Excitations of Atomic Vibrations in Amorphous Solids
aa r X i v : . [ c ond - m a t . d i s - nn ] N ov Excitations of Atomic Vibrations in Amorphous Solids
Li Wan ∗ Department of Physics, Wenzhou University, Wenzhou 325035, P. R. China (Dated: November 19, 2020)We study excitations of atomic vibrations in the reciprocal space for amorphous solids. Thereare two kinds of excitations we obtained, collective excitation and local excitation. The collectiveexcitation is the collective vibration of atoms in the amorphous solids while the local excitation isstimulated locally by a single atom vibrating in the solids. We introduce a continuous wave vectorfor the study and transform the equations of atomic vibrations from the real space to the reciprocalspace. We take the amorphous silicon as an example and calculate the structures of the excitationsin the reciprocal space. Results show that an excitation is a wave packet composed of a collection ofplane waves. We also find a periodical structure in the reciprocal space for the collective excitationwith longitudinal vibrations, which is originated from the local order of the structure in the realspace of the amorphous solid.
Keywords: atomic vibrations; amorphous solid; excitation; wave packet; local order
I. INTRODUCTION
Atomic vibrations have been well understood in crys-talline solids both theoretically and experimentally .In the crystalline solids, atoms vibrate around theirequilibrium positions and excitations of the lattice vi-brations are known as phonons. Each phonon has a welldefined crystal wave vector and a vibration frequency.To get phonons, the equation of lattice dynamics needto be transformed from the real space to the reciprocalspace. In the transformation, lattice plane waves areintroduced, and play their roles as lattice Fouriertransformations. By using the lattice periodicity in thecrystalline solids, the lattice Fourier transformationsdiagonalize the dynamics matrix to get the phonons.One lattice plane wave in the transformation gets onekind of phonons. However, in the amorphous solids,the periodicity of the lattices is broken, which makesphonons have no definition in the solids . Fromthe quantum mechanical view of point, the lack ofthe periodical lattices in the amorphous solids makesthe crystal wave vector not a good quantum numberany longer and the lattice Fourier transformationsbased on the crystal wave vector fail to diagonalizethe Hamiltonian of the system. Such invalidity of theFourier transformations breaks the bridge between thereal space and the reciprocal space for the amorphoussolids. Therefore, how to get excitations in the reciprocalspace for the amorphous solids still remains as a problem.To study atomic vibrations of the amorphous solids,various techniques have been applied. In the techniques,molecular dynamics (MD) simulation as a powerful toolhas been widely used . The MD simulation is to solvea set of equations of motion by the Newton’s law forthe positions and velocities of atoms in the amorphoussolids. After a long time evolving, the atoms reach theirequilibrium positions. Besides generating the amorphoussolids, the MD technique has been widely used to studythe thermal transport in the solids . The calcula- tions of the thermal conductivity by MD are based onthe Green-Kubo formula. The informations of latticedynamics obtained from MD simulations are in the realspace and can not provide the structures of the ex-citations in the reciprocal space for the amorphous solids.Besides the MD technique, lattice dynamics(LD)method is a different way to get the informations of theatomic vibrations . LD method has been generalizedto the amorphous solids and is used to diagonalizethe dynamics matrix to get frequency spectrum. Thediagonalizing is conducted in the real space for the solidsand the density of state then can be obtained. To makethe results close to the real amorphous solids, the LDmethod normally is applied on a supercell. Philip Allen,Joseph Feldman and others conducted the LD calcula-tions on a supercell of amorphous silicon . Basingon the calculations, they classified the vibrational statesinto three categories: propagons, diffusons and locons.Propagons occupy the bottom of the frequency spectrumand are considered to be delocalized. Propagons canpropagate through out the whole disorder system andbehave like phonons in crystalline solids. Locons arehigh frequency modes and considered to be spatiallylocalized. The modes termed with diffusons have thefrequencies in the range between the propagons and thelocons. Diffusons are not spatially localized and theycontribute to the thermal transport through diffusiveprocesses rather than the propagation of the propagons.Such taxonomy introduced by Allen, Feldman and theircolleagues opened up a new perspective on vibrationalmodes in amorphous solids and provides insights intothe thermal transport in the