Experimental observation of moving intrinsic localized modes in germanium

arXiv:1311.4269v2 [cond-mat.mtrl-sci] 13 May 2015 Experimental observation of moving intrinsic localized modes in germanium Juan F. R. Archilla, Sergio M.M. Coelho, F. Danie Auret, Cloud Nyamhere, Vladimir I. Dubinko, and Vladimir Hizhnyakov 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Germanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Phonons in Ge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4 Defects and their detection with DLTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 Experiment of plasma-induced annealing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6 ILM hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7 Thermal annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 8 Comparison of thermal and plasma-induced annealing . . . . . . . . . . . . . . . . . . . . . . . . . 16 9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Abstract Deep level transient spectroscopy shows that defects created by alpha irradiation of germanium are annealed by low energy plasma ions up to a depth of several thousand lattice units. The plasma ions have energies of 2-8eV and therefore can deliverenergies of the order of a few eV to the germaniumatoms. The most abundant defect is identified as the E-center, a complex of the dopant antimony and a vacancy with and annealing energy of 1.3eV as determined by our measurements. The inductively coupled plasma has a very low density and a very low flux of ions. This implies that the ion impacts are almost isolated both in time and at the surface of the semiconductor.We conclude that energy of the order of an eV is able to travel a large distance in germanium in a localized way and is delivered to the defects effectively. The most likely candidates are vibrational nonlinear wave packets known as intrinsic localized modes, which exist for a limited range of energies. This property is coherent with the fact that more energetic ions are less efficient at producing the annealing effect. Key words: Germanium, ILM, discrete breathers, quodons, defects, DLTS J.F.R. Archilla Group of Nonlinear Physics, Universidad de Sevilla, ETSI Inform´atica, Avda. Reina Mercedes s/n 41011, Seville, Spain, e-mail: [email protected] S.M.M. Coelho and F.D. Auret Department of Physics, University of Pretoria, Lynnwood Road, Pretoria 0002, South Africa, C. Nyamhere Physics Department, Midlands State University, P. Bag 9055, Gweru, Zimbabwe V.I. Dubinko NSC Kharkov Institute of Physics and Technology, Akademicheskya Str. 1, Kharkov 61108, Ukraine V. Hizhnyakov Institute of Physics, University of Tartu, Ravila 14c, 50411 Tartu, Estonia 1 2 JFR Archilla, SMM Coelho, FD Auret, C. Nyamhere, VI Dubinko, V Hizhnyakov 1 Introduction In science like in many other aspects of human activity, there are often fortunate coincidences that orientate research in unexpected directions. In 2012 there was an international workshop in Pretoria, South Africa, called NEMI 2012 1. Several theoreticians and nonlinear physicists attended, among them there were two of the authors. Several talks were intended for non specialist in order that physics students could be able to understand them. One of the subjects was nonlinear localized excitations that travel along a periodic media without losing energy and keeping their shape. They are called intrinsic localized modes (ILMS) or discrete breathers(DBs). The first name emphasizes the internal character of the phenomenon and reminds us of the linear vibration modes or phonons. The latter name comes from the observation of the internal vibration they experience that can be compared with the breathing of a living being. They were first obtained as an exact solution for the continuous sine-gordon equation [26]. Simulations using molecular dynamics are able to reproduce them in several solids with energies of the order of a few tenths or a few units of an eV. Among the attendants was a PhD student, part of a research group of the University of Pretoria working on defects in semiconductors, particularly in germanium. They have obtained unexpected results while treating Ge with low energy (2-8eV) plasma ions. Those energies are known as subthreshold because the threshold energy to produce displacements of atoms in germanium is between 11.5 for the h111i direction and 19.5eV for the h100idirection [12]. However, they had observed that something was penetrating at least two µ m inside the germanium wafer and was able to anneal several defects, in particular, the most abundant one, the E-center. The energy for annealing an E-center is about 1.3eV, according to our measurements and theoretical calculations [23]. On