Berman Thermal Conduction in Solids by Umv8VmhZ

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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)
                                                             CT
                                                               =>   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 sinka  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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