solids .Similar to the MD simulation, LD method identifiesthe vibration modes not in the reciprocal space. Thewave vector is not involved in the LD method, whichstill can not provide the excitations in the reciprocalspace. The propagons, diffusons and locons are notthe elementary excitations in the amorphous solids.The classifying of the propagons, diffusons and loconswith clear range boundaries is still under debates. Tofigure out the informations of atomic vibrations fromthe reciprocal space, Moon et.al defined a dynamicstructure factor to study the crossover frequency fromthe propagating excitations to diffusive vibrations .In their work, a supercell is considered to be periodicaland vibration frequencies used for the dynamic structurefactor are all at Γ. The relation between the wave vectorand the frequency discovered by Moon et.al is not thedispersion relation of elementary excitations. Similarwork is performed by Seyf and Henry by defining variousversions of the structure factor .In this study, we transform the equations of theatomic vibrations from the real space to the reciprocalspace and provide the excitations in the reciprocal spacefor amorphous solids. The transformations are rigorous.The wave vector involved in the transformations cancover the whole reciprocal space. And the dispersionrelation of the excitations can be clearly shown in thereciprocal space, which paves a way for the study ofthermal transport in amorphous solids. II. THEORY
We consider an amorphous solid of three dimensionalstructure. The total number of atoms in the solid is N . Each atom in the solid is vibrating around its ownequilibrium position. For the l -th atom with the massof M l , the equilibrium position is denoted by ~R l and thedisplacement away from ~R l for the vibration is by ~r l .We fix a rectangle coordinate system in the solid withthe three axises denoted by x , y and z respectively. Dueto the random arrangement of atoms in the amorphoussolid, it is meaningless to specify concrete directionsfor the axises like what can be done in crystallinesolids. We use the coordinate system only to denote thecomponents of vectors conveniently. The component of ~R l along a coordinate axis, say x axis, is denoted by R l,x . Similar notation is applied on ~r .In the vibration, each atom follows the dynamics equa-tion of M l ¨ r l,α = − X p,β ∂ Ψ ∂R l,α ∂R p,β r p,β . (1)In the above equation, the double dots on the top of r means the second order derivative of r with respect totime t . The total potential energy of the amorphoussolid is denoted by Ψ, which is functional of equilib-rium positions ~R of atoms. In the subscripts, l and p are the indexes to label atoms. α and β both are thecoordinates x , y or z . The second order partial deriva-tive of Ψ with respect to R l,α and R p,β actually is theforce constant between the l -th atom and the p -th atom when the l -th atom moves along α direction and the p -thatom along β direction. For convenience, we introduceΦ α,βl,p = ( ∂ Ψ ∂R l,α ∂R p,β )( √ M l M p ) for the force constant. Wealso introduce a quantity ~u l = √ M l ~r l to simplify thedynamics equation (1) as¨ u l,α = − X p,β Φ α,βl,p u p,β , (2)which is general for solids consisting of multi-specieswith various masses.Our goal is to transform Eq.(2) from the real spaceto the reciprocal space to get excitations. It is knownthat the lattice Fourier transformation that has beenwidely used in the crystalline solids to get phononsnow is invalid in the amorphous solid due to the lackof the lattice periodicity. Thus, we need a transforma-tion different to the lattice Fourier transformation forour goal. Before we introduce the transformation toachieve our goal, we note that there exist two kinds ofexcitations in the amorphous solid, collective excitationand local excitation. The collective excitation is forthe collective vibrations of the atoms, while the localexcitation is stimulated by a single atom. The localexcitation is spatially localized around the simulatingatom and decays its intensity in propagating awayfrom the atom. The decaying of the local excitationis due to the random scattering by the other atomsaround the stimulating atom. We will introducetwo transformations on Eq.(2) for the collective excita-tion and the local excitation respectively in the following. A. Collective Excitation