the other hand the maximum energy that an Ar ion of 4eV can transmit to a Ge atom is 3.6eV, therefore the energies were precisely what was expected for ILMs. A line of collaboration was started that joined nonlinear theory, computer simulations, plasma physics and semiconductor physics that eventually confirmed ILMs as the most likely cause of the annealing [1] and also suggested them as the explanation for other long-distance effects such as the modification of defects by electron beam deposition, where the energy transmitted was below 1.3eV [5]. In this workd we will try to give an explanation of the different branches of the physics involved and to analyze the reasoning that leads to the ILM explanation and the consequences both for semiconductor physics and nonlinear physics. Experimental observation of moving intrinsic localized modes in germanium 3 2 Germanium The diamond structure of germanium is well known, each atom has covalent bonds with the four nearest neighbours at the vertices of a tetrahedron as shown if Fig. 1. Normally a conventional cubic unit cell comprised of 8 atoms is used. The diamond structure can be seen as an fcc lattice with two atoms at points (0,0,0)and at 1/4 of the diagonal [2]. The lattice unit is a=5.66˚ A for Ge, slightly larger than 5.43˚ A for Si and even larger than 3.57˚ A for C diamond. The diamond structure is not the best for moving ILMs because there is no chain of nearest neighbours forming a straight line. This is a reason for which, although stationary ILMs have been constructed with molecular dynamics[24], the attempts to construct moving ILMs have failed so far. Several lines of research seem promising, one option is to construct ILMs in the next neighbourdirections such as h100iwhere there is a straight line of atoms. Other is to study polarizations as the ones observed for ballistic phonons in germanium or silicon [13, 19] which can travel distances of 160nm. It seems also possible that ILMs can be nonlinear perturbations of linear optical modes with high energy, high velocity, short wavelength and low dispersion, such as at the middle of the Brillouin zone for optical branches. The number of Ge atoms per unit volume can be obtained as nGe =8/a3= 4.42×1022 cm3. Other properties of interest are atomic number 32, atomic mass M=72.61amu, density ρ =5.323g/cm3, sound velocity cs=5400m/s, Debye temperature TD=360K, Einstein temperature TE=288K, covalent radius 1.22 ˚ A, atomic radius 1.52 ˚ A, melting point 1210.55K, 1st ionization energy 7.899eV and specific heat 0.32J/gK at 300K. 1NEMI 2012: 1st International Workshop: Nonlinear effects in materials under irradiation, March 12-17, 2012, Pretoria, South Africa. P. Selyshev, chairman Fig. 1 Diamond structure of germanium. Each atom is bonded with four nearest neighbours at the vertices of a tetrahedron. The conventional cubic unit cell usually used is also shown. It includes 8 atoms and can be seen as an fcc lattice with two atoms at 0 and at 1/4 of the diagonal. The primitive cell has these two atoms as a basis and the primitive vectors have their origin at 0 and end at the center of each adjacent face 4 JFR Archilla, SMM Coelho, FD Auret, C. Nyamhere, VI Dubinko, V Hizhnyakov 3 Phonons in Ge The objective of this subsection is to review the well known concepts of lattice dynamics and to see how they apply to Ge and to justify subsequent calculations. Phonons are the usual means for energy transport in a crystal and the responsible party for thermal annealing of defects. With this review we want to demonstrate that they cannot be th responsible for the annealing of the E-center defect during Ar plasma bombardment. We will frequently use general concepts of lattice dynamics and the reader can consult any textbook, for example Refs. [2,7]. In classical mechanics for a crystal with nat per unit volume, there are 3nat degrees of freedom. In the harmonic approximation the substitution of uk, ω = Aexp(ik·r− ω t)in the equation of movementleads to 3nat differentlinear modes of frequency ω , wave number k, phase velocity c= ω /kand polarization A. They are organized in branches ω = ω (k), three of them are acoustic, that is ω vanishes linearly with kin the long wavelength limit. If the crystal has a basis of patoms or ions in each primitivecell, there are also 3(p−1)optical branches,that are boundedfrom below. In Ge with two Ge atoms in the unit cell, there are three optical branches. Each branch has nGe/2 modes. In the classical description, each mode can have any energy Ewith a probability at temperature Tgiven by Maxwell-Boltzmann equation P(E) = exp(−E/kBT)/kBT, which leads to an average energy kBTthat is identical for each mode. Therefore, it is