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Magnon Another Carrier of Thermal Conductivity Final Presentation for ME 381R Nov. 30 2004 Park, Keeseong Ha, Eun Contents • Review of Thermal Conductivity -Insulator -Metal • Unusual Data for Thermal Conductivity • Magnon - Definition - Thermal Conductivity from Magnon • More Data for Magnon’s Thermal Conductivity • Summary General Behavior of Thermal Conductivity (Insulator) 1 κ Cvl 3 v is const. assumed from Debye model. At high T (D) .. Interactions among phonons are dominant - l T-1 ; C = const. => T-1 At intermediate T .. - l exp(T*/T); C decreases as T goes down and T3 where T*= a fraction of the Debye Temp. D => exp(T*/T) At low T ( << D) .. - l = const. (depending on the shape and size of the specimen) R. Berman, Thermal Conduction in Solids, 1976. Chap 3. C T3 => T3 General Behavior of Thermal Conductivity (Metal) • Wiedemann-Franz-Lorenz law κ L T where L 2 . 4 5 1 0- 8 (W / deg 2 ) σ At high T ( TF ) .. Interactions among phonons are dominant - l T-1 ; C T. => = const At intermediate T .. - l increases ; C decreases as T goes down => T-2 At low T (~1 to ~100 K << TF) .. - l = const. (depending on the imperfections) CT => T R. Berman Thermal Conduction in Solids, 1976. Chap 3. Unusual Thermal Conductivity T.Lorenz, Nature 418,614 (2002) Bachgaard Salt ..Magnetic insulator No electron’s contribution Phonon contribution acoustic ..Tmax << D = 60K optical .. 0.1~0.2W/Km Magnetic Excitation (Magnon)? Large magnetic exchange interaction (J ~ 500 K) Dominating contribution at high T (J >> D ) Magnon? • Magnon – Quantized spin wave Qausi-One Dimensional Systems Exchange Coupling (J) a Elastic Coupling (k) Comparison with phonon Magnon Phonon • Density of state • Density of modes 1 h 3 2 VK2 1 D 1 2 D 4 2 2 JSa2 2 2 d dK • Energy of a mode • Energy of crystal vibration 1 k nk h k 1 2 n h • Number of magnons excited in 2 the mode k • Thermal equilibrium occupancy 1 1 nk n exp hw k k B T 1 exp hw k B T 1 • Energy of each magnon excited • Energy of each phonon h 2 JSa2 k2 hω • Total energy • Total energy 1 1 El n h El nk h 2 p K 2 p K Explanation for the TC in 1-D systems Debye model for magnon scattering 2 Jππ2 k BT x 2e x κ mag T s B T 2n k a where x=/kBT, and π e 0 x 1 l ( , T )dx 2 s l s ,i 1 1 ls Each ls,i represents an independent channel including spinon phonon scattering, spinon defect scattering .. Debye model for phonon scattering 3 Θ D /T k k x 4e x κ ph B B T3 2π 2 v e 0 x 1 2 τ(ω,T)dx where τ p,i 1 1 τp Each p,i represents an independent channel such as Boundary, Point defects, Phonon- phonon, dislocation, resonance scattering 1-D Anti-Ferromagnetic System Sr14-xCaxCu24O41 A.V Sologubenko PRB. 64, 054412 (2001) A.V Sologubenko PRL. 84, 2714 (2000) Strong 180O Cu-O-Cu Coupling SrCuO2 & Sr2CuO3 .. J~ 2100-3000K (=J’/J~10-5) Sr14-xCaxCu24O41 .. J~ 1500 K (~0.55) Two peaks in chain direction Low T peak .. Phonon fitting Second peak .. Spin excitations (Spinon) 1-D Anti-Ferromagnetic System (Spin-Peierls system) Low T peak .. Strong suppression with high magnetic fields Phonon scattering by defects and by spin excitation Y. Ando, PRB 58, R2913 (1998) High T peak.. Almost unchanged with increasing magnetic fields. Interaction between Magnons Jnn ~ 120 K , energy gap = 25K *1% increase if spin energy gap > Zeeman Energy Tsp = 14.08K, Peaks at T=5.5K, 22K Here, 25K > 18.6K for 14T *C. Hess PRB 64 184305 TC of 1-D chain system H J S i S i 1 Heisenberg Hamiltonian i Antiferromagntic system (J>0) Ferromagntic system (J<0) ε(k) J sinka ka ε(k) J(1- Cos(ka)) ka 2 Heisenberg chain .. Sr2CuO3 (J~2500 K) No Data !!! Spin ladder system.. : Sr14Cu24O41(J~1500 K) ..Anisotropy, Double peaks in the chain direction Summary • Magnon .. Magnetic Excitation • Magnon’s contribution to Thermal conductivity of 1-D Anti-ferromagnetic Systems • Properties - Maximum Peak at High Temperature - Big Anisotropy - Big magnetic Exchange interaction • Ferromagnetic Systems? Questions?
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