We start from Eq.(2), and introduce a wave vector ~κ for a transformation. The wave vector ~κ is continuousand is different from the crystal wave vector that is dis-crete for the crystalline solids. We denote the imaginaryunit by i and define a transformation for an normalcoordinate Q α~κ = (1 / √ N ) P l e − i~κ · ~R l u l,α in the spirit ofFourier transformation. The wave vector ~κ is continuousbecause of the absence of the lattice periodicity in theamorphous solid, and ranges from −∞ to + ∞ alongany direction. The collective excitation(CE) is thecollective vibration of all the atoms and propagatewithout decaying its intensity. Thus, the wave vector ~κ introduced for the CE must be real. Or, the imaginarypart of the wave vector will decay the intensity of theCE. The complex wave vector will be applied for thelocal excitation, but not for the CE.Physically, the normal coordinate Q ~κ defines a planewave with the wave vector of ~κ . The term of u l,α e − i~κ · ~R l in Q ~κ shows the magnitude and phase of the plane waveat the equilibrium position of the l -th atom. In the crys-talline solids, such plane wave defines a phonon. Thefrequency of the phonon corresponds to the wave vectorthrough the dispersion relation. All possible solutionsto the dispersion relation are curves for the crystallinesolids. However, it is not the case in the amorphoussolid, where the curves in the dispersion relation are re-organized, broadening and even dispersive. For a givenwave vector ~κ , we multiple the both sides of Eq.(2) by(1 / √ N ) e − i~κ · ~R l and sum the both sides over the totalatoms. Then we get a new equation, reading¨ Q α~κ = − X β V ~κ ′ Z F α,β~κ,~κ ′ Q β~κ ′ d~κ ′ (3)with F α,β~κ,~κ ′ = P l,p e − i~κ · ~R l Φ α,βl,p e i~κ ′ · ~R p . On the left handside of Eq.(3), we have used the definition of the normalcoordinate Q and kept the second order derivative withrespect to time. On the right hand side of Eq.(3), V ~κ ′ is the volume for the integration R d~κ ′ in the reciprocalspace. The details for the derivation of Eq.(3) has beenshown in Appendix A. Now we have transformed thedynamics equation from the real space to the reciprocalspace by Eq.(3).To go further, we express Eq.(3) in matrix form. Wearrange Q α~κ in one column as a vector Q . Each entry in Q is indexed by both of ~κ and α . We arrange the forceconstant Φ α,βl,p in the dynamics matrix Φ. The row of Φis indexed by both of l and α while the column of Φ isby p and β . Since α or β represents three perpendiculardirections ( x , y and z ), Φ α,βl,p is a 3 × α and β for a given pair of l and p . Then, wedefine a matrix ξ for e − i~κ · ~R l . The row of ξ is indexed byboth of ~κ and β while the column is indexed by both of l and α , even though the indexes α and β do not appear inthe element e − i~κ · ~R l of ξ . By varying the indexes of α and β for each given pair of row index ~κ and column index l in ξ , we have a 3 × ξ is a 3 × e − i~κ · ~R l . In this way, thesizes of the matrices are consistent for the matrix productand Eq.(3) is still hold in the matrix form. According toEq.(3), ξ † is in between Φ and Q . Then, we discrete theintegral of Eq.(3). We set the infinitesimal volume ∆ ~κ ′ for the reciprocal space and divide the total volume V ~κ ′ by ∆ ~κ ′ to get the total discrete number N for the wavevector. Finally, we introduce a matrix F = N ξ · Φ · ξ † forthe component (1 /V ~κ ′ ) F α,β~κ,~κ ′ . Eq.(3) then is expressed inthe matrix form, reading¨ Q = − N ξ · Φ · ξ † · Q = − F · Q, (4)which is equivalent to Eq.(3) in the limit of ∆ ~κ ′ ap-proaching zero. The dot between two matrices representsthe matrix product. It could be found that matrix F is not diagonal, mean-ing that the plane waves u l,α e − i~κ · ~R l with different wavevectors ~κ interact with each other. It is the nature ofthe disorder system, comparing to the case of crystallinesolids in which F is diagonal simultaneously after theFourier transformation and one plane wave leads to onephonon. In order to get the CE for the amorphous solid,we need to diagonalize the matrix F to get decoupled ex-citations. We find an unitary matrix U for the diagonaliz-ing. After that, we obtain a diagonal matrix Ω = U · F · U † and then define a new vector P = U · Q . We multiply theboth sides of Eq.(4) to the left by the matrix U . And weinsert the identity matrix U † · U in between the matrices F and Q of Eq.(4). In this way, we get an equation¨ P = − Ω · P (5)with Ω diagonal. Now we are at the position to solvethe equation (5). We take the p -th mode of the vector P as an example. The p -th mode is at the p -th rowof P and is denoted by P p . We also denote the entryat the p -th row and the p -th column of the diagonalmatrix Ω by Ω p,p . Then, the equation (5) is reduced tobe ¨ P p = − Ω p,p P p for the mode. We set P p have a timephase of e i πω p t with ω p as the vibration frequency forthe p -th mode. We substitute the time phase into thereduced equation of P p . We solve out that the frequency2 πω p is the square root of Ω p,p with a positive and realvalue.The physical meaning of P is the key element toget the CE. In the vector Q , each entry representsan organization of all the atoms in the real space toform a plane wave. The plane waves have various wavevectors and interfere with each other in the amorphoussolid. They are not decoupled. The interference of theplane waves is reflected by the non-diagonal matrix F in Eq.