# 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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