trivial to obtain the energy per unit volume Fig. 2 Phonon dispersion and density of states for Ge. Experimental values are shown as circles and theoretical calculation are shown as solid lines. Modes about the center of some optical bands with high frequency, large group velocity, short wavelength and low dispersion may convert into ILMs when the amplitude enters the nonlinear range. Reproduced with permission from: Wei, S., Chou, M.Y.: Phonon dispersion of silicon and germanium from first principles calculations. Phys. Rev. B 50, 2221 (1994). Copyright (1994) by American Physical Society Experimental observation of moving intrinsic localized modes in germanium 5 Fig. 3 Primitive Wigner-Seitz reciprocal cell for an fcc lattice such as Ge, showing the directions in k-space and points that appear in the spectrum shown in Fig.2. The point Γ corresponds to wave number k=0, where the three acoustical bands originate. The Wigner-Seitz cell is the region of k-space that is closer to (0,0,0) than to any other point of the lattice. Modes with wave vectors about the middle of Γ -L, Γ -K and Γ -X may convert into ILMs when their amplitude increases. Axes are the same as in Fig. 1 u=3kBTnGe and the specific heat at constant volume cV= ∂ u/ ∂ T=3kBnat, a result known as the Dulong-Petit law. This result is approximateat room temperatures and above but fails spectacularly at lower temperatures, which led to the quantum description of the harmonic crystal. The classical description of the linear modes of the crystal remains valid but the statistics are quite different. In quantum mechanics a linear oscillator with frequency ω can only have energies given by En=1 2¯ h ω +n¯ h ω , where nis the excitation or occupation number. As the ground state energy 1 2¯ h ω cannot be lost we will usually suppress it and use En=nE,with E=¯ h ω ,(1) where E=¯ h ω is the quantum of energy, also called the energy level. At a given temperature T, the average values hniand hEnican be obtained with Bose-Einstein statistics. They are hni=1 eE/kbT−1,hEni=hniE=E eE/kbT−1,(2) where kB=8.617×10−5eV/K is the Boltzmann constant. In a solid with 3nat degrees of freedom and therefore the same number of linear modes, each one is equivalent to a linear oscillator with a given frequency ω . It is usual to describe them as phonons or quasi-particles and to use the expression nphonons of a particular type with energy E=¯ h ω instead of a linear mode or state with frequency ω and excitation number n. We will also follow this convention although in some instances it may be more convenient to revert to the original terminology. As the number of frequenciesis very large and they are very close, ω and E=¯ h ω are considered as quasi-continuousvariables. Most energylevelsare degenerate,i.e., 6 JFR Archilla, SMM Coelho, FD Auret, C. Nyamhere, VI Dubinko, V Hizhnyakov 0 10 20 30 40 0 0.05 0.1 0.15 0.2 E (meV) f(E) (meV−1) Fig. 4 Comparison with the number function or normalized density of states for germanium: obtained from Fig. 2 and Ref. [25] (—); Einstein model with TE=288 (vertical —) ; Debye model with TD=360 (···). The mean mode energy is for the first two Eph =kBTE≃25meV. The Debye model by definition has a maximum energy KBTD=31meV with average energy 23.3meV equivalent to 270K (vertical ···) smaller than the Einstein’s one. The acoustic and optical bands can be seen although they overlap there is more than one mode with that energy, and in the quasi-continuous description there are very many in an interval [E,E+dE]. A variable density of states (DOS) g(E)is introduced, also some times called the density of levels. It is defined such as g(E)dEis the number of linear modes or quantum phonon states per unit volume with energies between Eand E+dE. For a discrete system the phonon spectrum is always bounded from above, that is, there is a maximum frequency and energy ω Mand EM=¯ h ω M, therefore ZEM 0g(E)dE=3nat .(3) A rough estimate of the maximum value of energy level for the acoustic modes can be obtained using the fact that the minimum value of the wavelength is twice the lattice unit of the primitive cell da, then EM,ac ≃¯ h ω M,ac =¯ hc2 π /2da, with c the speed for the mode. For Ge, da=a/√2=4.00 ˚ A and using cs=5400m/s, we obtain EM,ac =28meV and fM,ac =6.7THz similar at the observed values in Figs. 2 and 3. However,such a simple estimate for the optical modes isnot possible because the phase velocity tends to infinity when k→0. Generally speakingthere is no minimum frequencyor energy as explainedabove, however,when consideringonly a partof the system, it can be described assubjected to an external potential representing the interaction with the rest of the crystal. In this case the phonon