(4). Then, we use the matrix U to reorganize theplane waves to form wave packets in the reciprocal space.Each entry in the vector P represents a wave packetreorganized by U . The wave packets are decoupled toeach other and they are exactly the CEs we want. Eachwave packet is composed of a collection of plane waveswith various wave vectors but only one same vibrationfrequency, like ω p in our example for the wave packet P p . Physically, one CE is the collective vibration of theatoms in the real space and the atomic vibrations havevarious wave vectors but one vibration frequency.It is clear now that a CE, say P p , is a wave packet.Explicitly, we have the expression of P p = P q U pq Q q forthe CE basing on the definition of P . The contributionof each plane wave Q q to the wave packet P p is exactlyrevealed by the entry U pq which is at the p -th row and the q -th column in U . Generally, U pq is a complex numberincluding the informations of intensity and phase of theplane wave Q q . B. Local Excitation
Compared to the CE that is for the collective vibrationof global atoms, the local excitation (LE) is localizedaround a single atom that stimulates the LE. Theintensity of the LE decays when the LE propagates awayfrom the center. In the amorphous solid, the disorderarrangement of atoms makes every atom be the centerto stimulate LEs. Without lose of generality, we take the0-th atom as an example to stimulate a LE and thinkabout the l -th atom which the LE can reach with l = 0.We set the equilibrium position of the 0-th atom by ~R and that of the l -th atom by ~R l . The displacement ~u l of the l -th atom for the LE vibration must decaywhen the distance | ~R l − ~R | between the 0-th and the l -th atoms increases. For simplicity, we neglect theanisotropic decaying along different directions for LEs.To show the decaying, we introduce a complex wavevector κ = κ r + iκ i with κ r and κ i both real scalars. Thedisplacement for the vibration follows ~u l = ~A κ e iκ | ~R l − ~R | with κ i be positive to guarantee the decaying of theLE. The coefficient ~A κ can be written inversely as ~A κ = ~u l e − iκ | ~R l − ~R | . Basing on such statement, we candefine a transformation.Similar to the case of CE, we define the transforma-tion to get a quantity Q ακ = P l √ N e − iκ | ~R l − ~R | u l,α . Wemultiple both sides of Eq.(2) by √ N e − iκ | ~R l − ~R | and sumover all the atoms on the both sides. After some algebra,we get a new equation from Eq.(2) by using the definitionof Q ακ . The equation reads¨ Q ακ = − X β L κ ′ r L κ ′ i Z F α,βκ,κ ′ Q βκ ′ dκ ′ (6)with F α,βκ,κ ′ = P l,p e − iκ | ~R l − ~R | Φ α,βp,l e iκ ′ | ~R p − ~R | . Note that dκ ′ = dκ ′ r dκ ′ i in Eq.(6) for convenience. And, L κ ′ r is thetotal length for κ ′ r in the reciprocal space since we haveignored the direction of the wave vectors. L κ ′ i then isthe total length for κ ′ i . The detail derivation could befound in Appendix B.In the following, we express Eq.(6) in matrix form aswe have done for the CE. We arrange the components Q ακ in a column as a vector Q indexed by κ r , κ i and α . To discrete the integral, we replace dκ ′ r and dκ ′ i bythe infinitesimal length ∆ κ ′ r and ∆ κ ′ i respectively. Andthen define L κ ′ r / ∆ κ ′ r = N r for the discrete number ofthe real part κ ′ r in the reciprocal space. Similarly, we set L κ ′ i / ∆ κ ′ i = N i for the imaginary part κ ′ i . Note that κ ′ i must be positive while κ ′ r can be both of positive andnegative. Finally, we define a matrix F for F α,βκ,κ ′ by ab-sorbing the factor 1 / ( N r N i ). Then, we have an equationtransformed from Eq.(6), reading¨ Q = −F · Q . (7) Similar to the matrix F for CE, local vibrations withvarious wave vectors in F are interfered with each other.We need diagonalize F to get decoupled LEs. We findan unitary matrix U for the diagonalizing and then geta diagonal matrix Λ = U · F · U † . We define a vector by P = U · Q for the LE. Finally, we have an equation forthe LE, reading ¨ P = − Λ · P . (8)To solve this equation, we still take the time phase e i πω p t for the entry P p at the p -th row in P with ω p the vibration frequency. Then from Eq.(8), 2 πω p isthe square root of Λ pp that is at the p -th row and p -thcolumn in Λ.Each entry in P represents an LE. Each LE comprises acollection of decaying plane waves. Those decaying planewaves for the LE are with various complex wave vectorsbut only one vibration frequency. The imaginary part ofthe complex wave vectors decays the LE away from thestimulating center. The contributions of the decayingplane waves to an LE are calculated from the matrix U , as we have specified for the CE by using the matrix U . III. COMPUTATIONAL DETAILS
We study CEs and LEs in an amorphous silicon asan application of our theory. The amorphous silicon isgenerated by MD. The Eigen library is implemented inour code to diagonalize matrices. In the following, wespecify the computational details.
