7 Lattice Vibrations. 7.1 Introduction

Transcription

1 7 Lattice Vibrations 7.1 Introduction Up to this point in the lecture, the crystal lattice (lattice vectors & position of basis atoms) was always assumed to be immobile, i.e., atomic displacements away from the equilibrium positions of a perfect lattice were not considered. This allowed us to develop formalisms to compute the electronic ground state for an arbitrary periodic but static atomic configuration and therewith the total energy of our system. It is obvious that the assumption of a completely rigid lattice does not make a lot of sense neither at finite temperatures nor at 0K, where it even violates the uncertainty principle. Nonetheless, we obtained with this simplification a very good description for a wide range of solid state properties, among which were the electronic structure (band structure), the nature and strength of the chemical bonding, and low temperature cohesive properties, e.g., the stable crystal lattice and its lattice constants. The reason for this success is that the internal energy U of a solid (or similarly of a molecule) can be written as U = E static + E vib, (7.1) where E static is the electronic ground state energy with a fixed lattice (on which we have focused so far) and E vib is the additional energy due to lattice motion. In general (and we will substantiate this below), E static E vib, so that many properties are described in qualitatively correct fashion and often even semi-quantitatively even when E vib is neglected. There is, however, also a multitude of solid state properties, for which the consideration of the lattice dynamics is crucial. In this respect, dynamics is already quite a big word, because for many of these properties it is sufficient to take small vibrations of the atoms around their (ideal) equilibrium position into account. As we will see below, these vibrations of the crystal can be described as quasi-particles so called phonons. The most salient examples are: High-Accuracy Predictions at 0 K: Even at 0 K, the nuclei are always in motion due to quantum-nuclear effects. In turn, this alters the equilibrium properties at 0 K (lattice constants, bulk moduli, band gaps, etc.). For high-accuracy predictions, the nuclear motion can thus not be neglected. Thermodynamic Equilibrium Properties for T > 0 K: In a rigid and immobile lattice, the electrons are the only degrees of freedom. In metals, only the free-electron-like conduction electrons can take up small amounts of energy (of the order of thermal energies). The specific heat capacity due to these conduction electrons was computed to vary linearly with temperature for T 0 in exercise 4. In 181

2 insulators with the same rigid perfect lattice assumption, not even these degrees of freedom are available, since thermal excitations across the band gap E g k B T exhibit a vanishingly small probability exp( E g /k B T ). In reality, one observes, however, that for both insulators and metals the specific heat capacity varies predominantly with T 3 at low temperatures, in clear contradiction to the above cited rigid lattice dependency of T. The same argument holds for the pronounced thermal expansion typically observed in experiments: Due to the negligibly small probability of electronic excitations over the gap, there is nothing left in a rigid lattice insulator that could account for it. These expansions are, on the other hand, key to many engineering applications in material science (different pieces simply have to fit together over varying temperatures). Likewise, a rigid lattice offers no possibility to describe or explain structural phase transitions upon heating and the final melting. Thermodynamic Non-Equilibrium Properties for T > 0 K: In a perfectly periodic potential, Bloch-electrons suffer no collisions, since they are eigenstates of the many-body Hamiltonian. Formally, this leads to infinite transport coefficients. The thermodynamic nuclear motion breaks this perfect periodicity and thus introduces scattering events and finite mean free paths, which in turn explain the really measured, and obviously finite electric and thermal conductivity of metals and semiconductors. 1 Computing and understanding these transport properties and other related non-equilibrium properties (propagation of sound, heat transport, optical absorption, etc.) is of fundamental importance for many real devices (like transistors, LED s, lasers, etc.). 7. Vibrations of a Classical Lattice 7..1 Adiabatic (Born-Oppenheimer) Approximation Throughout this chapter we assume that the electrons follow the atoms instantaneously, i.e., the electrons are always in their ground-state at any possible nuclear configuration {R I }. In this case, the potential-energy surface (PES) on which the nuclei move is given by V BO ({R I }) = min Ψ E({R I};Ψ) (7.) for any given atomic structure described by atomic positions {R I } (I nuclei). This is nothing else but the Born-Oppenheimer approximation, which we discussed at length in the first chapter, and that is why the so defined PES is often also called Born-Oppenheimer surface. Qualitatively, this implies that there is no memory effect stored in the electronic motion (e.g. due to the electron cloud lacking behind the atomic motion), since V BO ({R I }) is conservative, i.e., its value at a specific configuration {R I } does not depend on how this configuration was reached by the nuclei. 1 Please note that defects and boundaries break the perfect symmetry of a crystal as well. However, they are not sufficient to quantitatively describe these phenomena. AbreakdownoftheBorn-Oppenheimerapproximationsoftenmanifests itself as friction: Inthatcase, the actual value of V ({R I }) depends not only on {R I },butonthewholetrajectory.seeforinstance Wodtke, Tully, and Auerbach, Int. Rev. Phys. Chem. 3, 513(004). 18

