1. Magnon Another Carrier of Thermal Conductivity
Final Presentation for ME 381R
Nov
Park, Keeseong
Ha, Eun

2. 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

3. General Behavior of Thermal Conductivity (Insulator)
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)
C  T3
=>   T3
V may be regarded as independent of temperature ~ 5*10^5 cm/sec sound velocity (from kittel)
V has the same when T< D, v is average when T> D.
Quartz l = 40 Angstrong (at 0 Celsius degree) and I= 540 (at –190)
NaCl l=23 (at 0) and l=100 (-190)
exp(T*/T) is Boltzman factor.
R. Berman, Thermal Conduction in Solids, Chap 3.

4. General Behavior of Thermal Conductivity (Metal)
Wiedemann-Franz-Lorenz law
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
At low T, L tends to decrease because two mean free paths are different.
At high T, phonons block the motion of electrons so the electric conductivity decreases but the thermal conductivity remains by phonon interactions
At low T, longer lattice wavelengths are more important, which can be 100 atomics spacings at 1K.
These waves are almost unaffected by disorders on an atomic scale, but are scattered on the external boundaries of a sample.
The energies of the conduction electrons in ordinary metals are little dependent on T. Electrons are strongly scattered by imperfections of atomic dimensions.
R. Berman Thermal Conduction in Solids, Chap 3.

5. Unusual Thermal Conductivity T.Lorenz, Nature 418,614 (2002)
Bachgaard Salt ..Magnetic insulator
No electron’s contribution
Phonon contribution
acoustic ..Tmax << D = 60K
optical ~0.2W/Km
Magnetic Excitation (Magnon)?
Large magnetic exchange interaction
(J ~ 500 K)
Dominating contribution at high T
(J >> D )

6. Magnon?
Magnon
Quantized spin wave

7. Qausi-One Dimensional Systems
Exchange Coupling (J) a
Elastic Coupling (k)

8. Comparison with phonon
Magnon
Phonon
Density of state
Energy of crystal vibration
Thermal equilibrium occupancy
Energy of each phonon hω
Total energy
Density of modes
Energy of a mode
Number of magnons excited in the mode k
Energy of each magnon excited
Total energy

9. Explanation for the TC in 1-D systems
Debye model for magnon scattering
where x=/kBT, and
Each ls,i represents an independent channel including spinon phonon scattering, spinon defect scattering ..
Debye model for phonon scattering
where
Each p,i represents an independent channel such as Boundary, Point defects, Phonon-phonon, dislocation, resonance scattering

10. 1-D Anti-Ferromagnetic System
A.V Sologubenko PRB. 64, (2001)
Sr14-xCaxCu24O41
A.V Sologubenko PRL. 84, 2714 (2000)
Neel T?
Strong 180O Cu-O-Cu Coupling
SrCuO2 & Sr2CuO3 .. J~ K (=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)

11. 1-D Anti-Ferromagnetic System (Spin-Peierls system)
Y. Ando, PRB 58, R2913 (1998)
Kink show the sp transition because it is due to spin
If H>Hc(=spin gap/g ub), k will increase because the gap would be collapsed and closure of this gap allows the conduction of heat by magnons. R. Jin, PRL 91, (2003)
Spin-Peierls transition.. Exponential drop of the magnetic susceptibility. Lattice dimerization along the a and c axies.
The energy gained by splitting the degeneracy of the original magnetic ground state exceeds the lattice deformation energy
The exchange constant between neighbering spins on the chain alternates.
For the occurrence of a spin-Peierls transition, several perconditions are necessary: first of all, a crystal must contain (quasi) one-dimesional antiferromagnetic spin chains of half-integer spin, i.e. the exchange coupling between neighbouting spin along one crystal direction has to be much larger than those perpendicular to this direction. Secondly, a finite magneto-elastic coupling is necessary, i.e. the exchange interaction depends on the distance between neighbouring sites. In a strictly one-dimensional spin chain no long-range antiferromagnetic order can develop, even at T = 0 K. However, the magnetic energy can be lowered by a spin-Peierls transition, which may, in a simplified picture, be described as follows: below a certain transition temperature Tsp the distances between neighbouring spins are no longer uniform but alternate a1,2 = a0 (1((). Due to the magneto-elastic coupling this leads to an alternation of the exchange coupling and each pair of stronger-coupled spins is forming a spin singlet. This so-called dimerization leads to a gain of magnetic energy which overcompensates the loss of elastic energy arising from the alternating structural distortion along the spin chains. In this sense, the spin-Peierls transition is a three-dimensinal structural phase transition which is driven by the one-dimensional magnetism. From the experimental point of view there are several characteristic features which signal the spin-Peierls transition. Below Tsp, superstructure reflections can be observed due to the structural distortion leading to the doubling of the unit cell. In addition, there is a drop of the magnetic susceptibility due to the formation of non-,agnetic spin singlets. Finally, Tsp shows a very characteristic magnetic field dependence. The name 'spin-Peierls transition' arises from the fact that this transition is the magnetic analogue of the 'Peierls transition', which is a metal insulator transition occurring in (quasi-) one-dimensional metals. Within this analogy, the spn-Peierls transition in zero magnetic field corresponds to a Peierls transition of a one-dimensional metal with exactly half-filled conduction band and the application of an external magnetic field corresponds to a continuous change of the bandfilling.
Low T peak ..
Strong suppression with high magnetic fields
Phonon scattering by defects and by spin excitation
High T peak..
Almost unchanged with increasing magnetic fields.
Interaction between Magnons
*1% increase if spin energy gap > Zeeman Energy
Here, 25K > 18.6K for 14T *C. Hess PRB
Jnn ~ 120 K , energy gap = 25K
Tsp = 14.08K, Peaks at T=5.5K, 22K

12. TC of 1-D chain system Heisenberg Hamiltonian No Data !!!
Antiferromagntic system (J>0)
Ferromagntic system (J<0)
Heisenberg chain .. Sr2CuO3 (J~2500 K)
No Data !!!
Spin ladder system.. : Sr14Cu24O41(J~1500 K)
..Anisotropy, Double peaks in the chain direction

13. 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?

14. Questions?