spectrum becomes optical, i.e., bounded from below. The energy of the solid per unit volume is given by Experimental observation of moving intrinsic localized modes in germanium 7 300 350 400 0 0.5 1 1.5 2 2.5 3 T (K) 〈n〉 300 350 400 0 10 20 30 40 T (K) 〈n〉E (meV) Fig. 5 (Left): Average number of phonons with respect to temperature between room temperature TR=300K and E-center annealing temperature TA=423K. From bottom to top: optical phonons with Eop =35meV; Einstein phonons with EE=24.9meV; average number of phonons with Ge DOS; acoustic phonons with Eac=10. (Right): Average energy for different phonons, from bottom to top: optical phonons with Eop =35meV; Einstein phonons with EE=24.9meV (TE=288K) indistinguishable from the one obtained with Ge DOS; acoustic phonons with Eac =10meV and average classical energy kBT. It can be seen that the acoustic modes have more phonons and more energy than theoptical ones and at room temperature and above a quantum description is necessary uE=ZET 0hn(E)iEg(E)dE=ZET 0 E eE/kbT−1g(E)dE.(4) We will also use the number density or normalized density of states f(E) = g(E)/3nGe, with the property that as f(E)dEis the fraction of modes with energies between Eand E+dEand therefore the normalization condition and average phonon energy Eph are given by ZET 0f(E)dE=1,Eph =ZET 0E f(E)dE.(5) There are two approximationsfrequently used for the density of states: the Debye and the Einstein models. In the Debye model, all phonon modes are substituted by three acoustic branches with dispersion relation ω =ck, with the same c, which is an average velocity. These acoustic branches lead to a density of modes or states per unit volume gD(E) = 3/(2 π 2¯ h3)c3E2[2]. Then, fD(E) = g(E)/nat = α DE2, with the constant α Ddepending on the particular solid through cand nat. The energy has a cutoff value EDsuch that the condition of normalization RED 0fD(E)dE=1 is fulfilled. Therefore, α DE3 D/3=1. The values EDand TD=ED/kBare known as the Debye energy and temperature, respectively. Therefore there is only one unknown, either cor TD, either of which cannot be measured as they do not correspond to real magnitudes. What is done is to choose TDsuch that the specific heat cv(T) 8 JFR Archilla, SMM Coelho, FD Auret, C. Nyamhere, VI Dubinko, V Hizhnyakov fits the measurements. For Ge, a value of TD=360Kor ED=kBTD=31.1meV is usually given,whichcorrespondsto c=3420m/s. This velocityis not a real quantity but coherently it is approximately the mean of the velocities of the two transversal modes, ≃2500m/s, and the longitudinal one, 5400m/s [15]. The Debye dispersion relation works, of course, better for the acoustic branches and small wave vectors. The Einstein model supposes that there are 3nGe modes with the same frequency ω E, being EE=¯ h ω Eand TE=EE/kB, the Einstein energy and temperature, respectively. The value of TEis chosen so as to fit the specific heat of the solid, being EEan average energyof the phononsin the crystal. For germanium its value is TE=288K and will be used in this work. In this model the mean energy per unit volume at temperature Tin germanium is simply uE=3nGe eEE/kBT−1(6) The actual phonon dispersion relation and the density of states have been obtained and checked with experimental ones in Ref. [25]. Both magnitudes are shown in Fig. 2. The normalized density of states f(E)can be obtained from it but as the resolution is poor for low energies we have substituted that part by the Debye one. The Ge density of states is shown in Fig. 3 together with the corresponding one for 0 100 200 300 400 500 600 0 0.2 0.4 0.6 0.8 1 T (K) cv/3kB Fig. 6 Comparison of the experimental specific heat per degree of freedom: (–) using germanium density of states [25]; (--) the Einstein model with TE=288; (−·−) the Debye model with TD=360; (◦) and ( ∆ ) experimental values from Refs. [3,17], respectively. The horizontal line corresponds to the classical Dulong-Petit law. The Einstein and the Debye model are slightly better at intermediate temperatures, because they have been fitted for that. At high temperatures the experimental cvbecomes larger because actual frequencies also increase with temperature. The two separated dotted curves correspond to two Einstein models with energies Eac =10 and Eop =35meV, the upper and lower curves, respectively. These values are representative of the acoustic and optical branches Experimental observation of moving intrinsic localized modes in germanium 9 the Debye and Einstein modelfor comparison. For g(E)two concentrationsof states appear near the top and near the bottom of the spectrum, with a drastic simplification we can describe them as an optical band around Eop =35meV and an acoustic one around