Thermodynamics of crystalline states


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Thermodynamics of crystalline states

  1. 1. Chapter 2 Phonons Disregarding point symmetry, we can simplify the crystal structure by the space group [1, 2], representing the thermodynamic state in equilibrium with the sur- roundings at given values of p and T. In this approach, the restoring forces secure stability of the lattice, where the masses at lattice points are in harmonic motion. In this case, we realize that their directional correlations in the lattice are ignored so that a possible disarrangement in the lattice can cause structural instability. In this chapter, we discuss a harmonic lattice to deal with basic excitations in equilibrium structure. Lattice vibrations in periodic structure are in propagation, specified by frequencies and wavevectors in virtually continuous spectra. Quantum mechanically, on the other hand, the corresponding phonons signify the dynamical state in crystals. In strained crystals, as modulated by correlated constituents, low- frequency excitations dominate over the distorted structure, which is however thermally unstable as discussed in this chapter. 2.1 Normal Modes in a Simple Crystal A crystal of chemically identical constituent ions has a rigid periodic structure in equilibrium with the surroundings, which is characterized by translational symmetry. Referring to symmetry axes, physical properties can be attributed to the translational invariance, in consequence of energy and momentum conservations among constituents. Constituents are assumed to be bound together by restoring forces in the lattice structure. Considering a cubic lattice of N3 identical mass particles in a cubic crystal in sufficiently large size, we can solve the classical equation of motion with nearest- neighbor interactions. Although such a problem should be solved quantum mechan- ically, classical solutions provide also a useful approximation. It is noted that the lattice symmetry is unchanged with the nearest-neighbor interactions, assuring M. Fujimoto, Thermodynamics of Crystalline States, DOI 10.1007/978-1-4614-5085-6_2, # Springer Science+Business Media New York 2013 11
  2. 2. structural stability in this approach. In the harmonic approximation, we have linear differential equations, which can be separated into 3N independent equations; this one-dimensional equation describes normal modes of N constituents in collective motion along the symmetry axis x, y, or z [3]. Denoting the displacement by a vector qn from a site n, we write equations of motion for the components qx;n; qy;n and qz;n independently, that is, €qx;n ¼ o2 qx;nþ1 þ qx;nÀ1 À 2qx;n À Á ; €qy;n ¼ o2 qy;nþ1 þ qy;nÀ1 À 2qy;n À Á and €qz;n ¼ o2 qz;nþ1 þ qz;nÀ1 À 2qz;n À Á ; where o2 ¼ k m= and k and m are the mass of a constituent particle and the force constant, respectively. As these equations are identical, we write the following equation for a representative component to name qn for brevity: €qn ¼ o2 qnþ1 þ qnÀ1 À 2qn À Á ; (2.1) which assures lattice stability along any symmetry direction. Defining the conjugate momentum by pn ¼ m _qn, the Hamiltonian of a harmonic lattice can be expressed as H ¼ XN n¼0 p2 n 2m þ mo2 2 qnþ1 À qn À Á2 þ mo2 2 qn À qnÀ1ð Þ2 & ' : (2.2) Each term in the summation represents one-dimensional infinite chain of identi- cal masses m, as illustrated in Fig. 2.1a. Normal coordinates and conjugate momenta, Qk and Pk , are defined with the Fourier expansions qn ¼ 1 ffiffiffiffi N p XkN k¼0 Qk exp iknað Þ and pk ¼ XkN k¼0 Pk exp iknað Þ; (2.3) Fig. 2.1 (a) One- dimensional monatomic chain of the lattice constant a. (b) A dispersion curve o vs: k of the chain lattice. 12 2 Phonons
