1. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr Ch4. Phonons Ⅰ Crystal Vibrations 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 4.1 Vibrations of Crystals with Monatomic Basis (1) → Primitive cell 내의 하나의 원자 ( 한 종류의 원자 ; monatomic) 로 된 crystal 의 “elastic vibration” 을 고려하자 왜 ? 외부로부터 wave ( 예 : 전자기파 ) 나 원자들을 진동시켜 주는 힘 (F) 가 가해지면 → “vibration” 측정 시료 ( 즉 Matter) 의 물리적 성질에 따라 “ 다양한 vibration” → Phonon 이란 ? Energy of a lattice vibration is quantized. The quantum of energy is “Phonon” → Wave 진행 방향에 대해서 원자들이 이루는 평면이.. 수평적 변위 longitudinal (wave) displacement 수직적 변위 transverse (wave) displacement one mode two mode & “ 동일 원자로 구성 ”

3. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 4.1 Vibrations of Crystals with Monatomic Basis (2) → 가정 : crystal 의 elastic 한 vibration response 가 가해준 힘에 linear 한 경우 ( 예 : F = kx ; Hooke’s law ) F s ∝ u s or E ∝ u s 2 ( 힘 ) ( 변위 ) [ 참고 ] Energy 가 u s 3 을 포함하는 경우 “high temp.” → Consider, C : 용수철 상수 ( 동일 ∵ monatomic ) M : 원자 질량 ( 동일 ∵ monatomic ) ※ 단 longitudinal 경우와 transverse 의 경우는 C 가 다름 ∴ S 평면상의 힘 F s = C(u s+1 -u s ) + C(u s-1 -u s ) + … + C(u s+p -u s ) +… = ∑ p=±1 C(u s+p -u s ) M usus u s+1 u s+p s s+1 s+p

4. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 4.1 Vibrations of Crystals with Monatomic Basis (3) → 단지 nearest-neighbor 상호작용만 dominant 한 경우를 고려, 즉 p=±1 ∴ F s = C(u s+1 -u s ) + C(u s-1 -u s ) Note : u s → 주기적 time dependent → exp(-iωt) 즉, u s = u s0 exp(-iωt) ∴ -Mω 2 u s = C(u s+1 +u s-1 -2u s )… ① → Let the general solution for u s as u s = u 0 exp(iksa) exp(-iωt)… 추측 : x = x 0 sin(kx- ωt) ②... and u s+p = u 0 exp{(ik(s+p)a)} exp(-iωt) lattice const.

5. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 4.1 Vibrations of Crystals with Monatomic Basis (4) → ②식을 ①식에 대입해서 풀면 ★ At the boundary of the 1 st B.Z. (k=±π/a) ★ The plot of ω vs. k ω k π/a -π/a Longitudinal Transverse ; C 가 다름 slope is zero Special signification of phonon wavevectors lies on the zone boundary

6. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 4.2 First Brillouin Zone  질문 : 원자의 진동으로 생기는 ‘elastic wave’ 가 물리적으로 중요한 범위 → “only the 1 st B.Z.” 왜 ? → 1 과 2 점의 physical property 는 동일 지역 ( 즉, 1 st B.Z.) 의 물리적 성질이 측정 시료의 성적을 대표 π/a ω k -π/a 2π/a -2π/a 0 1 2 “Extended zone”

7. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 4.3 Group Velocity → 정의 : elastic wave packet 의 전달속도 → “group velocity” 예 ) the velocity of energy propagation in the medium no propagation!! → the wave is standing wave → zero net transmission velocity (note) diffraction condition π/a k 0 vgvg

8. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 4.4 Long Wavelength Limit → “continium theory of elastic wave” π/a k 0 ω λ>>a λ=2a

9. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 4.5 Two Atoms per Primitive Basis (1)  Looking for phonon dispersion relation in crystals with 2 different atoms / primitive basis For each polarization mode, the dispersion relation (ω vs. k) “acoustic branch” (LA, TA) and “optical branch” (LO, TO)  Consider a cubic crystal with 2 atoms  Consider the interaction between the n, n, atoms basis associated with primitive cell Longitudinal or transverse mode u s-1 usus u s+1 M1 M1 M2 M2 v s-1 vsvs v s+1 … … a : lattice const. u s-1 usus u s+1 M1 M1 M2 M2 v s-1 vsvs v s+1 … … C

10. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 4.5 Two Atoms per Primitive Basis (2)  Looking for a solution in the form of a traveling wave ; 운동방정식에 대입 → eq. ① → eq. ② eq. ① … eq. ② …

11. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 4.5 Two Atoms per Primitive Basis (3) → 즉 ω 에 대한 4 차 방정식 쉽게 계산하기 위해 2 가지 극한적 경계에서 고려, 즉, ka<<1 ( 원점 근처에서 ) and ka=±π (zone boundary 에서 ) ① ② 즉, ka → 0 상수

12. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 4.5 Two Atoms per Primitive Basis (4)  Transverse mode 와 합쳐서 생각하면 (Fig.8a) Ge at 80K  For transverse optical branch, → → the atoms vibrate each other, but their C.O.M. is fixed → we may excite a motion of this type with el. field of a light wave → the branch is called “the optical branch”  For transverse acoustical branch, at a small k, u=v → the atoms (and their C.O.M.) move together, as in “the long wavelength acoustic vibrations” → “acoustic branch” 대입 (eq. ① and eq. ② ) TO & TA in a diatomic linear lattice

13. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 4.6 Quantization of Elastic Waves  The energy of a lattice vibration is “quantized”; the quantum of lattice vibration energy is called a “phonon” (similar to the photon of the EM wave) [Note] Phonon 의 실험적 측정 방법 1.Neutron scattering : can map entire B.Z., but poor resolution (needs a special places such as Nat’l Lab.) 2. Raman scattering : (polariton scattering) high resolution and for optical mode 3. Brillouin scattering : (photon scattering) very high resolution and for acoustic mode

14. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 4.7 Phonon Momentum  A phonon of wave vector K Interacts with photons, neutrons, and electrons  For most practical purposes, a phonon acts as if its momentum wave crystal momentum [Note] For a x-ray photon by a crystal (photon elastic scattering) momentum conservation reciprocal lattice vector incident photon wave vector scattered photon wave vector  If the scattering of the photon is inelastic ① with the creation of a phonon ( K ), then the wave vector selection rule is ② if a phonon K is absorbed in the process, ∴ For an inelastic scattering, ① ②

15. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 4.8 Inelastic Scattering by Phonons → the energy conservation is 입사되는 전자나 neutron 의 에너지 the energy of the phonon created (+), or absorbed (-) Kittel Ch.4 #1 and #5 due : one week later