{2}{3}}$ is due to zeropoint motion. The peak amplitude anticipated for first interference maxima is ≈2 times larger, A_{max} ≈ 10 pm (see Fig. 1c). The expected range of phonon amplitudes is indicated as vertical grayed area in Fig. 5a. The comparison of curves in Fig. 5a clearly shows that in order to justify the experimentally observed ~10 pm STM signals, a significant apparent oscillation asymmetry has to be present in the studied system. As it was noticed in ref. 36, asymmetric harmonic oscillators can develop due to dipolelike interactions. Because their energy levels (see lower inset of Fig. 5a) are also equidistant, this type of “anharmonicity” would be unnoticeable in the temperature dependence of Raman shifts^{5}. Generally, all types of atomic motions resulting in asymmetric probability distribution for oscillator are anticipated to produce linear h(A) dependence. We also cannot exclude the possibility that the “apparent” oscillation asymmetry is being enhanced by tunneling measurements due to asymmetric electronic response associated with periodic stretching and compression of WSe bonds^{37,38}. In general, the visibility of surface phonon oscillations for STM is anticipated to be enhanced due to phononmediated deformation potential modulating the electronic structure. Because the spatial radius of optical phonon deformation potential is comparable to unit cell size (≈3 Å), observed by STM patterns resemble slightly smoothed segmented rings. In view of the important role of deformation potential for STM detection of surface atomic oscillations, the contribution of another weakly dispersive, inplane polarized LO_{2} branch^{2,3} also cannot be ruled out.
To better understand the difference between the two observed types of standing wave patterns, one should take into account that for optical phonons the scattering centers can develop due to local change of the oscillation frequency in the presence of intercalation defects. For example, for ZO_{2} phonons local oscillation frequency introduced by molecular adsorption at the most favorable Hsite^{33} can be estimated (see Supplementary Note 1) from the increase of the oscillating mass
where M_{Se} is a mass of selenium atom, m is a mass of adsorbed molecule, and N = 3 is a coordination number for Hsite. The corresponding frequency shift, normalized to phonon frequency ω, is given by
For H_{2}O adsorption we find δω/ω = −2%, for CO adsorption δω/ω = −3%. The frequency shifts are small and comparable to intrinsic phonon linewidth ∆ω. The distribution of phonon scattering crosssections σ and scattering phase shifts θ for such resonant defects can be established from the scattering theory^{32,39,40} (see Supplementary Note 2)
where δω = ω_{R} − ω is determined by equation (3) is full width at half maximum (FWHM) of the defectinduced local resonance, and k is phonon wave vector. Equation (4) represents the 2D analogue of the BreitWigner formula^{40}. For negative values of δω, the scattering phase shift belongs to the interval π/2 < θ < π, and the dominant contribution to standing wave patterns arises from sine modes, as in equation (1). We also find that at these conditions σ(k) ∝ k^{−1}, i.e. resonant scattering crosssection of 2D phonon is proportional to its wavelength. As a result, in two dimensions the ∝k increase of the phase space area is compensated by k^{−1} dependence of phonon scattering probabilities. Consequently, for typeB patterns, most frequently observed on STM images, realspace distribution of phonon amplitudes represents a superposition of three (120° rotated) quasionedimensional oscillations described by equation (1). In Fig. 5b and in the inset of Fig. 5b we show the profile of typeB interference pattern and experimentally measured height distribution of interference maxima for such patterns. The fact that the interference maxima decay by x^{−1} law confirms our earlier conclusion that real h(A) dependence in our samples is linear. After subtraction of the background, the crosssectional profile in Fig. 5b strongly resembles 1D simulation from Fig. 1c, confirming that optical phonons in our samples form synchronized coherent superposition. The modeling simulations have shown that the background curve in Fig. 5b appears to be the result of quadratic “superposition” of all phonon modes (see Supplementary Fig. 1), possibly indicating that phonon quasibound states possess both coherent and incoherent components. Phase and frequency synchronization of phonons was earlier discussed in theory for term crossing^{24,25}, where it occurs due to anharmonic intermode coupling. In our samples, superposition involves phonon states with incommensurate wave vectors, and their coupling primarily manifests near