  3. 3. where a is the lattice constant. For each mode of qn and pn, the amplitudes Qk and Pk are related as QÀk ¼ Qk à ; PÀk ¼ Pk à and XN n¼0 exp i k À k0 ð Þna ¼ Ndkk0 ; (2.4) where dkk0 is Kronecker’s delta, that is, dkk0 ¼ 1 for k ¼ k0 , otherwise zero for k ¼ k0 . Using normal coordinates Qk and Pk, the Hamiltonian can be expressed as H ¼ 1 2m X2p a= k¼0 PkPk à þ QkQk à m2 o2 sin2 ka 2 ' ; (2.5) from which the equation of motion for Qk is written as €Qk ¼ Àm2 o2 Qk; (2.6) where ok ¼ 2o sin ka 2 ¼ 2 ffiffiffiffi k m r sin ka 2 : (2.7) As indicated by (2.7), the k-mode of coupled oscillators is dispersive, which are linearly independent from the other modes of k0 6¼ k . H is composed of N independent harmonic oscillators, each of which is determined by the normal coordinates Qk and conjugate momenta Pk. Applying Born–von Ka´rman’s boundary conditions to the periodic structure, k can take discrete values as given by k ¼ 2pn Na and n ¼ 0; 1; 2; . . . ; N. Figure 2.1b shows the dispersion relation (2.7) determined by the characteristic frequency ok: With initial values of Qkð0Þ and _Qkð0Þ specified at t ¼ 0, the solution of (2.7) can be given by QkðtÞ ¼ Qkð0Þ cos okt þ _Qkð0Þ ok sin okt: Accordingly, qnðtÞ ¼ 1 ffiffiffiffi N p XkN k¼0 X n0¼n;nÆ1 qn0ð0Þ cos ka n À n0 ð Þ À oktf g þ _qn0 ð0Þ ok sin ka n À n0 ð Þ À oktf g ! ; (2.8) where a n À n0 ð Þ represents distances between sites n and n0 so that we write it as x ¼ a n À n0 ð Þ in the following. The crystal is assumed as consisting of a large number of the cubic volume L3 where L ¼ Na, if disregarding surfaces. 2.1 Normal Modes in a Simple Crystal 13
  4. 4. The periodic boundary conditions can then be set as qn¼0ðtÞ ¼ qn¼NðtÞ at an arbitrary time t. At a lattice point x ¼ na between n ¼ 0 and N , (2.8) can be expressed as q x; tð Þ ¼ X k Ak cos Ækx À oktð Þ þ Bk sin Ækx À oktð Þ½ Š; where Ak ¼ qkð0Þ ffiffiffi N p and Bk ¼ _qkð0Þ ok ffiffiffi N p , and x is virtually continuous in the range 0 x L, if L is taken as sufficiently long. Consisting of waves propagating in Æ x directions, we can write q x; tð Þ conveniently in complex exponential form, that is, q x; tð Þ ¼ X k Ck exp i Ækx À okt þ jkð Þ; (2.9) where C2 k ¼ A2 k þ B2 k and tan jk ¼ Bk Ak . For a three-dimensional crystal, these one- dimensional k-modes along the x-axis can be copied to other symmetry axes y and z; accordingly, there are 3N normal modes in total in a cubic crystal. 2.2 Quantized Normal Modes The classical equation of motion of a harmonic crystal is separable to 3N indepen- dent normal propagation modes specified by kn ¼ 2pn aN along the symmetry axes. In quantum theory, the normal coordinate Qk and conjugate momentum Pk ¼ Àih @ @Qk are operators, where h ¼ h 2p and h is the Planck constant. For these normal and conjugate variables, there are commutation relations: Qk; Qk0½ Š ¼ 0; Pk; Pk0½ Š ¼ 0 and Pk; Qk0½ Š ¼ ihdkk0 ; (2.10) and the Hamiltonian operator is Hk ¼ 1 2m PkP y k þ m2 o2 kQkQ y k : (2.11a) Here, P y k and Q y k express transposed matrix operators of the complex conjugates Pk à and Qk à , respectively. Denoting the eigenvalues of Hk by ek, we have the equation HkCk ¼ ekCk: (2.11b) For real eigenvalues ek , Pk and Qk should be Hermitian operators, which are characterized by the relations P y k ¼ PÀk and Q y k ¼ QÀk , respectively. Defining operators 14 2 Phonons
  5. 5. bk ¼ mokQk þ iP y k ffiffiffiffiffiffiffiffiffiffi 2mek p and b y k ¼ mokQ y k À iPk ffiffiffiffiffiffiffiffiffiffi 2mek p ; (2.12) we can write the relation bkb y k ¼ 1 2mek m2 o2 kQ y k Qk þ P y k Pk þ iok 2ek Q y k P y k À PkQk ¼ Hk ek þ iok 2ek QÀkPÀk À PkQkð Þ: From this relation, we can be derive Hk ¼ hok b y k bk þ 1 2 ; if ek ¼ 1 2 hok: (2.13) Therefore, Hk are commutable with the operator b y k bk, that is, Hk; b y k bk h i ¼ 0; and from (2.12) bk0 ; b y k h i ¼ dk0k; bk0 ; bk½ Š ¼ 0 and b y k0 ; b y k h i ¼ 0: Accordingly, we obtain Hk; b y k h i ¼ hokb y k and bk; Hk½ Š ¼ hokbk: Combining with (2.11b), we can derive the relations Hk b y k Ck ¼ ek þ hokð Þ b y k Ck and Hk bkCkð Þ ¼ ek À hokð Þ bkCkð Þ; indicating that b y k Ck and bkCk are eigenfunctions for the energies ek þ hok and ek À hok , respectively. In this context, b y k and bk are referred to as creation and annihilation operators for the energy quantum hok to add and subtract in the energy ek; hence, we can write b y k bk ¼ 1: (2.14) Applying the creation operator b y k on the ground state function Cknk-times, the eigenvalue of the wavefunction b y k nk Ck can be given by nk þ 1 2 À Á hok, generating a 2.2 Quantized Normal Modes 15