A. Amorphous Silicon
We use MD to generate an amorphous silicon. TheMD simulations were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator(LAMMPS) with a time step of 0.5 f s . In thesimulations, the Stilling-Web interatomic potential wasimplemented and the periodical boundary condition wasapplied . We follow the simulation steps in Ref.(18) forLAMMPS . We started from a crystalline silicon with 8cells along [001], [010] and [100] directions, containing4096 silicon atoms in total, and melt the structure at3500 K for 500 ps in an NVT ensemble. Next, wequenched the liquid silicon to 1000 K with the quenchrate of 100 K/ps , followed by annealing the structure at1000 K for 25 ns . Finally, we quenched the structure ata rate of 100 K/ps to 300 K and then equilibrated thestructure at 300 K for 10 ns in an NVT ensemble. Afteran additional equilibration at 300 K in an NVE ensemblefor 500 ps , we obtain the amorphous silicon we need. B. Dynamics Matrix
We consider the force constant Φ α,βl,p =( ∂ Ψ ∂R l,α ∂R p,β )( √ M l M p ) for the l -th atom vibratingalong α direction and the p -th atom along β directionwith l = p . We fix the l -th atom at some positionthat is shift away from its equilibrium position along α direction. And then shift the p -th atom away fromits equilibrium position along β direction. In shiftingthe p -th atom, we calculate the difference of the totalinteratomic potential Ψ at various positions of the p -thatom to get the derivative ∂ Ψ /∂R p,β . Then, we fixthe l -th atom at a new position along α direction andrepeat shifting the p -th atom to get a new derivative of ∂ Ψ /∂R p,β . Basing on the difference of the derivative ∂ Ψ /∂R p,β for the l -th atom at various positions, weget the second order of the derivative ( ∂ Ψ ∂R l,α ∂R p,β ) forthe force constant. In the calculation, the Stilling-Webinteratomic potential is applied . For the case of l = p , the force constant Φ α,βl,l = − P p = l Φ α,βl,p has beenwell defined to guarantee that no force is applied onthe l -th atom along α direction when all the atomsmove by an identical distance along β direction. Basedon the above statement, the dynamics matrix is obtained.In diagonalizing matrices F and F , the unitarymatrices U and U obtained are complex, including theinformations of intensity and phase of waves for theexcitations. The frequencies of the excitations must bereal and positive as well. Thus, after the diagonalizing,we only take the real and positive entries from thematrices Ω and Λ for the frequencies of the excitations.In the numerical calculation, we set a small value ǫ andselect the entries with their imaginary parts in the rangeof ( − ǫ, ǫ ) from the matrices Ω and Λ. C. Structure of Excitations
We focus on the physical structure of the excitations.We take the p -th CE as an example, which is the p -th entry of the vector P and has the expression of P p = P q U pq Q q . As we have discussed, U pq is the con-tribution of the plane wave Q q to the CE of P p . U pq is acomplex number, have the informations of intensity andphase of the wave Q q . In this study, we take the absolutevalue of U pq to show the intensity of the wave Q q , andneglect the phase information. By solving Eq.(5), we getthe frequency ω p for the CE of P p . We note the wavevector of Q q by κ q . By manipulating the relation of ω p , κ q and the intensity | U pq | of Q q , we can investigate thestructure of the CE P p in the reciprocal space. Suchstatement can be applied on LE with the same treatment.For a given wave vector, say along x direction,there exists two types of CEs. One type of the CE is transverse, in which atoms have the vibrational directionperpendicular to the wave vector. The transverse CE hastwo degenerate states since there exist two orthogonaldirections y and z both normal to x . The other typeCE is longitudinal, in which atoms have the vibrationaldirection parallel to the wave vector. To illustrate ourtheory, we vary the wave vector along only one directionsuch as only along x direction. In this case, the CEscomposed of Q y~κ (or Q z~κ ) is the transverse CEs and theCEs composed of Q x~κ are for the longitudinal CEs.In this calculation,we normalize the mass of one siliconatom by 10 − kg , and the length by 1˚ A . The energy isnormalized by 10 KJ/mol , which is 1 . × − J peratom. Thus, the wave vector is normalized by ˚ A − andthe frequency for the normalization is 2 . T Hz . IV. RESULTS