defects. For typeB patterns, this corresponds to R_{B} ≈ 1.5 nm radius regions, most likely due to enhanced intermode coupling associated with resonant defects. There are several reasons that can justify the resonant enhancement of phononphonon interactions at such defects: (a) higher scattering crosssections and correspondingly higher probabilities of “finding” a scattered phonon inside such defects^{31,39,40}, (b) nonvanishing oscillation amplitudes at resonant defects owing to additional to equation (1) contribution of cosine modes (see Fig. 6a). For standing wave packets produced by coherent superposition of cosine modes, oscillation amplitudes follow x^{−1} sin Kx dependence and vanish at all atomic sites, except for defect location. As we show in Fig. 6a,b, the amplitude associated with this “zero” feature can be estimated in convenient for data analysis terms
On STM crosssections, this feature is manifested as ~3 pm flattening of the central interference minima, which can also be observed in Figs 3c and 5b. For typeA patterns, such evident “zero” features were not found most likely because of less resonant scattering conditions. From this data, we find that θ_{A} − θ_{B} ≈ 40°. This method for estimating phonon scattering phase shifts is different from the methods earlier used for electrons^{21,41}. Only the first interference maximum from equation (1) can be observed for typeA patterns, indicating a smaller R_{A} ≈ 0.7 nm effective radius of quasibound states for such defects.
Since the typical timescale of our STM experiment exceeds both the lifetime and the dephasing time of optical phonons by at least 12 orders of magnitude, a manybody condensation^{27,28,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58} mechanism capable to synchronize all thermal phonons within ~1 oscillation period is required for their interference patterns to be observable using STM. Synchronization of phonons essentially implies that instead of each of them possessing an independent oscillation phase φ_{k}(t), all φ_{k}(t) become equal to Φ(t) (condensate phase), the behavior well known earlier for systems of coupled mechanical oscillators^{42} and for spin precession in cold vapors^{51}. Due to finite lifetime of optical phonons, the content of the condensate is anticipated to be dynamically renewed, and all newly emerging phonons around the defects are anticipated to join the condensate droplet being affected by its collective oscillation field e^{iΦ(t)}. At low temperatures, the coherent collective oscillation field most likely disappears when the average number of thermal phonons inside the interference area decreases to ~1. This can explain the disappearance of interference signals at low temperatures in Fig. 4a (see Supplementary Figure 4). The comparison of typeA and typeB condensate droplets in Fig. 7 clearly indicates that increasing the strength of the scattering resonance causes the increase of the effective radius without affecting the condensate fractions inside the droplets.
We anticipate that defectmediated phonon condensates can become useful components of quantum computers. The observed effects may also be important for understanding the room temperature coherence mechanism of biological systems^{53}. Although strong anharmonicity can explain large observed interference signal amplitudes, the possibility of unusually high phonon populations inside the droplets anticipated for realspace BoseEinstein condensate^{57,58} also cannot be excluded.
Methods
The WSe_{2} films for our study were prepared using MOCVD technique on top of several layers of epitaxial graphene on SiC(0001) substrate. The growth details, including preliminary Raman, TEM, and XPS characterization, were described in the earlier publications^{59,60}. The STM measurements were performed using ultrahigh vacuum system UHV300 from RHK Technology, with base pressure 7 × 10^{−11} Torr. Before STM measurements, the samples were in situ annealed to 350 °C for few hours in order to eliminate the adsorbed water from their surfaces. The sample temperature was estimated using Ktype thermocouple. For STM measurements, we used commercial PtIr STM tips from Bruker Corp. that were in situ cleaned using electron beam heating technique.
Additional Information
How to cite this article: Altfeder, I. et al. Scanning Tunneling Microscopy Observation of Phonon Condensate. Sci. Rep. 7, 43214; doi: 10.1038/srep43214 (2017).
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 1
Lui, C. H. et al. Observation of interlayer phonon modes in van der Waals heterostructures. Phys. Rev. B 91, 165403 (2015).
 2
Sahin, H. et al. Anomalous Raman spectra and thicknessdependent electronic properties of WSe2 . Phys. Rev. B 87, 165409 (2013).