  6. 6. state of nkquanta plus 1 2 hok. Considering a quantum hok like a particle, called a phonon, such an exited state with nk identical phonons is multiply degenerate by permutation nk! Hence, the normalized wavefunction of nk phonons can be expressed by 1ffiffiffiffi nk! p b y k nk Ck. The total lattice energy in an excited state of n1; n2; ::::: phonons in the normal modes 1, 2,. . ... can be expressed by U n1; n2; :::::ð Þ ¼ Uo þ X k nkhok; (2.15a) where Uo ¼ P k hok 2 is the total zero-point energy. The corresponding wavefunction can be written as C n1; n2; :::::ð Þ ¼ b y 1 n1 b y 2 n2 ::::: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1! n2! ::::: p C1C2:::::ð Þ; (2.15b) which describes a state of n1; n2; :::: phonons of energies n1hok1 ; n2hok2 ; :::::. The total number N ¼n1 þ n2 þ ::::: cannot be evaluated by the dynamical theory; however, we can determine the value in thermodynamics, as related to the level of thermal excitation at a given temperature. 2.3 Phonon Field and Momentum In a one-dimensional chain of identical mass particles, the displacement mode qk is independent from each other’s modes, and hence representing normal modes in a three-dimensional crystal. However, this model is only approximate, in that these normal modes arise from the one-dimension harmonic chain model, where mutual interactions between different normal modes are prohibited. For propagation in arbitrary direction, the vibrating field offers more appropriate approach than the normal modes, where quantized phonons move in any direction like free particles in the field space. Setting rectangular coordinates x; y; z along the symmetry axes of an orthorhom- bic crystal in classical theory, the lattice vibrations are described by a set of equations px;n1 2 2m þ k 2 qx;n1 À qx;n1þ1 À Á2 þ qx;n1 À qx;n1À1 À Á2 n o ¼ ex;n1 ; py;n2 2 2m þ k 2 qy;n2 À qy;n2þ1 À Á2 þ qy;n2 À qy;n2À1 À Á2 n o ¼ ey;n2 ; and pz;n3 2 2m þ k 2 qz;n3 À qz;n3þ1 À Á2 þ qz;n3 À qz;n3À1 À Á2 n o ¼ ez;n3 ; (2.16) 16 2 Phonons
  7. 7. whereex;n1 þ ey;n2 þ ez;n3 ¼ en1n2n3 is the total propagation energy along the direction specified by the vector q n1; n2; n3ð Þ and k is the force constant. The variables qx;n1 ; qy;n2 ; qz;n3 in (2.16) are components of a classical vector q n1; n2; n3ð Þ , which can be interpreted quantum theoretically as probability amplitudes of components of the vector q in the vibration field. We can therefore write the wavefunction of the displacement field as C n1; n2; n3ð Þ ¼ qx;n1 qy;n2 qz;n3 , for which these classical components are written as q x; tð Þ ¼ X kx Ck;x exp i Ækxx À okx t þ jkx À Á ; q y; tð Þ ¼ X ky Cky exp i Ækyy À oky t þ jky ; q z; tð Þ ¼ X kz Cky exp i Ækzz À okz t þ jkz À Á ; and hence, we have C n1; n2; n3ð Þ ¼ X k Ak exp i Æk:r À n1ex;n1 þ n2ey;n2 þ n3ez;n3 h t þ jk : Here, Ak ¼ Ckx Cky Ckz ; jk ¼ jkx þ jky þ jkz , and k ¼ kx; ky; kz À Á are the ampli- tude, phase constant, and wavevector of C n1; n2; n3ð Þ , respectively. Further writing n1ex;n1 þ n2ey;n2 þ n3ez;n3 h ¼ ok n1; n2; n3ð Þ ¼ ok; (2.17a) the field propagating along the direction of a vector k can be expressed as C k; okð Þ ¼ Ak exp i Æk:r À okt þ jkð Þ; (2.17b) representing a phonon of energy hok and momentum Æ hk. For a small kj j, the propagation in a cubic lattice can be characterized by a constant speed v of propaga- tion determined by ok ¼ v kj j, indicating no dispersion in this approximation. The phonon propagation can be described by the vector k, composing a recipro- cal lattice space, as illustrated in two dimensions in Fig. 2.2 by kx ¼ 2pn1 L ; ky ¼ 2pn2 L and kz ¼ 2pn3 L ; 2.3 Phonon Field and Momentum 17
  8. 8. in a cubic crystal, where aN ¼ L. A set of integers n1; n2; n3ð Þ determines an energy and momentum of a phonon propagating in the direction of k, where ok ¼ ok n1; n2; n3ð Þ and kj j ¼ 2p L ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 x þ n2 y þ n2 z q . In the reciprocal space, all points on a spherical surface of radius kj jcorrespond to the same energy hok , representing a sphere of a constant radius kj j and energy ek. Quantum mechanically, we can write the phonon momentum as p ¼ hk to supplement the energy hok, characterizing a phonon particle. 