3 The motion of the atoms is then governed by the Hamiltonian H = T nu + V BO ({R I }), (7.3) which does not contain the electronic degrees of freedom explicitly anymore. The complete interaction with the electrons (i.e. the chemical bonding) is instead contained implicitly in the functional form of the PES V BO ({R I }). For a perfect crystal, the minimum of this PES corresponds to the equilibrium configuration {R I } of a Bravais lattice (lattice vectors & position of the basis atoms). In the preceding chapters, we have exclusively discussed these equilibrium configurations {R I } in the rigid lattice model. Now, we will also discuss deviations from these equilibrium configurations {R I }: In principle, this boils down to evaluating the PES at arbitrary positions {R I }. Practically, this can become rapidly intractable both analytically and computationally. In many cases, however, we do not need to consider all arbitrary configurations {R I }, but only those that are very close to the (pronounced) minimum corresponding to the ideal lattice {R I }.Thisallowsusto write R I = R I + s I with s I a (a: lattice constant). (7.4) For such small displacements around a minimum, the PES can be approximated by a parabola using the harmonic approximation. 7.. Harmonic Approximation The fundamental assumption of the harmonic approximation consists in limiting the description to small displacements s I from the equilibrium configuration {R I }, i.e., to vibrations with small amplitudes. Accordingly, this is a low-temperature approximation. In this case, the potential in which the particles are moving can be expanded in a Taylor series around the equilibrium geometry, whereby only the first leading terms are considered. We illustrate this first for the one-dimensional case: The general Taylor expansion around a minimum at x 0 yields [ ] V (x) = V (x 0 )+ x V (x) s + 1 [ ] x 0 x V (x) s + 1 [ ] 3 x 0 3! x V (x) s , (7.5) 3 x 0 with the displacement s = x x 0. The linear term vanishes, because x 0 is a stationary point of the PES, i.e., the equilibrium geometry in which no forces are active. In the harmonic approximation, cubic (and higher) terms are neglected, which yields V (x) V (x 0 )+ 1 [ ] x V (x) s = V (x 0 )+ 1 x 0 cs. (7.6) [ ] Here, we have introduced the spring constant c = V (x) in the last step, which x x 0 shows that we are indeed dealing with a harmonic potential that yields restoring forces that are proportional to the actual displacement s. F = V (x) = cs (7.7) x 183

4 A corresponding Taylor expansion can also be carried out in three dimensions for the Born-Oppenheimer surface of many nuclei, which leads to V BO ({R I }) = V BO ({R I }) + 1 M,M 3,3 [ ] s µ I sν J R µ V BO ({R I }). (7.8) I Rν J {R I } I=1, J=1 µ=1, ν=1 Here, µ, ν = x, y, z are the three Cartesian coordinates, IJrun over the individual nuclei, and the linear term has vanished again for the exact same reasons as in the one-dimensional case. By defining the (3M 3M) matrix, i.e., the Hessian [ ] Φ µν IJ = R µ V BO ({R I }), (7.9) I Rν J {R I } the Hamilton operator for the M nuclei of the solid can be written as H = M I=1 PI + V BO ({R M I})+ 1 M,M nu,i I=1 J=1 3,3 µ=1 ν=1 s µ I Φµν IJ sν J, (7.10) where M nu,i is the mass of nucleus I. Please note that the contributions of the N electrons is still implicitly contained in V BO ({R I }) and Φµν IJ. As a matter of fact, V BO ({R I }) corresponds to the energy of the rigid lattice discussed in previous chapters. Given that it is a constant, it does however not affect the motion of the nuclei, which only depends on Φ µν IJ in the harmonic approximation. This also sheds some light on our earlier claim that generally V BO ({R I })=Estatic E vib : The thermodynamically driven E vib is typically 0.5 ev/atom, e.g., approx. 50 mev/atom at room temperature. In comparison, the static term is often much larger, especially in the case of a strong (ionic or covalent) bonding, which often feature cohesive energies > 1 ev/atom (see previous chapter). However, E vib can become a decisive contribution whenever there is only a weak bonding or when there are (even strongly bound) competing phases, the energy difference of which is in the order of E vib. To evaluate the Hamiltonian H defined in Eq. (7.10) we need the (3M) second derivatives of the PES at the equilibrium geometry with respect to the 3M coordinates s µ I.Analytically, this is very involved, since the Hellmann-Feynman theorem cannot be applied here: In exercise 1, we had seen that the forces acting on a nucleus J can be generally expressed as F J = RJ V BO ({R I })= RJ Ψ H Ψ (7.11) = 0 Ψ RJ H Ψ Ψ H RJ Ψ. (7.1) Here, H is the full many-body Hamiltonian and Ψ=Ψ {RI }({r i }) is the exact electronic many-body wavefunction in the Born-Oppenheimer approximation (see Chap. 1). Accordingly, evaluating the forces at a specific configuration {R I } only requires to know the wavefunction (density) at this specific configuration, but not its derivative. This is no longer true for higher-order derivatives of V BO ({R I }), e.g., Φ µν IJ = dfµ I dr ν J H = Ψ R µ I Rν J Ψ Ψ/ R ν J H R µ I Ψ Ψ H R µ I (7.13) Ψ/ R ν J. (7.14) 184