 3
Cai, Y., Lan, J., Zhang, G. & Zhang, Y. W. Lattice vibrational modes and phonon thermal conductivity of monolayer MoS2 . Phys. Rev. B 89, 035438 (2014).
 4
Tonndorf, P. et al. Photoluminescence emission and Raman response of monolayer MoS2, MoSe2, and WSe2 . Opt. Express 21, 4908 (2013).
 5
Late, D. J., Shirodkar, S. N., Waghmare, U. V., Dravid, V. P. & Rao, C. N. R. Thermal expansion, anharmonicity and temperaturedependent Raman spectra of single and fewlayer MoSe2 and WSe2, ChemPhysChem 15, 1592 (2014).
 6
Schwab, K., Henriksen, E. A., Worlock, J. M. & Roukes, M. L. Measurement of the quantum of thermal conductance. Nature 404, 974 (2000).
 7
Fransson, J. & Balatsky, A. V. Surface imaging of inelastic Friedel oscillations. Phys. Rev. B 75, 195337 (2007).
 8
Fasolino, A., Los, J. H. & Katsnelson, M. I. Intrinsic ripples in graphene. Nature Materials 6, 858 (2007).
 9
Klein, M. V. Phonon scattering by lattice defects. Phys. Rev. 131, 1500 (1963).
 10
Altfeder, I., Voevodin, A. A. & Roy, A. K. Vacuum phonon tunneling, Phys. Rev. Lett. 105, 166101 (2010).
 11
Balandin, A. & Wang, K. L. Significant decrease of the lattice thermal conductivity due to phonon confinement in a freestanding semiconductor quantum well. Phys. Rev. B 58, 1544 (1998).
 12
Altfeder, I., Matveev, K. A. & Voevodin, A. A. Imaging the electronphonon interaction at the atomic scale. Phys. Rev. Lett. 109, 166402 (2012).
 13
Gambetta, A. et al. Realtime observation of nonlinear coherent phonon dynamics in singlewalled carbon nanotubes. Nature Physics. 2, 515 (2006).
 14
Gawronski, H., Mehlhorn, M. & Morgenstern, K. Imaging phonon excitation with atomic resolution. Science 319, 930 (2008).
 15
RodriguezNieva, J. F., Saito, R., Costa, S. D. & Dresselhaus, M. S. Effect of 13C isotope doping on the optical phonon modes in graphene: localization and Raman spectroscopy. Phys. Rev. B 85, 245406 (2012).
 16
Brown, R. A. Electron and phonon bound states and scattering resonances for extended defects in crystals. Phys. Rev. 156, 889 (1967).
 17
Khadjai, G. Ya., Merisov, B. A. & Sologubenko, A. V. Some peculiarities of phonon scattering by point defects in layered crystals. Physica Status Solidi (b) 200, 413 (1997).
 18
Vermeersch, B., Mohammed, A. M. S., Pernot, G., Koh, Y. R. & Shakouri, A. Thermal interfacial transport in the presence of ballistic heat modes. Phys. Rev. B 90, 014306 (2014).
 19
Gustafsson, M. V., Aref, T., Kockum, A. F., Ekström, M. K., Johansson, G. & Delsing, P. Propagating phonons coupled to an artificial atom. Science 346, 207 (2014).
 20
Moreau, A. & Ketterson, J. B. Detection of ultrasound using a tunneling microscope. J. Appl. Phys. 72, 861 (1992).
 21
Hasegawa, Y. & Avouris, Ph. Direct observation of standing wave formation at surface steps using scanning tunneling spectroscopy. Phys. Rev. Lett. 71, 1071 (1993).
 22
Panaggio, M. J. & Abrams, D. M. Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity 28, R67 (2015).
 23
Bastidas, V. M., Omelchenko, I., Zakharova, A., Schöll, E. & Brandes, T. Quantum signatures of chimera states. Phys. Rev. E 92, 062924 (2015).