Eac =10meV. The mean phonon energyRET 0f(E)EdEis approximately equal to the Einstein energy. Figure 3 represents the number of phonons and the average energies as a function of temperature for acoustic phonons, optical phonons, Einstein phonons, and average values obtained with the density of states g(E). It can be seen that the classical statistics is not valid at the temperatures of interest in this work and that there are significant differences between optical and acoustic phonons. The energy in the acoustic modes is larger than in the optical ones in spite of having less energy but with more phonons. It can also be seen that the average number of phonons hniis smaller or closer to one which indicates that the classical description is not good at room temperatures and above. Figure 3 represents the specific heat at constant volume obtainedfrom these models. There is no significant difference at the temperatures of interest in this work between room temperature TR=300K and the annealing temperature of the E-center TA=423K. This justifies the use of the Einstein density of states as a good approximation for calculations. The specific heats for two Einstein models with Eac and Eop are also represented for comparison. 4 Defects and their detection with DLTS Point defects in the structure or the type of atoms of the semiconductor can appear with some probability due to the temperature but they can also be created by radiation. In the experiments described in this work most of the defects are created by 5MeV alpha radiation [14,21] produced in the decay of the americium isotope 241Am. A Ge sample with dimensions 3×5×0.6mm is brought into contact with americium foil for 30 minutes. Defects can be of many types, some simple examples are shown in Fig. 7, such as a vacancy, a substitutional atom, a self-interstitial, a foreign interstitial, a Frenkel pair that is, a combination of a vacancy and a self-interstitial and an E-center, which is a combination of a dopant substitutional atom and a vacancy. The germanium sample used in this work is doped with antimony (Sb), with a dopant concentration nSb =1.3×1015cm−3. Dopant atoms as Sb atoms occupy substitutional positions but are not considered defects as they are an essential part of the semiconductor electrical properties. The main defect appearing after α irradiation is the E-center already described. There are many others types such as vacancy complexes like the di-vacancy (V-V), the tri-vacancy (V3), the tetra-vacancy (V4) and combinations of interstitials as di or tri-interstitials (I2, I3). Also, hydrogen (H), due to its small size is able to penetrate almost everywhere and can combine with other defects forming complexes such as VHn, where n is an integer with values from 1−4. A variant of the E-center is the A-center, a complex of an oxygen interstitial and a vacancy. 16 JFR Archilla, SMM Coelho, FD Auret, C. Nyamhere, VI Dubinko, V Hizhnyakov 7 Thermal annealing In this section we review thermal annealing and apply it to Ge in order to compare the temperature and energy needed to obtain the same rate of thermal annealing as with Ar ions. Thermal annealing of defects in semiconductors is given by a first order kinetic dNT dt=−KNT,(17) where K, known as the reaction rate constant is given by an Arrhenius type law K=Ae−Ea/kBT,(18) where Eais known as the annealing energy and Aas the pre-exponential factor. Eacan be interpreted as the potential barrier which is necessary to surmount in order that the transformation or diffusion process for annealing takes place. The exponential term can be seen as the probability for an accumulation of energy of magnitude Ea. The pre-exponential term Ahas units of frequency and it is also known as the frequency factor. It is related to the number of attempts per unit time that the system tries to pass the barrier and with the curvature of the energy with respect to the reaction coordinate. Amay also depend on the temperature but in a much weaker way than the exponential term. It also depends on the entropy change. The integrationof Eq. (17) leadsto the exponentialdecay NT(t) = N(0)exp(−Kt) and comparing the experimental data with ln(N(t)) = ln(N(0)) −Kt it is possible to obtain K. Several data for E-center annealing have been published [11,18]. Here we will use the results obtained by some of the authors according to the procedure described in Ref. [20] and using the same dopant and defect concentration as in this work. Figure 7(a) shows the exponential decay at 165◦C and Figure 7(b) represents ln(NT)with respect to time for three temperatures. The approximate linear dependence can be seen. From the slopes, three values of the reaction rate constant are obtained and in Fig. 7(c) ln(K)is represented with respect to 1000/Tand the linear dependence can be observed. Comparing with ln(K) = ln(A)−Ea/kBTthe values A=5.5×1011s−1and Ea=1.3eV are obtained. These numbers should be treated with caution as the experimental procedure is very sensitive to the details of the experimental technique. The sample has to be cooled and reheated to measure the defect concentration. 