2.4 Thermal Equilibrium In thermodynamics, a crystal must always be in thermal contact with the surround- ings. At constant external pressure p, the quantized vibration field can be in equilibrium with the surroundings at a given temperature T. A large number of phonons are in collision-free motion, traveling in all directions through the lattice, colliding with surfaces to exchange their energy and momentum with the surround- ings. Assuming the crystal volume as unchanged, the average of phonon energies can be calculated with the Boltzmann probability at T. In equilibrium, the total energy of a crystal can be expressed asU þ Us, whereUs is the contribution from the heat reservoir and Urepresents the energy of a stable crystal. In this case, the total energy U þ Us should be stationary with any thermodynamic variation around equilibrium. Using probabilities w and ws for keeping the crystal in equilibrium with the surroundings, the product wws should be calculated as maximizing U þ Us to determine the most probable value. Setting this variation problem as 2πny ky = a 2πnx kx = a dk k 4 3 2 1 0 1 2 3 4 Fig. 2.2 Two-dimensional reciprocal lattice. A lattice point is indicated by kx; ky À Á . Two quarter-circles of radii k and k + dk show surfaces of constant ek and ekþdk for small kj j in kxky plane. 18 2 Phonons
  9. 9. d wwsð Þ ¼ 0 and d U þ Usð Þ ¼ 0 for arbitrary variations dw and dws, these variations can be calculated as wsdw þ wdws ¼ 0 and dU þ dUs ¼ 0; respectively. We can therefore write d ln wð Þ dU ¼ d ln wsð Þ dUs ; which is a common quantity between U and Us. Writing it as equal to b ¼ 1 kBT , we can relate b to the conventional absolute temperature T. Therefore, d ln wð Þ dU ¼ b ¼ 1 kBT or w ¼ wo exp À U kBT : (2.18) Here, w is called the Boltzmann probability; wo is the integration constant that can be determined by assuming U ¼ 0 at T ¼ 0 K, where kB is the Boltzmann constant. Quantum theoretically, however, T ¼ 0 is fundamentally unreachable, as stated in the third law of thermodynamics. Accordingly, we write U ¼ 0 þ Uo at 0 K, where Uo ¼ 1 2 Nhoo is the zero-point energy. Although we considered only vibrations so far, physical properties of a crystal are also contributed by other variables located at lattice points or at interstitial sites. Though primarily independent of lattice vibrations, these variables can interact with the lattice via phonon scatterings. If accessed by random collisions of phonons, energiesei of these variables are statistically available with Boltzmann’s probabilities wi, so we can write equations U ¼ X i ei and w ¼ Piwi; where wi ¼ wo exp À ei kBT and X i wi ¼ 1; (2.19) as these wi are for exclusive events. In this case, the function Z ¼ P i exp À ei kBT , called the partition function, is useful for statistical calculation. In such a system as called microcanonical ensemble, thermal properties can be calculated directly with Z. The Boltzmann statistics is a valid assumption for a dynamical system under the ergodic hypothesis. Despite of the absence of rigorous proof, the Boltzmann statis- tics can usually be applied to phonon gas in a fixed volume. Thermodynamically, 2.4 Thermal Equilibrium 19
  10. 10. however, it is only valid for isothermal processes, because the volume is not always constant in adiabatic processes of internal origin. The anharmonic lattice cannot be ergodic in strict sense, whereas the harmonicity is essentially required for stable crystals at constant volume and pressure. 2.5 Specific Heat of a Monatomic Crystal The specific heat at a constant volume CV ¼ @U @T À Á V is a quantity measurable with varying temperature under a constant external pressure p. The phonon theory is adequate for simple monatomic crystals, if characterized with no structural changes. For such a crystal, the specific heat and internal energy are given by quantized phonon energies ek ¼ nk þ 1 2 À Á hok , for which the wavevector k is distributed virtually in all directions in the reciprocal space. Assuming 3N phonons in total, the energies ek are degenerate with the density of k-states written as gðkÞ, which is a large number as estimated from a spherical volume of radius kj j in the reciprocal space. In this case, the partition function can be expressed as Zk ¼ gðkÞ exp À ek kBT ¼ gðkÞ exp À hok 2kBT X1 nk¼0 exp À nkhok kBT ; where the infinite series on the right converges, if hok kBT 1. In