5 In this case, the derivative or response of the wavefunction Ψ/ R ν J to a displacement enters explicitly. Formally, this can be generalized in the n +1 theorem [Gonze and Vigneron, Phys. Rev. B 39,1310(1989)]:Evaluatingthe(n+1) th derivative of the Born- Oppenheimer surface requires to know the n th derivative of the wavefunction (density). Essentially, this problem can be tackled by means of perturbation theory [Baroni, de Gironcoli, and Dal Corso, Rev. Mod. Phys. 73, 515 (001).] In this lecture, we will discuss an alternative route that exploits the three-dimensional analog to Eq. (7.7), i.e., Φ µν IJ =0 {}}{ F µ I (R 1,...,R J + δsµ J,...,R M ) F µ I (R 1,...,R J,...,R M). (7.15) δs J,µ In this finite difference approach, the second derivative can be obtained numerically by simply displacing an atom slightly from its equilibrium position and monitoring the arising forces on this and all other atoms. This approach will be discussed in more detail in the exercises. At this stage, this force point of view highlights that many elements of the second derivative matrix will vanish: A small displacement of atom I will only lead to non-negligible forces on the atoms in its immediate vicinity. Atoms further away will not notice the displacement, and hence the second derivative between these atoms J and the initially displaced atom I will be zero: 3 Φ µν IJ = 0 for R I R J a 0, (7.16) where a 0 is the lattice parameter. Very often, one even finds that the values of Φ µν IJ become negligible for R I R J > a 0. Accordingly, only a finite number of differences (7.15), i.e., only a finite number of neighbours J need to be inspected for each atom I. Despite these simplifications, this numerical procedure is at this stage still unfortunately intractable, because we would have to do it for each I of the M atoms in the solid (i.e. atotalof3m times). With M 10 3, this is unfeasible, and we need to exploit further properties of solids to reduce the problem. Obviously, just like in the case of the electronic structure problem a much more efficient use of the periodicity of the lattice must be made. Although the lattice vibrations break the perfect periodicity of the static crystal, we can still exploit the translational invariance of the lattice by assuming that periodic images of the unit cell behave in a similar fashion, i.e., break the local symmetry in a similar fashion. Most clearly, this can be understood and derived by inspecting the equations of motion following from the general form of Eq. (7.10) Classical Equations of Motion Before further analyzing the quantum-mechanical problem, it is instructive and more intuitive to first discuss the classical Hamilton function corresponding to Eq. (7.10). We will see that the classical treatment emphasizes more the wave-like character of the lattice vibrations, whereas a particle (corpuscular) character follows from the quantum-mechanical 3 In the case of polar crystals with ionic bonding and thus long-range Coulomb interactions this assumption does not hold. In that case, long-range interactions need to be explicitly accounted for, see for instance Baroni, de Gironcoli, and Dal Corso, Rev. Mod. Phys. 73, 515(001). 185

6 treatment. This is not new, but just another example of the wave-particle dualism known, e.g., from light (electromagnetic waves viz. photons). In the classical case and knowing the structure of the Hamilton function from Eq. (7.10), the total energy (as sum of kinetic and potential energy) can be written as E = M I=1 M nu,i ṡ I + V BO ({R I})+ 1 M,M I=1, J=1 3,3 µ=1, ν=1 s µ I Φµν IJ sν J (7.17) in the harmonic approximation. Accordingly, the classical equations of motion for the nuclei can be written as M nu,i s µ I = F µ I = J,ν Φ µν IJ sν J, (7.18) where the right hand side represents the µ-component of the force vector acting on atom I of mass M nu,i and the double dot above s represents the second time derivative. As a first step, we will get rid of the time dependence by using the well-known ansatz s µ I (t) = uµ I eiωt. (7.19) for this harmonic second-order differential equation. Inserting Eq. (7.19) into Eq. (7.18) leads to M ion,i ω u µ I = Φ µν IJ uν J. (7.0) J,ν This is a system of coupled algebraic equations with 3M unknowns (still with M 10 3!). As already remarked above, it is hence not directly tractable in this form, and we exploit that the solid has a periodic lattice structure in its equilibrium geometry (just like in the electronic case). Correspondingly, we decompose the coordinate R I into a Bravais lattice vector L n (pointing to the unit cell) and a vector R α pointing to atom α of the basis within the unit cell, R µ I = L µ n + R µ α. (7.1) Formally, we can therefore replace the index I over all atoms in the solid by the index pair (n, α) running over the unit cells and different atoms in the basis. With this, the matrix of second derivatives takes the form Φ µν IJ = Φµν αβ (n, n ). (7.) And in this form, we can now start to exploit the periodicity of the Bravais lattice. From the translational invariance of the lattice follows immediately that Φ µν αβ (n, n ) = Φ µν αβ (n n ). (7.3) This suggests, that u I may be describable by means of a vibration u µ α(l n ) = 1 Mnu,α c µ αe iqln (7.4) 186

