Phonon Softening, Chaotic Motion, and Order-Disorder Transition in Sn=Ge111 D. Farı ´as, 1 W. Kamin ´ski, 2,3 J. Lobo, 1 J. Ortega, 2 E. Hulpke, 4 R. Pe ´rez, 2 F. Flores, 2 and E. G. Michel 1 1 Departamento de Fı ´sica de la Materia Condensada and Instituto Nicola ´s Cabrera, Universidad Auto ´noma, 28049 Madrid, Spain 2 Departamento de Fı ´sica Teo ´rica de la Materia Condensada, Universidad Auto ´noma, 28049 Madrid, Spain 3 Institute of Experimental Physics, University of Wroclaw, plac Maksa Borna 9, 50-204 Wroclaw, Poland 4 Max-Planck-Institut fu ¨r Stro ¨mungsforschung, Bunsenstrasse 10, 37073 Go ¨ttingen, Germany (Received 24 January 2003; published 3 July 2003) The phonon dynamics of the Sn=Ge111 interface is studied using high-resolution helium atom scattering and first-principles calculations. At room temperature we observe a phonon softening at the K point in the   3 p   3 p R30  phase, associated with the stabilization of a 3  3 phase at low temperature. That phonon band is split into three branches in the 3  3 phase. We analyze the character of these phonons and find out that the low- and room-temperature modes are connected via a chaotic motion of the Sn atoms. The system is shown to present an order-disorder transition. DOI: 10.1103/PhysRevLett.91.016103 PACS numbers: 68.35.Ja, 68.35.Rh, 71.15.Nc, 73.20.–r Low dimensional materials exhibit a wide range of phenomena that cannot be observed in three dimensional systems. The different reconstructions that a crystalline surface may exhibit, and the phase transitions that relate them, are a prominent example [1,2]. These topics are widely investigated because of their implications in many different fields of solid state physics. The  phase of Sn on Ge(111) undergoes a temperature-induced phase transi- tion [3] that has received attention in recent years. The room-temperature (RT) phase has a   3 p   3 p R30  structure (in the following  3 p ) that becomes 3  3 at low temperature (LT). The 3  3 phase is well under- stood: out of the three Sn adatoms (on T 4 sites) per 3  3 unit cell, one is displaced outwards and the other two inwards, with a total vertical distortion of 0:3  A [4–6]. However, in spite of the efforts made, the nature of the RT  3 p phase, the driving force underlying the phase transi- tion and its character— order-disorder vs displacive —are still open questions [3–12]. Different models have been put forward to explain the structural and electronic properties of the phase transi- tion [3–12]. In the dynamical fluctuations model [4], Sn adatoms fluctuate at RT between ‘‘up’’ and ‘‘down’’ posi- tions, with a correlated motion that keeps locally the 3  3 structure, explaining the apparent contradiction between electronic and structural evidences [3,4,8]. STM studies have shown that Ge defects have a signifi- cant influence on the phase transition [9,10], stabilizing 3  3-ordered regions around them. Recent theoretical work has proposed that a soft phonon may be at the heart of the phase transition [13]. This surface phonon would be associated with the dynamical fluctuations of the Sn atoms, and it may also be responsible for the 3  3 ordering around defects [14]. Therefore, measuring the surface phonons and understanding their behavior seems to be of crucial importance to clarify the driving force behind this controversial phase transition. In this Letter, we report a combined experimental and theoretical study on the phonon dynamics of both the  3 p and 3  3 structures. The experiments show that the  3 p phase ex- hibits a soft surface phonon at the zone edge. This branch gives rise to three bands in the LT structure, associated with the vertical motion of the Sn adatoms. A molecular dynamics (MD) analysis was used to understand the  3 p phase and its phonon spectrum. We find that at RT Sn adatoms occupy up positions in a chaotic sequence and that the phase transition is of the order-disorder type. The experiments were performed in a high-resolution helium atom scattering (HAS) spectrometer previously described [15]. For details on the sample preparation we refer the reader to previous Letters [4,5]. The 3  3 phase was prepared by monitoring the intensity of a  3 p He-diffraction peak along the Sn deposition at 500 K, which exhibits a maximum at a coverage of 1=3 ML (monolayer). Because of the low deposition rate used the error in the coverage is 0:01 ML. The quality of the reconstruction was judged from the elastic HAS spec- tra recorded along the 11 2 direction ( K), which exhib- ited sharp diffraction peaks indicating the formation of a well-ordered and low-corrugated surface structure [16]. In the time-of-flight (TOF) spectra we observed, in addi- tion to several phonon-inelastic signatures, a significant quasielastic peak indicating the presence of defects at the surface, in agreement with previous STM results [9]. Figure 1 shows TOF spectra recorded along the K direction as a function of temperature, but for the same scattering conditions. The temperature range shown spans the  3 p $3  3 phase transition. The largest peak cor- responds to the diffuse elastic peak. Right and left of it, we find sharp inelastic features related to the scattering of He atoms by the different phonon modes discussed below. Collecting such data for different scattering conditions and converting them into energy loss spectra permits one to draw the experimental points depicted in Fig. 2. An analysis of these data reveals several phonon branches for the 3  3 phase. From a comparison with the slope of the bulk phonon dispersion curves, we identify the mode at lower energies with Q  0:0–0:2  A 1 as the Rayleigh PHYSICAL REVIEW LETTERS week ending 4 JULY 2003 VOLUME 91, NUMBER 1 016103-1 0031-9007= 03=91(1)=016103(4)$20.00  2003 The American Physical Society 016103-1