fact, this condition is satisfied at any practical temperature T lower than the melting point so that Zk is expressed as Zk ¼ gðkÞ exp À hok 2kBT 1 À exp À hok kBT : The total partition function is given by the product Z ¼ PkZk , so that ln Z ¼P k ln Zk; the free energy can therefore be calculated as the sum of ln Zk, namely, F ¼ kBT P k ln Zk, where ln Zk ¼ À hok 2 þ kBT ln gðkÞ À kBT ln 1 À exp À hok kBT ' : By definition, we have the relation F ¼ U À TS ¼ U þ T @F @T À Á V from which we can derive the formula U ¼ ÀT2 @ @T F T À Á . Using the above ln Zk, we can show that the internal energy is given by U ¼ Uo þ X k hok exp hok kBT À 1 and Uo ¼ 1 2 X k hok: (2.20) 20 2 Phonons
  11. 11. The specific heat at constant volume can then be expressed as CV ¼ @U @T V ¼ kB X k hok kBT 2 exp hok kBT exp hok kBT À 1 2 : (2.21) To calculate CV with (2.21), we need to evaluate the summation with the number of phonon states on energy surface ek ¼ h ok þ 1 2 À Á in the reciprocal space. In anisotropic crystals, such a surface is not spherical, but a closed surface, as shown in Fig. 2.3a. In this case, the summation in (2.21) can be replaced by a volume integral over the closed surface, whose volume element is written as d3 k ¼ dk:dS ¼ dk? dSj j. Here, dk?is the component of k perpendicular to the surface element dSj j ¼ dS, as illustrated two-dimensionally in Fig. 2.3b. We can write dok ¼ dek h ¼ 1 h gradke kð Þj jdk?; where 1 h gradke kð Þj j ¼ vg represents the group velocity for propagation, and hence dk? ¼ dok vg . Using these notations, we can reexpress (2.21) as CV ¼ kB ð ok hok kBT 2 exp hok kBT exp hok kBT À 1 2 D okð Þdok; (2.22a) where a b n kx dS dS 0 kz kx ky ky Fig. 2.3 (a) A typical constant-energy surface in three-dimensional reciprocal space, wheredSis a differential area on the surface. (b) The two-dimensional view in the kxky-plane. 2.5 Specific Heat of a Monatomic Crystal 21
  12. 12. D okð Þ ¼ L 2p 3 þ S dS vg (2.22b) is the density of phonon states on the surface S. Tedious numerical calculations performed in early studies on representative crystals resulted in such curves as shown in Fig. 2.4a, for example, of a diamond crystal. However, in relation with dispersive longitudinal and transversal modes, the analysis was extremely difficult to obtain satisfactory comparison with experi- mental results. On the other hand, Einstein and Debye simplified the functionD okð Þ independently, although somewhat oversimplified for practice crystals. Neverthe- less, their models are proven to be adequate in many applications to obtain useful formula for Uand CV for simple crystals [4]. 2.6 Approximate Models 2.6.1 Einstein’s Model At elevated temperatures T, we can assume that thermal properties of a crystal are dominated by n phonons of energy hoo. Einstein proposed that the dominant mode at a high temperature is of a single frequency oo, disregarding all other modes in the vibration spectrum. In this model, using the expression (2.22b) simplified as D ooð Þ ¼ 1, we can express the specific heat (2.22a) and the internal energy as 6 ω a b 4 2 0 01 2 2 4 6 8 ωD ω 2π D (ω) trans. trans. long. long. 3 2π k Fig. 2.4 (a) Examples of practical dispersion curves. Longitudinal and transverse dispersions are shown by solid and broken curves, respectively. (b) The solid curve shows an example of an observed density function, being compared with the broken curve of Debye model. 22 2 Phonons
  13. 13. CV ¼ 3NkB x2 exp x exp x À 1ð Þ2 and U ¼ 3NkB 1 2 x þ x exp x À 1 ; (2.23) respectively, where x ¼ YE T ; the parameter YE ¼ hoo kB is known as the Einstein temperature. It is noted that in the limit x ! 0, we obtain CV ! 3NkB. At high temperatures, U can be attributed to constituent masses, vibrating independently in degrees of freedom 2; hence, the corresponding thermal energy is 2 Â 1 2 kBT, and U ¼ 3NkBT and CV ¼ 3NkB: (2.24) This is known as the Dulong–Petit law, which is consistent with Einstein’s model in the limit of T ! 1. 2.6.2 Debye’s Model At lower temperatures, longitudinal vibrations at low frequencies are dominant modes, which are characterized approximately by a nondispersive relation o ¼ vgk. The speed vg is assumed as constant on a nearly spherical surface for constant energy in the k-space. Letting vg ¼ v, for brevity, (2.22b) can be expressed as D oð Þ ¼ L 2p 3 4po2 v3 : (2.25) Debye assumed that with increasing frequency, the density D oð Þ should be terminated at a frequency o ¼ oD, called Debye cutoff frequency, as shown by the broken curve in Fig. 2.4b. In this case, the density function D oð Þ / o2 can be normalized as RoD 0 D oð Þdo ¼ 3N, so that (2.25) can be replaced by D oð Þ ¼ 9N o3 D o2 : (2.26) Therefore, in the Debye model, we have U ¼ 3NkBT ðoD 0 ho 2 þ ho exp ho kBT À 1 0 @ 1 A 3o2 do o3 D and 2.6 Approximate Models 23