7 where c is the amplitude of displacement and q is a vector in reciprocal space. 4 Using this ansatz for Eq. (7.0), we obtain ω c µ α = βν { } 1 Φ µν αβ Mα M (n n )e iq(ln L n ) c ν β. (7.5) n β Note that the part in curly brackets is only formally still dependent on the position R I (through the index n). Since all unit cells are equivalent, R I is just an arbitrary zero for the summation over the whole Bravais lattice. The part in curly brackets, D µν αβ (q) = 1 Mα M β n Φ µν αβ (n n )e iq(ln L n ), (7.6) is called dynamical matrix, and its definition allows us to rewrite the homogeneous linear system of equations in a very compact way ω c µ α = βν D µν αβ (q)cν β ω c = D(q)c. (7.7) In the last step, we have introduced the vectorial notation c =(c x 1,c y 1,c z 1,c x, ) to highlight that we are dealing with an eigenvalue problem here. Compared to the original equation of motion, cf. Eq. (7.0), a dramatic simplification has been reached. Due to the lattice symmetry we achieved to reduce the system of 3M p M equations to 3M p equations, where M p is the number of atoms in the basis of the unit cell. The price we paid for it is that we have to solve the equations of motion for each q anew. This is comparably easy, though, since this scales linearly with the number of q-points and not cubic like the matrix inversion, and has to be done only for few q (since we will see that the q-dependent functions are quite smooth). From Eq. (7.7), one gets a condition for the existence of solutions to the eigenvalue problem (i.e. solutions must satisfy the following:) ( ) det ˆD(q) ω ˆ1 =0. (7.8) This problem has 3M p eigenvalues, which are functions of the wave vector q, ω = ω i (q), i =1,,...,3M p (7.9) and to each of these eigenvalues ω i (q), Eq. (7.7) provides the eigenvector c i (q) =(c x i,1(q),c y i,1 (q),cz i,1(q),c x i,(q), ). (7.30) These eigenvectors are determined up to a constant factor, and this factor is finally obtained by requiring all eigenvectors to be normalized (and orthogonal to each other). In conclusion, we find that the displacement s I (t) of atom I =(n, α) away from the equilibrium position at time t is given by s µ I (t) = 1 A i (q)c µ i,α (q)ei(qln ω i(q)t) Mnu,α i,q, (7.31) 4 To distinguish electronic and nuclear degrees of freedom, q and not k is used in the description of lattice vibrations. 187

8 whereby the amplitudes A i (q) are determined by the initial conditions. With this, the system of 3M three-dimensional coupled oscillators is transformed into 3M decoupled (but collective ) oscillators, and the solutions (or so-called normal modes) correspond to waves which propagate throughout the whole crystal. The resemblance of these waves to what we think of as oscillators traditionally is thus only formal. The normal modes form a basis for the description of lattice vibrations, i.e., any arbitrary displacement situation of the atoms can now be written as an expansion in these special solutions Comparison between ϵ n (k) and ω i (q) We have derived the normal modes of lattice vibrations exploiting the periodicity of the ideal crystal lattice in a similar way as we did for the electron waves in chapters 4 and 5. Since it is only the symmetry that matters, the dispersion relations ω i (q) will therefore exhibit a number of properties equivalent to those of the electronic eigenvalues ϵ n (k). We will not derive these general properties again, but simply list them here (check chapter 4 and 5 to see how they emerge from the periodic properties of the lattice): 1. ω i (q) is periodic in q-space. The discussion can therefore always be restricted to the first Brillouin zone.. ϵ n (k) and ω i (q) have the same symmetry within the Brillouin zone. Additionally to the space group symmetry of the lattice, there is also time inversion symmetry, yielding ω i (q) =ω i ( q). 3. Due to the use of periodic boundary conditions, the number of q-values is finite. When the solid is composed of M/M p unit cells, then there are M/M p different q- values in the Brillouin-zone. Since i =1...3M p, there are in total 3M p (M/M p )= 3M values for ω i (q), i.e., exactly the number of degrees of freedom of the crystal lattice. When speaking of solids, M 10 3 so that q and the dispersions relations ω i (q) become essentially continuous. For practical calculation using supercells of finite size, this aspect has however to be kept in mind, as discussed in more detail in the exercises. 4. ω i (q) is an analytic function of q within the Brillouin zone. Whereas the band index n in ϵ n (k) can take arbitrarily many values, i in ω i (q) is restricted to 3M p values. We will see below that this is connected to the fact that electrons are fermions and lattice vibrations (phonons) bosons Simple One-dimensional Examples The difficulty of treating and understanding lattice vibrations is primarily one of bookkeeping. The many indices present in the general formulae of section 7.. and 7..3 let one easily forget the basic (and in principle simple) physics contained in them. It is therefore very useful (albeit academic) to consider toy model systems to understand the idea behind normal modes. The simplest possible system representing an infinite periodic lattice (i.e. with periodic boundary conditions) is a linear chain of atoms, and we will see that with the conceptual understanding obtained for one dimension it will be straightforward to generalize to real three-dimensional solids. 188

9 unit cell a M s(n-1) s(n) s(n+1) s(n+) Figure 7.1: Linear chain with one atomic species (Mass M, and lattice constant a) Linear Chain with One Atomic Species Consider an infinite one-dimensional linear chain of atoms of identical mass M, connected by springs with (constant) spring constant f as shown in Fig The distance between two atoms in their equilibrium position is a, ands(n) the displacement of the n th atom from this equilibrium position. In analogy to eq. (7.18), the Newton equation of motion for this system is written as M.. s n = f(s(n) s(n +1))+f(s(n 1) s(n)), (7.3) i.e., the sum over all atoms breaks down to left and right nearest neighbor, and the second derivative of the PES corresponds simply to the spring constant. We solve this problem as before by treating the displacement as a sinusoidally varying wave: s(n) = 1 M ce i(qan ωt), (7.33) yielding ω M = f ( e iqa e iqa). (7.34) Solving for ω, one obtains f ω(q) = M ( qa ) sin. (7.35) Alternatively, one can solve this problem using the dynamical matrix formalism given by Eq. (7.7) and discussed in the previous chapter. Formally, the potential energy terms that depend on s n are which yields the second derivatives V = f (s(n 1) s(n)) + f (s(n) s(n +1)), (7.36) V s(n) s(n 1) = V s(n) s(n +1) = f and V s(n) s(n) 189 =f. (7.37)