 24
Gornostyrev, Yu. N., Katsnelson, M. I., Platonov, A. P. & Trefilov, A. V. Phase synchronization in a heat bath and the lattice dynamics of metals under Fermiresonance conditions. J. Exp. Theor. Phys. 80, 525 (1995).
 25
Katsnelson, M. I. & Trefilov, A. V. Synchronization of phonon frequencies and quasistatic atomic shifts in crystals. J. Exp. Theor. Phys. 70, 1067 (1990).
 26
Manzano, G., Galve, F., Giorgi, G. L., HernándezGarcía, E. & Zambrini, R. Synchronization, quantum correlations and entanglement in oscillator networks. Scientific Reports 3, 1439 (2013).
 27
Szymańska, M. H., Keeling, J. & Littlewood, P. B. Nonequilibrium quantum condensation in an incoherently pumped dissipative system. Phys. Rev. Lett. 96, 230602 (2006).
 28
Kalafati, Yu. D. & Safonov, V. L. Theory of quasiequilibrium effects in a system of magnons excited by incoherent pumping. J. Exp. Theor. Phys. 73, 836 (1991).
 29
Chen, L. et al. Screwdislocationdriven growth of twodimensional fewlayer and pyramidlike WSe2 by sulfurassisted chemical vapor deposition. ACS Nano 8, 11543 (2014).
 30
Melios, C., Panchal, V., Giusca, C. E., Strupiński, W., Silva, S. R. P. & Kazakova, O. Carrier type inversion in quasifree standing graphene: studies of local electronic and structural properties. Sci. Rep. 5, 10505 (2015).
 31
Wilson, J. A. & Yoffe, A. D. The transition metal dichalcogenides discussion and interpretation of the observed optical, electrical and structural properties. Adv. Phys. 18, 193 (1969).
 32
Walton, D. PhononDefect Interaction Ch. 5, 393–440 Point Defects in Solids: Vol. 2 Semiconductors and Molecular Crystals. (Eds Crawford, J. H. & Slifkin, L. M. ) Plenum Press, New York (1975).
 33
Yue, Q., Shao, Z., Chang, S. & Li, J. Adsorption of gas molecules on monolayer MoS2 and effect of applied electric field. Nanoscale Res. Lett. 8, 425 (2013).
 34
Altfeder, I., Eichfeld, S. M., Naguy, R. D., Robinson, J. A. & Voevodin, A. A. Scanning tunneling microscopy of atomic scale phonon standing waves in quasifreestanding WSe2 monolayers. MRS Advances, doi: 10.1557/adv.2016.170 1, 1645 (2016).
 35
Schouteden, K., Lievens, P. & Van Haesendonck, C. Fouriertransform scanning tunneling microscopy investigation of the energy versus wave vector dispersion of electrons at the Au(111) surface. Phys. Rev. B 79, 195409 (2009).
 36
Palma, G. & Raff, U. The onedimensional harmonic oscillator in the presence of a dipolelike interaction, Am. J. Phys. 71, 247 (2003).
 37
Nayak, A. P. et al. Pressureinduced semiconducting to metallic transition in multilayered molybdenum disulphide. Nature Communications 5, 3731 (2014).
 38
Chang, C. H., Fan, X., Lin, S. H. & Kuo, J. L. Orbital analysis of electronic structure and phonon dispersion in MoS2, MoSe2, WS2, and WSe2 monolayers under strain. Phys. Rev. B 88, 195420 (2013).
 39
Landau, L. D. & Lifshitz, E. M. Quantum Mechanics. Pergamon, London (1958).
 40
ChaosCador, L. & GarcıaCalderon, G. Theory of resonant scattering in two dimensions. Journal of Physics A: Mathematical and Theoretical 43, 035301 (2010).
 41
Altfeder, I. B., Narayanamurti, V. & Chen, D. M. Imaging subsurface reflection phase with quantized electrons. Phys. Rev. Lett. 88, 206801 (2002).
 42
Zanette, D. H. & Mikhailov, A. S. Condensation in globally coupled populations of chaotic dynamical systems. Phys. Rev. E 57, 276 (1998).
 43
Eastham, P. R. Mode locking and mode competition in a nonequilibrium solidstate condensate. Phys. Rev. B 78, 035319 (2008).