  14. 14. CV ¼ 3NkB ðoD 0 exp ho kBT exp ho kBT À 1 2 ho kBT 2 3o2 do o3 D : Defining Debye temperature hoD kBT ¼ YD and ho kBT ¼ x, similar to Einstein’s model, these expressions can be simplified as U ¼ 9 8 NkBYD þ 9NkBT T YD 3 ð YD T 0 x3 exp x À 1 dx and CV ¼ 9NkB T YD 3 ð YD T 0 x4 exp x exp x À 1ð Þ2 dx: Introducing the function defined by Z YD T ¼ 3 T YD 3 ð YD T 0 x3 dx exp x À 1 ; (2.27) known as the Debye function, the expression CV ¼ 3NkBZ YD T À Á describes temperature-dependent CV for TYD. In the limit of YD T ! 1, however, these U and CV are dominated by the integral ð1 0 x3 dx exp x À 1 ¼ p4 15 ; and hence the formula U ¼ 9 8 NkBYD þ 9NkBT T YD 3 p4 15 and CV ¼ 9NkB T YD 3 p4 15 : (2.28) 24 2 Phonons
  15. 15. can be used at lower temperatures than YD. In the Debye model, we have thus the approximate relation CV / T3 for TYD, which is known as Debye T3 -law. Figure 2.5 shows a comparison of observed values of CV from representative monatomic crystals with the Debye and Dulong–Petit laws, valid at low and high temperatures, respectively, showing reasonable agreements. 2.7 Phonon Statistics Part 1 Quantizing the lattice vibration field, we consider a gas of phonons hok; hkð Þ. A large number of phonons exist in excited lattice states, behaving like classical particles. On the other hand, phonons are correlated at high densities, owing to their quantum nature of unidentifiable particles. Although dynamically unspecified, the total number of phonons is thermodynamically determined by the surface boundaries at T, where phonon energies are exchanged with heat from the sur- roundings. In equilibrium, the number of photons on each k-state can be either one of n ¼ 1; 2; . . . ; 3N. Therefore, the Gibbs function can be expressed by G p; T; nð Þ, but the entropy fluctuates with varying n in the crystal. Such fluctuations can be described in terms of a thermodynamic probability g p; T; nð Þ, so that we consider that two phonon states, 1 and 2, can be characterized by probabilities g p; T; n1ð Þ and g p; T; n2ð Þ in an exclusive event, in contrast to the Boltzmann statistics for indepen- dent particles. At constant p, the equilibrium between the crystal and reservoir can therefore be specified by minimizing the total probability g p; T; nð Þ ¼ g p; T; n1ð Þ þ g p; T; n2ð Þ, considering such binary correlations dominant under n ¼ n1 þ n2 ¼ constant, leaving all other niði 6¼ 1; 2Þ as unchanged. Applying the variation principle for small arbitrary variations dn1 ¼ Àdn2, we can minimize g p; T; nð Þ to obtain Cv T T3 -law 3R 0.6 0.4 0.2 0 .2 .4 .6 .8 1.0 1.2 Al Cu Ag Pb 95K 215K 309K 396K Dulong-Petit ΘD Fig. 2.5 Observed specific heat CV 3R= against T YD= for representative metals. R is the molar gas constant. In the bottom-right corner, values of D for these metals are shown. The T3 -law and Dulong–Petit limits are indicated to compare with experimental results. 2.7 Phonon Statistics Part 1 25
  16. 16. dgð Þp;T ¼ @g1 @n1 p;T dn1 þ @g2 @n2 p;T dn2 ¼ 0; from which we derive the relation @g1 @n1 p;T ¼ @g2 @n2 p;T : This is a common quantity between g1 and g2, which is known as the chemical potential. Therefore, we have equal chemical potentials m1 ¼ m2 in equilibrium against phonon exchange. Writing the common potential as m, a variation of the Gibbs potential G for an open system at equilibrium can be expressed for an arbitrary variation d n as dG ¼ dU À TdS þ pdV À mdn; (2.29) where dn represents a macroscopic variation in the number of phonons n. Consider a simple crystal, whose two thermodynamic states are specified by the internal energy and phonon number, Uo; Noð Þ and Uo À e; No À nð Þ , which are signified by probabilities go and g, as related to their entropies S Uo; Noð Þ and S Uo À e; No À nð Þ, respectively. Writing the corresponding Boltzmann relations, we have go ¼ exp S Uo; Noð Þ kB and g ¼ exp S Uo À e; No À nð Þ kB : Hence, g go ¼ exp S Uo À e; No À nð Þ=kBf g exp S Uo; Noð Þ=kBf g ¼ exp DS kB ; where DS ¼ S Uo À e; No À nð Þ À S Uo; Noð Þ ¼ À @S @Uo No e À @S @No Uo n: Using (2.29), we obtain the relations @S @Uo No ¼ 1 T and @S @No Uo ¼ À m T so that 26 2 Phonons