10 .0 ω(q) / (f/m) 1/ π/a 0 +π/a q Figure 7.: Dispersion relation ω(q) for the linear chain with one atomic species. Accordingly, the full Hessian matrix has the structure f f f Φ= f f f f f f (7.38) Please note that we omitted the cartesian degrees of freedom µ, ν in this one-dimensional case. Since this system has only one atom in the unit cell, it is sufficient to focus on one line of the Hessian (α =1). This yields the dynamical matrix with one entry: Accordingly, we get the dispersion relation f ω(q) = M D(q) = f M ( eiqa e iqa ). (7.39) ( qa ) sin, (7.40) the eigenvector c =1, and thus the displacement field s(n, t) = 1 A(q)e i(qan ω(q)t). (7.41) M q As expected, ω is a periodic function of q and symmetric with respect to q and q. The first period lies between q = π/a and q =+π/a (first Brillouin zone), and the form of the dispersion is plotted in Fig. 7.. Equation (7.41) gives the explicit solutions for the displacement pattern in the linear chain for any wave vector q. They describe waves propagating along the chain with phase velocity ω/q and group velocity v g = ω/ q. Let us look in more detail at the two limiting cases q 0 and q = π/a (i.e. the center and the border of the Brillouin zone). For q 0 we can approximate ω(q) given by Eq. (7.35) with its leading term in the small q Taylor expansion (sin(x) x for x 0) ( ) f ω a q. (7.4) M 190

11 s(n-1) = -s(n) s(n) s(n+1) = -s(n) s(n+) = s(n) s(n+3) = -s(n) s(n+4) = s(n) Figure 7.3: Snap shot of the displacement pattern in the linear chain at q =+π/a. ω is thus linear in q, and the proportionality constant is simply the group velocity v g = a f/m, which is in turn identical to the phase velocity and independent of frequency. For the displacement pattern, we obtain s(n, t) = A(q) M e iq(an vgt) (7.43) in this limit, i.e., at these small q-values (long wavelength) the phonons propagate simply as vibrations with constant speed v g (the ordinary sound speed in this 1D crystal!). This is the same result we would have equally obtained within linear elasticity theory, which is understandable because for such very long wavelength displacements the atomic structure and spacing is less important (the atoms move almost identically on a short length scale). Fundamentally, the existence of a branch of solutions whose frequency vanishes as q vanishes results from the symmetry requiring the energy of the crystal to remain unchanged, when all atoms are displaced by an identical amount. We will therefore always obtain such types of modes, i.e., ones with ω(0) = 0, regardless of whether we consider more complex interactions (e.g. spring constants to second nearest neighbors) or higher dimensions. Since their dispersion is close to the center of the Brillouin zone linear in q, which is characteristic of sound waves, these modes are commonly called acoustic branches. 5 One of the characteristic features of waves in discrete media, however, is that such a linear behaviour ceases to hold at wavelengths short enough to be comparable to the interparticle spacing. In the present case ω falls below v g q as q increases, the dispersion curve bends over and becomes flat (i.e. the group velocity drops to zero). At the border of the Brillouin zone (for q = π/a) we then obtain s(n, t) = 1 M e iπn e iωt. (7.44) Note that e iπn =1for even n,ande iπn = 1 for odd n. This means that neighboring atoms vibrate against each other as shown in Fig This also explains, why in this case the highest frequency occurs. Again, the flattening of the dispersion relation is determined entirely by symmetry: From the repetition in the next Brillouin zone and with kinks forbidden, v g = ω/ q =0must result at the Brillouin zone boundary, regardless of how much more complex and real we make our model. 191

12 a -a/4 +a/4 M1 M s(1,n) s(,n) s(1,n+1) s(,n+1) Figure 7.4: Linear chain with two atomic species (Masses M1 and M). The lattice constant is a and the interatomic distance is a/ Linear Chain with Two Atomic Species Next, we consider the case shown in Fig. 7.4, where there are two different atomic species in the linear chain, i.e., the basis in the unit cell is two. The equations of motion for the two atoms follow as a direct generalization of Eq. (7.3).. M 1 s 1 (n) = f (s 1 (n) s (n) s (n 1)) (7.45).. M s (n) = f (s (n) s 1 (n +1) s 1 (n)). (7.46) Hence, also the same wave-like ansatz should work as before s 1 (n) = 1 M1 c 1 e i(q(n 1 4 )a ωt) (7.47) s (n) = 1 M c e i(q(n+ 1 4 )a ωt). (7.48) This leads to ω c 1 = f M1 M c 1 + ω c = f M1 M c + f ( qa ) c cos M1 M f ( qa ) c 1 cos M1 M (7.49), (7.50) i.e., in this case the frequencies of the two atoms are still coupled to each other. The eigenvalues follow from the determinant f M1 M ω f M1 M cos ( ) qa f M1 M cos ( ) qa f M1 M ω = 0, (7.51) which yields the two solutions ( 1 ω± = f + 1 ) ( 1 ± f + 1 ) 4 ( qa ) sin M 1 M M 1 M M 1 M = f M 1 ± f M 4 ( qa ) sin, (7.5) M 1 M 5 Notable exceptions do exist: For instance, graphene exhibits aparabolicdispersioninitsacoustic modes close to Γ. 19