We use LAMMPS to generate the amorphous silicon.Fig.(1a) is the Radial Distribution Function (RDF) ofthe liquid silicon melted at 3500 K as the first step inthe LAMMPS simulation. The RDF of the amorphoussilicon at 300 K is plotted in Fig.(1b), in which the firstpeak is located at 2 . A and shows the local order of thestructure. The second peak of the RDF in Fig.(1b) issplit, which is the main feature of the amorphous soliddifferent from the liquid RDF in Fig.(1a) . The splitof the second peak has been indicated by an arrow inFig.(1b) for clarity, meaning that the silicon we study isreally an amorphous solid. A. Collective Excitation
As we have mentioned in Section(III C), the solutionsto Eq.(5) give us the following informations for the p -thmode CE, the frequency ω p , the wave vector κ q for thewave Q q contributing to the CE, and the intensity | U pq | of Q q . In the following plots, we drop off the subscripts p and q for clear notation. And we use the phrase ofintensity | U | referred to the intensity | U pq | of Q q inshort. We manipulate the relation of ω , κ and intensity | U | to show the structures of CEs. In Fig.(2a), we fixthe frequency ω = 20 × . T Hz for a transverse CE andplot intensity | U | as functional of the wave vector κ .Here 2 . T Hz is the frequency normalization as we havementioned in Section(III C). In the figure, we get twowave packets and the peaks of the two wave packets arelocated at − . A − and 0 . A − respectively. Thesetwo wave packets are symmetric about κ = 0 and theyare the forward and the backward waves respectively.Due to the symmetry, we focus on only one wave packetat the peak of 0 . A − . The width of the half intensityof the peak is about 0 . A − , which is corresponding toa wave packet with a scale of 2 π/ . ≅ A in the realspace. Then, we fix κ = 0 . A − in the reciprocal space RD F Radial Distance (Å) 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7(b) amorphous silicon RD F Radial Distance (Å)
FIG. 1: Radial Distribution Function (RDF) of silicon.(a) Liquid silicon is melted at 3500 K . (b) Amorphoussilicon is obtained at 300 K by LAMMPS. The split ofthe second peak in (b) has been indicated by an arrow.and plot the intensity | U | in Fig.(2b) by scanning thefrequency ω . We still get a peak, meaning that for awell defined wave vector κ there exist many waves Q that have various vibration frequencies. Comparably,in crystalline solids, an excitation of lattice vibrationor a phonon has only one well defined wave vectorcorresponding to one frequency. Such difference ofthe excitation structure of the atomic vibrations in aamorphous solid and a crystalline solid is due to thedisorder arrangement of atoms in the former.We use a color bar to show the intensity | U | of theCEs and plot the relation of the frequency ω and thewave vector κ in Fig.(3). Fig.(3a) is for the transverseCE and Fig.(3b) is for the longitudinal CE. Resultsshow that the frequency ω of CEs is no more than42 × . T Hz . In Fig.(3a), we observe two branchesstart from ω = 0 T Hz to ω = 25 × . T Hz , behavinglike the acoustic branches in the crystalline silicon. Thebranches in Fig.(3a) are linear-like due to the disorderarrangement of atoms erasing the anisotropic scatteringof waves in the amorphous silicon. Each branch has abroadening line width, which shows that the CEs arewave packets as we have given the example in Fig.(2).Once the frequency is larger than ω = 25 × . T Hz , the FIG. 2: Intensity | U | of transverse CEs. (a) For antransverse CE with the frequency fixed at 20 × . T Hz ,the intensity of the CE shows two wave packets in thereciprocal space. (b) For transverse CEs with the wavevector κ fixed at 0 . A − , the intensity of the CEsshows a wave packet by scanning the frequency.two branches end into a band in Fig.(3a). The upperfrequency of the band is ω = 37 × . T Hz . In the band,the structure of the transverse CEs is random ratherthan a wave packet. This is because the transverse CEsin the band have large wave vectors and short wavelengths. Those transverse CEs with short wave lengthscan see the discreteness of the atoms in the solid and canbe scattered by the atoms easily. What is more, thosetransverse CEs vibrate with the directions perpendicularto the wave vector. The random arrangement of atomsalong the vibration directions of the transverse CEsmulti-scatters the CEs for the propagation along thewave vector, which brings the random structure of thetransverse CEs in the reciprocal space. On the otherside, the occurrence of the branches in Fig.