  17. 17. g ¼ go exp mn À e kBT : (2.30) For phonons, the energy e is determined by any wavevector k, where kj j ¼ 1; 2; ::::; 3N, and N can take any integral number. The expression (2.30) is the Gibbs factor, whereas for classical particles, we use the Boltzmann factor instead. These factors are essential in statistics for open and closed systems, respectively. For phonons, it is convenient to use the notation l ¼ exp m kBT , with which (2.30) can be written as g ¼ goln exp À e kBT . The factor l here implies a probability for the energy level e to accommodate one phonon adiabatically [5], whereas the conven- tional Boltzmann factor exp À e kBT is an isothermal probability of e at T. Origi- nally, the chemical potential m was defined for an adiabatic equilibrium with an external chemical agent; however, for phonons l is temperature dependent as defined by l ¼ exp m kBT . Here, the chemical potential is determined as m ¼ À @G @n À Á p;T from (2.29), which is clearly related with the internal energy due to phonon correlations in a crystal. For phonon statistics, the energy levels are en ¼ nho, and the Gibbs factor is determined by e ¼ ho and n. The partition function can therefore be expressed as ZN ¼ XN n¼0 ln exp À ne kBT ¼ XN n¼0 l exp À e kBT 'n : Consideringl exp À e kBT 1, the sum ofthe infinite series evaluated for N ! 1is Z ¼ 1 1 À l exp À e kBT : With this so-called grand partition function, the average number of phonons can be expressed as nh i ¼ l @ ln Z @l ¼ 1 1 l exp À e kBT À 1 ¼ 1 exp e À m kBT À 1 : (2.31) This is known as the Bose–Einstein distribution. It is noted that the energy e is basically dependent on temperature, whereas the chemical potential is small and temperature independent. Further, at elevated temperatures, we consider that for e m; (2.31) is approximated as nh i % exp mÀe kBT % exp Àe kBT , which is the Boltzmann factor. However, there should be a critical temperature Tc for nh i ¼ 1 to be determined by e Tcð Þ ¼ m, which may be considered for phonon condensation. 2.7 Phonon Statistics Part 1 27
  18. 18. So far, phonon gas was specifically discussed, but the Bose–Einstein statistics (2.31) can be applied to all other identical particles characterized by even parity; particles obeying the Bose–Einstein statistics are generally characterized by even parity and called Bosons. Particles with odd parity will be discussed in Chap. 11 for electrons. 2.8 Compressibility of a Crystal In the foregoing, we discussed a crystal under a constant volume condition. On the other hand, under constant temperature, the Helmholtz free energy can vary with a volume change DV, if the crystal is compressed by DF ¼ @F @V T DV; where p ¼ À @F @V À Á T is the pressure on the phonon gas in a crystal. At a given temperature, such a change DFmust be offset by the external work À pDV by applying a pressure p, which is adiabatic to the crystal. It is realized that volume-dependent energies need to be included in the free energy of a crystal in order to deal with the pressure from outside. Considering an additional energy Uo ¼ UoðVÞ, the free energy can be expressed by F ¼ Uo þ 9NkBT T YD 3 ð YD T 0 x 2 þ ln 1 À exp Àxð Þf g ! x2 dx ¼ Uo þ 3NkBTZ YD T ; (2.32) where Z YD T À Á is the Debye function defined in (2.27), for which we have the relation @Z @ ln YD T ¼ À @Z @ ln T V ¼ À T YD @Z @T : (2.33) Writing z ¼ z T; Vð Þ ¼ TZ YD T À Á for convenience, we obtain @z @V T ¼ À g V @z @ ln YD T ¼ gT V @Z @ ln T V ; where the factor 28 2 Phonons