8 Comparison of thermal and plasma-induced annealing Comparing the equations for thermal annealing Eq. (17) and ion-induced annealing Eq. (12) we can observe that if both process have the same rate of annealing Experimental observation of moving intrinsic localized modes in germanium 17 0 2000 4000 6000 8000 0 2 4 6 8x 1013 Time (s) NT (cm−3) 0 2000 4000 6000 8000 43 43.5 44 44.5 45 45.5 46 Time (s) ln(NT) 155°C 160°C 165°C (a) (b) (c) 2.25 2.3 2.35 2.4 −9 −8.5 −8 −7.5 −7 1000/T (K−1) ln(K) Fig. 10 (a) Defect concentration versus annealing time at T=165◦C. (b) Semi-log plot of defect concentration versus annealing time at temperatures 155◦C, 160◦C and 165◦C from which the annealing rate constant, K, is calculated. (c) The Arrhenius plot from which EA=1.3eV, A=0.55THz and TA=423K are obtained. Lines are fitted curves, circles and triangles are experimental values, the square in (c) corresponds to a thermal annealing rate equal to ion-induced annealing. Details of the experimental procedure used can be read in Ref. [20] K= σ i Φ ior Ae−Ea/kbT= σ i Φ i.(19) From this equation, the value of TA=423K is obtained. The thermal energy at TAper unit volume using Ge density of states g(E)from Sect. 3 is given by uph =ZEm 0hniEg(E)dE.(20) 18 JFR Archilla, SMM Coelho, FD Auret, C. Nyamhere, VI Dubinko, V Hizhnyakov Note that the use of the Einstein model with TE=288K leads to very similar results. The increment in energy from room temperature TR=300K to TA=423K is given by ∆ uph =uph(TA)−uph(TR)≃2.9KJ/mol ≃30.1meV/atom.(21) The energy per unit volume of energy in ILMs is given by uILM = ρ ILMEILM ,(22) where ρ ILM is the density per unit volume of ILMs and EILM is the mean ILM energy. Both quantities are unknown but we can estimate both. The maximum flux of ILMs is the flux of ions Φ iand the maximum energy is the energy that a 4eV Ar ion can deliver to a Ge atom, 3.6eV. Let us suppose EILM ≃3eV and Φ ILM ≃ Φ i. The velocity of ILMs should be of the order of magnitudeof the d velocity of sound, vILM ≃cs=5400m/s. Then ρ ILM ≃ Φ ILM/vILM ≃105cm−3and the ILM energy per Ge atom is uILM nGe = Φ ILMILMEILM vILM nGe ≃7×10−15meV/atom.(23) This value is so small because there is only an ILM for every 4×1017 Ge atoms. Therefore the ratio uILM/ ∆ uph ≃10−16, which proves that an enormously larger amount ofenergyin phononsis needed in orderto producethe same annealingeffect than the Ar ions produced. Changes in the ILM energy, their speed, the number of them created by neutrals in the plasma and other factors cannot change their energy density by a factor of 1016. 9 Summary In this work we have described an experiment in which a low energy, low flux Ar plasma anneals defects in Sb-doped Ge up to a significant depth below the surface. The hypothesis advanced in Ref. [1] and continued here is that Ar ions produce some kind of travelling localized excitation with great efficiency. We call these entities intrinsic localized modes or ILMs because their energy and other properties indicates that their energyis vibrational, although this is by no means demonstrated. Some space has been dedicated to phonons in germanium in order to have a clear picture of them and their energies and so doing clarify that they cannot be responsible for the annealing effect, because the ILM energy density is much smaller than the phonon density which produces the same annealing rate. Also we think that the study of the dispersion relation can bring home ideas about how to construct ILMs in Ge, which will be the confirmation of the present hypothesis but seems to be a daunting challenge. The numbers are approximate, many hypotheses and estimations that have been advanced may be incorrect, however none of these problems can change the fact of the observation of long-range annealing in germanium produced by Ar plasma and that ILMs are the most promising cause. Experimental observation of moving intrinsic localized modes in germanium 19 Acknowledgments The authors were funded by MICINN, project FIS2008-04848; the South African National Research Foundation and the European Regional DevelopmentFund, Centre of Excellence Mesosystems: Theory and Applications. JFRA and VD acknowledges the Physics Institute in Tartu for their hospitality. References 1. 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