1. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr Ch 5. Phonons Ⅱ Thermal Properties Prof. J. Joo (jjoo@korea.ac.kr) Department of Physics, Korea University http://smartpolymer.korea.ac.kr Solid State Physics

2. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 5.1 Phonon Heat Capacity (1)  Phonon heat capacity ≡ C v (at constant vol.) cf. Electron heat capacity ≡ C p → The contribution of the phonons to the heat capacity is called the lattice heat capacity and C lattice  Total energy of the phonons at τ (≡k B T) → Sum over all phonon modes : Where K : phonon wave vector p : polarization index : thermal equilibrium occupancy of phonons of wave vector K and polarization p ( 열적 평형상태에서 phonon 의 점유도 ) 1.Phonon heat capacity 2.Thermal conductivity cf.) 전자에 의한 표현도 있음 → 6 장 more fundamental

3. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 5.1 Phonon Heat Capacity (2) → is given by the Plank distribution function Note> Phonon ① wavefunction is symmetric ② integer spin → follows the Plank distribution function where D p (ω) : ① the number of modes per unit freq. or ② the number of quantum states per unit energy → “Density of States (DOS)” D(ε)

4. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 5.2 DOS in 1-dimension  Consider the medium as unbounded, but require that the solution be periodic over a large distance L ; u(sa)=u(sa+L), Na=L (N 개의 particles) 2π/L K→K→ -2π/L 4π/L -4π/L 0 2π/L Suppose an arbitrary interval dK in K-space, and look for the # of modes in dK → or k ω positivenegative

5. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 5.3 Density of States in Three Dimensions  Apply periodic b.c. over N 3 primitive cells within a cubic of side L  There is one allowed value of K per volume (2π/L) 3 in K-space  Total number of modes with wavevector less than K  Total energy 계산을 위한 두 가지 modes : (ω 와 K 의 관계 ) ① Debye model : the collective lattice modes as a whole ② Einstein model : the individual atomic vibration

6. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 5.4 Debye Model for Density of States (1)  The collective lattice modes as a whole : represented by group motion 즉 “group velocity”  The thermal energy by phonon (collective modes) is where ω D (Debye cutoff freq. : Debye model 에서 K D 보다 큰 wavevector 의 mode 가 허용안됨 )

7. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 5.4 Debye Model for Density of States (2)  # of mode Assume that phonon velocity is independent of the polarization, so “3” independent modes

8. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 5.5 Debye T 3 Law  At very low temp., 즉 실험값, Ar → 극저온에서 실험값과 일치 → only long wavelength acoustic modes are thermally excited 참고 > Table 1 : Debye temp. for each element p.114 읽을 것 C V of a solidline according to Debye approximation At very low temp., C v ∝ T 3

9. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 5.6 Einstein Model of the D.O.S.  Consider N atoms oscillate independently of each other  N oscillators are the same freq. ω 0 in 1-dim. D(ω)=Nδ(ω-ω 0 )  The thermal energy of the system is  At high temp., 가정 온도에 무관하고 constant. “Dulong-Petit” value k B T>>ħω ; high temp. At high temp., the Einstein model has been a succes (optical phonon part)  At low temp., the Einstein model is not good. → C V ∝ exp[-ħω/τ]  but in experiment, C V ∝ T 3  Einstein model is often used to approximate the optical phonon part of the phonon spectrum.

10. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 5.7 General Result for D(ω) (1)  The number of states per unit frequency range  The number of allowed values of K between ω and ω+dω D(ω)dω = (L/2π) 3 ∫ shell d 3 K K-space 상에서 부피 one allowed value of K in K-space  i.e., “real problem” → obtain ∫ shell d 3 K → let dS ω : an element of area on the surface in K-space of the selected constant ω then ∫ shell d 3 K= ∫ dS ω dK ⊥ dωdω dS ω

11. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 5.7 General Result for D(ω) (2) 특성 1) taken over the area of the surface ω constant, in K space 2) also can use it in electron band theory 3) Van Hove singularities [Fig.14]

12. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 5.8 Thermal Conductivity  정의 : temperature gradient flux of thermal energy thermal conductivity ( 단위시간당 단위면적을 통과하는 thermal energy)  The energy diffuses through the specimen, suffering frequent collision  Thermal conductivity (K) : where C : heat capacity per unit vol. v : average particle velocity l : mean free path between collisions → For phonon, C, v, and l are for phonon Table 2 참고, 온도가 감소하면 l → 증가 ; C → 감소 ; K→ 증가 ( 극저온제외 ≤ ~10K) jUjU ▽T▽T high T low T

13. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr Reference.. 의 증명 ( → 생략가능 ) → 1 차원으로 부터 출발 The flux of particles in x-direction : * concentration of molecules * ave. value of velocity of particles → If C(small) is a heat capacity of a particle, a particle will give up an energy : C∆T when a particle moves from T+∆T to T → ∆T: temp. difference between the ends of a free path of a particle ave. time between collisions → The net flux of energy (particle flux 의 양면성 고려 ) : ∆T lxlx C

14. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 5.9 Thermal Resistivity of Phonon Gas (1)  Phonon mean free path (l) : determined by 2 processes ; ① geometrical scattering ( 예 ) crystal boundary lattice imperfections ② scattering by other phonons → “Anharmonic crystal interactions” : so far, harmonic oscillation → F ∝ U s or P.E. ∝ U s 2 P.E. ∝ U s n, where n>2 → 3-phonon process 1.one phonon causes a periodic elastic strain 2.crystal 내의 elastic constant 를 변조 3.a second phonon 이 이와 같은 elastic constant 의 변조를 받아들여 4.a third phonon 을 산란시킴 → 이와 같은, anharmonic case 에, “l (mean free path)” 는 다른 phonon 들의 coupling 에 의해서 제한 받는다. → : 즉, 온도가 증가함에 따라 충돌할 수 있는 phonon 의 숫자가 많아짐

15. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 5.9 Thermal Resistivity of Phonon Gas (2)  In anharmonic case, we should consider “a 3-phonon collision process”  그러나 Umklapp process 와 같은 3-phonon collision process 는 thermal resistivity 에 영향을 준다. 1 st B.Z. next slide

16. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 5.10 Umklapp Process “ 뒤로 전환 ”  Effective at high temp.  Important for the thermal resistivity by phonon → Exist K 3 inside the 1 st B.Z. K 3 phonon 이 K 1 과 K 2 의 충돌에 의해 발생 K 3 phonon 의 진행 방향은 original phonons K 1 과 K 2 의 “ 반대방향 ” → l 을 축소 thermal resistivity 에 영향 1 st B.Z.

17. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 5.11 Imperfection → Geometrical effects : limiting the mean free path [Fig.18] crystal bonding, isotropic, impurities, lattice imperfections, amorphous structure 참고 > 극저온에서 thermal conductivity 의 감소 ( ≤10K) l → 증가 (comparable with 시료 size) 시료의 size effect : de Haas and Biermasz 효과 #1 Kittel (6th Ed.) 5 장 4 번 Heat Capacity of layer lattice 극저온에서 K 의 급격한 감소 : size effect