(3a) is due tothe long wave length of the transverse CEs by which theCEs can not distinguish the discreteness of the atoms. Italso could be found in the figure that there exists a gapbetween the two branches. That means no transverseCEs can be stimulated in the gap.FIG. 3: Structure of collective CEs in the reciprocalspace. (a) Structure of the transverse CEs in thereciprocal space shows two branches starting from zerofrequency to ω = 25 × . T Hz and end into a frequencyband. (b) A periodical structure is found for thelongitudinal CEs in the reciprocal space.It is very interesting to find a periodical structurein Fig.(3b) for longitudinal CEs. In the figure, we stillcan find two branches go up from ω = 0 T Hz , but endat ω = 42 × . T Hz . And then the two branches godown with a zigzag structure. The zigzag structureis periodical with the period roughly about 2 . A − since we can not accurately locate the peaks for suchdisorder system. We think the phase for one period is2 π , and calculate the averaged lattice period in the realspace corresponding to the periodicity in the reciprocalspace. We find that the averaged lattice period isabout 2 . A , which is the location of the first peak inFig.(1b). Such observation reveals that the local orderof the structure in the amorphous solid plays its role asthe lattice parameter and makes the amorphous solidbehave like a crystalline solid for the longitudinal CEs.Thus, it is possible for us to define a quasi-Brillouinzone for the longitudinal CEs. The boundary of thefirst quasi-Brillouin zone is at ± π/a with a the location of the first peak in RDF. Then, we can go further todefine the reciprocal lattice vector by G = 2 π/a for thelongitudinal CEs, and map the longitudinal CEs to thephonons of crystalline solids for the study of physicalproperties. Such work is out of the scope of this paper. B. Local Excitation
To reveal the structures of LEs, we need to show therelation of the frequency ω , the wave vector κ , and theintensity |U| . Note that κ of LEs is a complex numberwith the real part κ r and the imaginary part κ i . Thatmeans we have four parameters at hand for the study.To clearly show the structure of the LEs, we use thecolor bar to show the intensity |U| and fix κ i for eachmap. We show the results in Fig.(4). In Fig.(4a), weshow the structure of LE with κ i = 0˚ A − . Zero κ i means the LE stimulated by a given atom can propagatewithout decaying. In the figure, the frequency for theLEs can reach as high as 170 × . T Hz , which is muchhigher than the maximum frequency of CEs. Those LEsstimulated locally by the atoms are distributed randomlyin the reciprocal space as shown in Fig.(4a). The LEsinterfere with each other in the amorphous solid to formCEs eventually as we have shown in Fig.(3). We alsofind a gap close to κ r = 0 in Fig.(4a), which originatesthe gap of the CEs in Fig.(3).In the case of κ i = 0, LEs will decay their intensitiesin propagating. The decaying length is approximated tobe 1 /κ i . Such as in Fig.(4b), we have κ i = 0 . A − andthe decaying length is about 20˚ A . It could be found thatthe maximum frequency in Fig.(4b) is 69 × . T Hz thatis much lower than the maximum frequency in Fig.(4a).The LEs with frequencies higher than 69 × . T Hz donot satisfy Eq.(8) when κ i = 0 . A − . This is becausethe LEs with a higher frequency have a larger energy andcan propagate further out of the decaying length of 20˚ A under the condition of κ i = 0 . A − . The LEs are stillrandomly distributed in the reciprocal space and the gapstill can be found in Fig.(4b). When κ i is increased to alarger value, the maximum frequency for LEs decreasesto a smaller value, as shown in Fig.(4c) and Fig.(4d). V. CONCLUSIONS