13 with the reduced mass M ( = ) 1 1. M M Again, one can also solve this problem using the dynamical matrix formalism given by Eq. (7.7) and discussed in the previous chapter. Formally, the potential energy terms that depend on s 1 (n) and s (n) are given by V = f [s (n 1) s 1 (n)] + f [s 1(n) s (n)] + f [s (n) s 1 (n +1)], (7.53) which yields the second derivatives V s 1 (n) s (n 1) V s 1 (n) s 1 (n) = = V s 1 (n) s (n) = V s (n) s (n) V s (n) s 1 (n) = V s (n) s 1 (n +1) = f =f. (7.54) Accordingly, the full Hessian matrix exhibits the structure f f f Φ= f f f f f f (7.55) Again, cartesian degrees of freedom µ, ν have been omitted for this one-dimensional case. In contrast to our earlier example, this system features two atoms in the unit cell, so that we now get a dynamical matrix with dimension two: D(q) = ( f f M 1 f M1 M (1 + exp(+iqa)) ) M1 M (1 + exp( iqa)). (7.56) The respective eigenvalues are the solutions of ( 1 ω 4 f + 1 ) ω + 4f ( qa ) sin =0 (7.57) M 1 M }{{ M } 1 M M 1 that yields the dispersion relation ω ± (q) = f M 1 ± f M 4 ( qa sin M 1 M f M ). (7.58) We therefore obtain two branches for ω(q), the dispersion of which is shown in Fig As apparent, one branch goes to zero for q 0, ω (0) = 0, as in the case of the monoatomic chain, i.e., we have again an acoustic branch. The other branch, on the other hand, exhibits interestingly a (high) finite value ω + (0) = f M for q 0 and shows a much weaker dispersion over the Brillouin zone. Whereas in the acoustic branch both atoms within the 193

14 .0 ω(q) / (f/m) 1/ π/a 0 +π/a q Figure 7.5: Dispersion relations ω(q) for the linear chain with two atomic species (the specific values are for M 1 =3M = M). Note the appearance of the optical branch not present in Fig. 7.. unit cell move in concert, they vibrate against one another in opposite directions in the other solution. In ionic crystals (i.e. where there are opposite charges connected to the two species), these modes can therefore couple to electromagnetic radiation and are responsible for much of the characteristic optical behavior of such crystals. Correspondingly, they are generally (i.e. not only in ionic crystals) called optical modes, and their dispersion is called the optical branch. f At the border of the Brillouin zone at q = π/a we finally obtain the values ω + = M 1 f and ω = M, i.e. as apparent from Fig. 7.5 a gap has opened up between the acoustic and the optical branch. The width of this gap depends only on the mass difference of the two atoms (M 1 vs. M ) in this case. 6 The gap closes for identical masses of the two species. This is comprehensible since then our model with equal spring constants and equal masses gets identical to the previously discussed model with one atomic species: The optical branch is then nothing but the upper half of the acoustic branch, folded back into the half-sized Brillouin zone Phonon Band Structures With the conceptual understanding gained within the one-dimensional models, let us now proceed to look at the lattice vibrations of a real three-dimensional crystal. In principle, the formulae to describe the dispersion relations for the different solutions are already given in section Most importantly, it is Eq. (7.8) that gives us these dispersions. In close analogy to the electronic case, the dispersions are called phonon band structure in their totality. The crucial quantity is the dynamical matrix. The different branches of the band structure follow from the eigenvalues after diagonalizing this matrix. This route is also followed in quantitative approaches to the phonon band structure problem, e.g., within density-functional theory. In the so-called direct, supercell, or frozenphonon approach, one exploits the expression of the forces as given in Eq. (7.15) for the computation of the dynamical matrix, since in principle it is nothing more than a massweighted Fourier transform of the force constants. In practice, the different atoms in the 6 In more realistic cases, varying spring constants, e.g., f 1 and f,wouldalsoaffectthegap. 194

15 Figure 7.6: DFT-LDA and experimental phonon band structure (left) and corresponding density of states (right) for fcc bulk Rh [from R. Heid, K.-P. Bohnen and W. Reichardt, Physica B 63, 43(1999)]. unit cell are displaced by small amounts and the forces on all other atoms are recorded. In order to be able to compute the dynamical matrix with this procedure at different points in the Brillouin zone, larger supercells are employed and the (in principle equivalent) atoms within this larger periodicity displaced differently, so as to account for the different phase behavior. Obviously, in this way only q-points that are commensurate with the supercell periodicity under observation can be accessed, and one runs into the problem that quite large supercells need to be computed in order to get the full dispersion curves. Approaches based on perturbation theory (linear response regime) are therefore also frequently employed: The details of the different methods go beyond the scope of the present lecture, but a good overview can for example be found in R.M. Martin, Electronic Structure: Basic Theory and Practical Methods, Cambridge University Press, Cambridge (004). Experimentally, phonon dispersion curves are primarily measured using inelastic neutron scattering and Ashcroft/Mermin dedicates an entire chapter to the particularities of such measurements. Overall, phonon band structures obtained by experimental measurements and from DFT-LDA/GGA simulations typically agree extremely well these days, regardless of whether computed by a direct or a linear response method. Instead of discussing the details of either measurement or calculation, we rather proceed to some practical examples. If one has a mono-atomic basis in the three-dimensional unit cell, we expect from the onedimensional model studies that only acoustic branches should be present in the phonon band structure. There are, however, three of these branches because the orientation of the polarization vector (i.e. the eigenvector belonging to the eigenvalue ω(q)), cf. Eqs. (7.30) and (7.31)) matters in three-dimensions. In the linear chain, the displacement was only considered along the chain axis and such modes are called longitudinal. If the atoms vibrate in the three-dimensional case perpendicular to the propagation direction of the wave, two further transverse modes are obtained, and longitudinal and transverse modes must not necessarily exhibit the same dispersion. Fig. 7.6 shows the phonon band structure for bulk Rh in the fcc structure and the phonon density of states (DOS) resulting from it (which is obtained analogously to the electronic structure case by integrating over the 195