  19. 19. g ¼ À d ln Y d ln V is known as Gru¨neisen’s constant. Using (2.30), the above relation can be reexpressed as @z @V T ¼ g V T @z @T V À z ' : From (2.29), we have NkBz T; Vð Þ ¼ F À Uo; therefore, this can be written as @ F À Uoð Þ @V ' T ¼ g V T @ F À Uoð Þ @T V À F þ Uo ' : (2.34) Noting Uo ¼ UoðVÞ, the derivative in the first term of the right side is equal to T @F @T À Á V ¼ ÀTS; hence, the quantity in the curly brackets is À U þ Uo ¼ Uvib that represents the energy of lattice vibration. From (2.31), we can derive the expression for pressure in a crystal, that is, p ¼ À dUo dV þ gUvib V ; (2.35) which is known as Mie–Gru¨neisen’s equation of state. The compressibility is defined as k ¼ À 1 V @V @p T ; (2.36) which can be obtained for a crystal by using (2.32). Writing (2.32) as pV ¼ ÀV dUo dV þgUvib and differentiating it, we can derive p þ V @p @V T ¼ À dUo dV À V d2 Uo dV2 þ g @Uvib @V T : Since the atmospheric pressure is negligible compared with those in a crystal, we may omit p, and also from (2.31) @Uvib @V T ¼ g V T @Uvib @T V À Uvib ' in the above expression. Thus, the compressibility can be obtained from 2.8 Compressibility of a Crystal 29
  20. 20. 1 k ¼ ÀV @p @V T ¼ dUo dV þ V d2 Uo dV2 À g2 V TCV À Uoð Þ; (2.37) where CV ¼ @Uvib @T À Á V is the specific heat of lattice vibrations. If p ¼ 0, the volume of a crystal is constant, that is, V ¼ Vo and dUo dV ¼ 0, besides Uvib ¼ const: of V. Therefore, we can write 1 ko ¼ Vo d2 Uo dV2 V¼Vo , meaning a hypothetical compressibility ko in equilibrium at p ¼ 0. Then with (2.34) the volume expansion can be defined as b ¼ V À Vo Vo ¼ kogUvib V : (2.38) Further, using (2.32) @p @T V ¼ g V @Uvib @T V ¼ gCV V ; which can also be written as @p @T V ¼ 1 V @V @T p À 1 V @V @p T ; and hence we have the relation among g; k, and b, that is, g ¼ À V CV b k . Such constants as Y; k; b, and g are related with each other and are significant parameters to characterize the nature of crystals. Table 2.1 shows measured values of YDby thermal and elastic experiments on some representative monatomic crystals. Exercise 2 1. It is important that the number of phonons in crystals can be left as arbitrary, which is thermodynamically significant for Boson particles. Sound wave propa- gation at low values of k and o can be interpreted for transporting phonons, which is a typical example of low-level excitations, regardless of temperature. Discuss why undetermined number of particles is significant in Boson statistics. Can there be any other Boson systems where the number of particles if a fixed constant? Table 2.1 Debye temperatures YD determined by thermal and elastic experimentsa Fe Al Cu Pb Ag Thermal 453 398 315 88 215 Elasticb 461 402 332 73 214 a Data: from Ref. [3] b Calculated with elastic data at room temperature 30 2 Phonons
  21. 21. 2. Einstein’s model for the specific heat is consistent with assuming crystals as a uniform medium. Is it a valid assumption that elastic properties can be attributed to each unit cell? What about a case of nonuniform crystal? At sufficiently high temperatures, a crystal can be considered as uniform. Why? Discuss the validity of Einstein’s model at high temperatures. 3. Compare the average number of phonons nh i calculated from (2.26) with that expressed by (2.31). Notice the difference between them depends on the chemi- cal potential: either m ¼ 0 or m 6¼ 0: Discuss the role of a chemical potential in making these two cases different. 4. The wavefunction of a phonon is expressed by (2.17b). Therefore in a system of many phonons, phonon wavefunctions should be substantially overlapped in the crystal space. This is the fundamental reason why phonons are unidentifiable particles; hence, the phonon system in crystals can be regarded as condensed liquid. For Boson particles 4 He, discuss if helium-4 gas can be condensed to a liquid phase at 4.2K. 5. Are the hydrostatic pressure p and compressibility discussed in Sect. 2.8 ade- quate for anisotropic crystals? Comment on these thermodynamic theories applied to anisotropic crystals. References 1. M. Tinkham, Group Theory and Quantum Mechanics (McGraw-Hill, New York, 1964) 2. R.S. Knox, A. Gold, Symmetry in the Solid State (Benjamin, New York, 1964) 3. C. Kittel, Quantum Theory of Solids, (John Wiley, New York, 1963) 4. C. Kittel, Introduction to Solid State Physics, 6th edn. (Wiley, New York, 1986) 5. C. Kittel, H. Kroemer, Thermal Physics (Freeman, San Francisco, 1980) References 31