We have studied the excitations of atomic vibrationsin the reciprocal space for amorphous solids. The exci-tations can be classified into two categories, collectiveexcitation and local excitation. The collective excitationis due to the collective vibrations of all the atoms in theamorphous solids while the local excitation is stimulatedby a single atom locally. The wave vector for thecollective excitations must be real while the wave vectorfor the local excitations is complex. The imaginary partFIG. 4: Structure of LEs in the reciprocal space. Theimaginary part κ i of the wave vector is fixed for eachmap. (a) κ i = 0 . A − . (b) κ i = 0 . A − .(c) κ i = 0 . A − . (d) κ i = 0 . A − . of the wave vector for the local excitations decays theexcitations.The excitations are wave packets in amorphous solids,comprising a collection of plane waves with variouswave vectors but one vibration frequency. The collectiveexcitation has two types, the transverse excitation andthe longitudinal excitation. It is interesting to find thatthe longitudinal excitation has a periodical structure inthe reciprocal space. The periodicity is originated fromthe local order of the structure in the real space. Thelocal excitation can also be found by our theory in thereciprocal space. Results show that the local excitationswith higher frequencies have larger decaying lengths.For the excitations, a gap can be found in the reciprocalspace where no excitation can occur.In this study, we didn’t touch two problems. The firstproblem is how to use the excitations in our theory toclassify the propagons, diffusons and locons. To definethe density of state of the excitations is the key to theproblem. The second problem is how to calculate thethermal conductivity of the amorphous solids by usingthe excitations. To solve this problem, we need to applythe Bose-Einstein statistics correctly for the excitations.To solve the two problems are our future works.The author kindly acknowledges Prof. Ning-Hua Tongfrom Renmin University of China for discussions. Appendix A
We multiple the both sides of Eq.(2) by (1 / √ N ) e − i~κ · ~R l and sum the both sides over the total atoms. Then weget a new equation reading¨ Q α~κ = − X l √ N e − i~κ · ~R l X p,β Φ α,βl,p X m δ m,p u m,β (A1)The left hand side of Eq.(A1) is Q α~κ as we have defined inthe text of this paper. On the right hand side of Eq.(A1),we have introduced the Kronecker Delta function δ m,p toreplace u p,β by P m δ m,p u m,β . The function δ m,p can beexpressed as δ m,p = 1 V ~κ ′ Z e i~κ ′ · ( ~R p − ~R m ) d~κ ′ . (A2)The integration is over the volume V ~κ ′ in the reciprocalspace. We substitute Eq.(A2) into Eq.(A1) and rewriteEq.(A1) as¨ Q α~κ = − X l,p,m,β √ N e − i~κ · ~R l Φ α,βl,p V ~κ ′ Z e i~κ ′ · ( ~R p − ~R m ) d~κ ′ u m,β = − X β V ~κ ′ Z X l,p e − i~κ · ~R l Φ α,βl,p e i~κ ′ · ~R p " √ N X m e − i~κ ′ · ~R m u m,β d~κ ′ . (A3)We define F α,β~κ,~κ ′ = P l,p e − i~κ · ~R l Φ α,βl,p e i~κ ′ · ~R p for the firstbracket and replace the term in the second bracket by Q β~κ ′ = √ N P m e − i~κ ′ · ~R m u m,β as we have defined. Then,we recover Eq.(3). Appendix B
We multiple both sides of Eq.(2) by √ N e − iκ | ~R l − ~R | andsum the both sides over all the atoms. Then, we have an equation, reading¨ Q ακ = − X l √ N e − iκ | ~R l − ~R | X p,β Φ α,βl,p X m δ m,p u m,β . (B1)Here, we have introduced the Kronecker Delta function δ m,p = 1 L κ ′ r L κ ′ i Z e − iκ ′ r ( | ~R m − ~R |−| ~R p − ~R | ) e κ ′ i ( | ~R m − ~R |−| ~R p − ~R | ) dκ ′ r dκ ′ i = 1 L κ ′ r L κ ′ i Z e − iκ ′ ( | ~R m − ~R |−| ~R p − ~R | ) dκ ′ . (B2)Here, κ ′ r is the real part of κ ′ while κ ′ i is the imag-inary part of κ ′ . We use dκ ′ to replace dκ ′ r dκ ′ i forshort notation. L κ ′ r is the length for κ ′ r in thereciprocal space while L κ ′ i is for κ ′ i . In Eq.(B2), L κ ′ r R e − iκ ′ r ( | ~R m − ~R |−| ~R p − ~R | ) dκ ′ r leads to the Kronecker function δ m,p . In the disorder solid, there is almost zeroprobability for more than 1 atoms have the same dis-tance to ~R . Therefore, δ m,p is a good result for theintegration of κ ′ r . Based on δ m,p , the integration of L κ ′ i R e κ ′ i ( | ~R m − ~R |−| ~R p − ~R | ) dκ ′ i gets unit. We substituteEq.(B2) into Eq.(B1) and we have¨ Q ακ = − X l √ N e − iκ | ~R l − ~R | X p,β Φ α,βl,p X m L κ ′ r L κ ′ i Z e − iκ ′ ( | ~R m − ~R |−| ~R p − ~R | ) dκ ′ u m,β = − X β L κ ′ r L κ ′ i Z X l,p e − iκ | ~R l − ~R | Φ α,βl,p e iκ ′ ( | ~R p − ~R | ) " √ N X m e − iκ ′ ( | ~R m − ~R | ) u m,β dκ ′ = − X β L κ ′ r L κ ′ i Z F α,βκ,κ ′ Q βκ ′ dκ ′ . (B3)We define F α,βκ,κ ′ for the first bracket and use the notation of Q for the second bracket on the second line of Eq.(B3).Then we recover Eq.(6). ∗ [email protected] M. Born and K. Huang,
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