16 Figure 7.7: DFT-LDA and experimental phonon band structure (left) and corresponding density of states (right) for Si (upper panel) and AlSb (lower panel) in the diamond lattice. Note the opening up of the gap between acoustic and optical modes in the case of AlSb as a consequence of the large mass difference. 8 cm 1 1 mev [from P. Giannozzi et al., Phys. Rev. B 43, 731 (1991)]. 196

17 Brillouin zone). Particularly along the lines Γ X and Γ L, which run both from the Brillouin zone center to the border, the dispersion of the bands qualitatively follows the form discussed for the linear chain model above and is thus conceptually comprehensible. Turning to a poly-atomic basis, the anticipated new feature is the emergence of optical branches due to vibrations of the different atoms against each other. This is indeed apparent in the phonon band structure of Si in the diamond lattice shown in Fig. 7.7, and we again roughly find the shape that also emerged in the diatomic linear chain model for these branches (check again the Γ X and Γ L curves). Since there are in general 3M p different branches for a basis of M p atoms, cf. section 7..4, 3M p 3 optical branches result (i.e. in the present case of a diatomic basis, there are 3 optical branches, one longitudinal one and two transverse ones). Although the diamond lattice has two inequivalent lattice sites (and thus represents a case with a diatomic basis), these sites are both occupied by the same atom type in bulk Si. From the discussion of the diatomic linear chain, we would therefore expect a vanishing gap between the upper end of the acoustic branches and the lower end of the optical ones. Comparing with Fig. 7.7 this is indeed the case. Looking at the phonon band structure of AlSb also shown in this figure, a huge gap can now, however, be discerned. Obviously, this is the result of the large mass difference between Al and Sb in this compound, showing that we can indeed understand most of the qualitative features of real phonon band structures from the analysis of the two simple linear chain models. In order to develop a rough feeling for the order of magnitude of lattice vibrational frequencies (or in turn the energies contained in these vibrations), it is helpful to keep the inverse variation of the phonon frequencies with mass in mind, cf. Eq. (7.6). For transition metals, we are talking about a range of 0-30 mev for the optical modes, cf. Fig. 7.6, while for lighter elements this becomes increasingly higher, cf. Fig First row elements like oxygen in solids (oxides) exhibit optical vibrational modes around mev, whereas the acoustic modes go down to zero at the center of the Brillouin zone in all cases. Thus, all frequencies lie in the range of thermal energies, which already tells us how important these modes will be for the thermal behavior of the solid. 7.3 Quantum Theory of the Harmonic Crystal In the preceding section we have learnt to compute the lattice vibrational frequencies when the ions are treated as classical particles. In practice, this approximation is not too bad for quite a range of materials properties. The two main reasons are the quite heavy masses of the ions (with the exception of H and He), and the overall rather similar properties of the quantum mechanical and the classical harmonic oscillator on which all of the harmonic theory of lattice vibrations is based. In a very rough view, one can say that if only the vibrational frequency ω(q) of a mode itself is relevant, the classical treatment gives already a good answer (which is ultimately why we could understand the experimentally measured phonon band structure with our classical analysis). As a matter of fact, the formalism introduced in the previous chapter has highlighted that the eigenvalues and -vectors are a property of the harmonic potential itself. If, on the other hand, the energy connected with this mode, ω(q), is important or, more precisely, the way one excites (or populates) this mode is concerned, then the classical picture usually fails. The prototypical example for this is the specific heat due to the lattice vibrations, which is nothing else but the measure of how effectively the vibrational modes of the lattice can 197

18 take up thermal energy. In order to understand material properties of this kind, we need a quantum theory of the harmonic crystal. Just as in the classical case, let us first recall this theory for a one-dimensional harmonic oscillator before we really address the problem of the full solid-state Hamiltonian One-dimensional Quantum Harmonic Oscillator Instead of just solving the quantum mechanical problem of the one-dimensional harmonic oscillator, the focus of this section will rather be to recap a specific formalism that can be easily generalized to the three-dimensional case of the coupled lattice vibrations afterwards. At first sight, this formalism might appear mathematically quite involved and seems to overdo for this simple problem, but in the end, the easy generalization will make it all worthwhile. In accordance with Eq. (7.6), the classical Hamilton function for the one-dimensional harmonic oscillator is given by H classic 1D = T + V = p m + 1 fs = p m + mω s, (7.59) where f = mω is the spring constant and s the displacement. From here, the translation to quantum mechanics proceeds via the general rule to replace the momentum p by the operator p = i s where p and s satisfy the commutator relation, (7.60) [p, s] = ps sp = i, (7.61) which is in this case nothing but the Heisenberg uncertainty principle. The resulting Hamiltonian preserves the form as in Eq. (7.59), but p and s have now to be treated as (non-commuting) operators. This structure can be considerably simplified by introducing thelowering operator and the raising operator a = a = mω s + i 1 mω p, (7.6) mω 1 s i mω p. (7.63) Note that the idea behind this pair of adjoint operators leads to the concept called second quantization, where a and a are denoted as annihilation and creation operator, respectively. The canonical commutation relation of Eq. (7.61) then implies [a, a ] = 1, (7.64) and the Hamiltonian takes the simple form ( H QM 1D = ω a a + 1 ). (7.65) 198