 44
Wouters, M. Synchronized and desynchronized phases of coupled nonequilibrium excitonpolariton condensates. Phys. Rev. B 77, 121302 (R) (2008).
 45
Eastham, P. R. & Rosenow, B. Disorder, synchronization and phase locking in nonequilibrium BoseEinstein condensates. ArXiv:1509.05264 (2015).
 46
Kasprzak, J. et al. BoseEinstein condensation of exciton polaritons. Nature 443, 409 (2006).
 47
Demokritov, S. O. et al. BoseEinstein condensation of quasiequilibrium magnons at room temperature under pumping. Nature 443, 430 (2006).
 48
Misochko, O. V., Hase, M., Ishioka, K. & Kitajima, M. Transient BoseEinstein condensation of phonons. Phys. Lett. A 321, 381 (2004).
 49
Fröhlich, H. Bose condensation of strongly excited longitudinal electric modes. Phys. Lett. A 26, 402 (1968).
 50
Griffin, A., Snoke, D. W. & Stringari, S. (eds) BoseEinstein Condensation Cambridge Univ. Press (1995).
 51
Oktel, M. Ö. & Levitov, L. S. Internal waves and synchronized precession in a cold vapor. Phys. Rev. Lett. 88, 230403 (2002).
 52
NowikBoltyk, P., Dzyapko, O., Demidov, V. E., Berloff, N. G. & Demokritov, S. O. Spatially nonuniform ground state and quantized vortices in a twocomponent BoseEinstein condensate of magnons. Sci. Rep. 2, 482 (2012).
 53
Fröhlich, H. Long range coherence and the action of enzymes. Nature 228, 1093 (1970).
 54
Rückriegel, A. & Kopietz, P. RayleighJeans condensation of pumped magnons in thinfilm ferromagnets. Phys. Rev. Lett. 115, 157203 (2015).
 55
Sun, C., Jia, S., Barsi, C., Rica, S., Picozzi, A. & Fleischer, J. W. Observation of the kinetic condensation of classical waves. Nat. Phys. 8, 470 (2012).
 56
Li, F., Saslow, W. M. & Pokrovsky, V. L. Phase diagram for magnon condensate in yttrium iron garnet film. Sci. Rep. 3, 1372 (2013).
 57
Masut, R. & Mullin, W. J. Spatial BoseEinstein condensation. Am. J. Phys. 47, 493 (1979).
 58
Dominici, L. et al. Realspace collapse of a polariton condensate. Nature Communications. 6, 8993 (2015).
 59
Eichfeld, S. M., Eichfeld, C. M., Lin, Y. C., Hossain, L. & Robinson, J. A. Rapid, nondestructive evaluation of ultrathin WSe2 using spectroscopic ellipsometry. APL Mat. 2, 092508 (2014).
 60
Eichfeld, S. M. et al. Highly scalable, atomically thin WSe2 grown via metalorganic chemical vapor deposition. ACS Nano 9, 2080 (2015).
Acknowledgements
We thank Y.C. Lin, A. J. Safriet, R.D. Naguy and M.E. McConney for help in preparing the experiment. We also acknowledge discussions with L. Levitov, I.A. Zaliznyak, D.L. Dorsey and A.M. Urbas. The AFRL team acknowledges the financial support from AFOSR. The work at PSU was funded by the Center for Low Energy Systems Technology. A.V.B. acknowledges the support from US DOE BES E304, VR and KAW.
Author information
Affiliations
Contributions
I.A. performed STM measurements and wrote the manuscript. S.M.E. and J.A.R. synthesized samples. A.V.B. formulated the theory. All authors contributed to analysis of results.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Altfeder, I., Voevodin, A., Check, M. et al. Scanning Tunneling Microscopy Observation of Phonon Condensate. Sci Rep 7, 43214 (2017). https://doi.org/10.1038/srep43214
Received:
Accepted:
Published:
Further reading

Singledefect phonons imaged by electron microscopy
Nature (2021)

Coulomb blockade effects in a topological insulator grown on a highTc cuprate superconductor
npj Quantum Materials (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.