19 From the commutation relation and the form of the Hamiltonian, it follows that the eigenstates n> (n =0, 1,,...) of this problem fulfill the following relations: a n> = (n +1) 1/ n +1>, (7.66) a n> = n 1/ n 1> (n 0), (7.67) a 0> = 0. (7.68) A derivation of these is given in any decent textbook on quantum mechanics. We see that the operators a and a lower or raise the excitation state of the system. This explains the names given to these operators, in particular when considering that each excitation can be seen as one quantum: The creation operator creates one quantum, while the annihilation operator wipes one out. With these recursion relations between the different states, the eigenvalues (i.e. the energy levels) of the one-dimensional harmonic oscillator are finally obtained as E n = <n H QM 1D n> = (n +1/) ω. (7.69) Note, that E 0 0, i.e., even in the ground state the oscillator has some energy (so called energy due to zero-point vibrations or zero-point energy), whereas with each higher excited state one quantum ω is added to the energy. With the above mentioned picture of creation and annihilation operators, one therefore commonly talks about the state E n of the system as corresponding to having excited n phonons of energy ω. This nomenclature accounts thus for the discrete nature of the excitation and emphasizes more the corpuscular/particle view. In the classical case, the energy of the system was determined by the amplitude of the vibration and could take arbitrary continuous values. This is no longer possible for the quantum mechanical oscillator Three-dimensional Quantum Harmonic Crystal For the real three-dimensional case, we will now proceed in an analogous fashion as in the simple one-dimensional model. To make the formulae not too complex, we will restrict ourselves to the case of a mono-atomic basis in this section, i.e., all ions have a mass M nu and M p =1. The generalization to polyatomic bases is straightforward, but messes up the mathematics without yielding too much further insight. In section 7.., we had derived the total Hamilton operator of the M nuclei in the harmonic approximation. It is given by H = M I=1 p I + V BO ({R M I})+ 1 M,M nu I=1 J=1 3,3 I=1 J=1 s µ I Φµν IJ sν J, (7.70) and the displacement field defined in Eq. (7.31) solves the classical equations of motion. For a mono-atomic basis, we can subsume all time-dependent terms into s i (q,t) and define the transformation to normal mode coordinates s µ I (t) = i,q s i (q,t)c µ i,α (q)ei(qln). (7.71) 199

20 Inserting this form of the displacement field into Eq. (7.70) yields H vib = = 3 p i (q) M nu ωi (q) s i (q,t) (7.7) M q nu i=1 q 3 [ p i (q) + M ] nuωi (q) s i (q,t), (7.73) M q nu }{{} i=1 i=1 H vib i (q) where we have introduced the generalized momentum p i = d/dt s i (q,t) and focused only on the vibrational part of the Hamiltonian, H vib, i.e., we discarded the contribution from the static minimum of the potential energy surface by setting V BO ({R I })=0. With this transformation, the Hamiltonian has decoupled to a sum over independent harmonic oscillators defined by one-dimensional Hamiltonians Hi vib (q) (compare with Eq. (7.59)!). It is thus straightforward to generalize the solution discussed in detail in the last section to this more complicated case. For this purpose, all previously defined operators need to become i-andq-dependent: We define annihilation and creation operators for a phonon (i.e. normal mode) with frequency ω i (q) and wave vector q as Mnu ω i (q) 1 a i (q) = s i (q) + i M nu ω i (q) p i(q), (7.74) a i (q) = Mnu ω i (q) 1 s i (q) i M nu ω i (q) p i(q). (7.75) Similarly, the Hamiltonian is brought to a form equivalent to Eq. (7.65), i.e., H vib = 3 ω i (q) i=1 q ( a i (q)a i(q)+ 1 ), (7.76) and the eigenvalues generalize to 3 E vib = ω i (q) i=1 q ( n i (q)+ 1 ). (7.77) The energy contribution of the lattice vibrations therefore stems from 3M harmonic oscillators with frequencies ω i (q). The frequencies are the same as in the classic case, which is (as already remarked in the beginning of this section) why the classical analysis permitted us already to determine the phonon band structure. In the harmonic approximation, the 3M modes are completely independent of each other: Any mode can be excited independently and contributes the energy ω i (q). While this sounds very familiar to the concept of the independent electron gas (with its energy contribution ϵ n (k) from each state (n, k)), the fundamental difference is that in the electronic system each state (or mode in the present language) can only be occupied with one electron (or two in the case of non-spin polarized systems). This results from the Pauli principle and we talk about fermionic particles. In the phonon gas, on the other hand, each mode can be excited with an arbitrary number of (indistinguishable) phonons, i.e., the occupation numbers n i (q) are 00