Post on 14-Dec-2015
Spin waves and magnons
Consider an almost perfectly ordered ferromagnet at low temperatures T<<Tc
x
y
z
nr
nrAt magnetic moment characterized by spin: , ,x y z
n n n nS S S S
Interaction between spins and magnetic field given by Hamiltonian
0( , )
n m nBn m n
H J S S g H S
J
Spin waves:Thermal properties of ferromagnet determined at T<<Tc by low energy excitations, quantized spin waves
Strategy similar to photons and phononsPhonons: classical dynamical problem provides correct eigenfrequencies of eigenmodes
classical spin wave dynamical problem provides correct eigenfrequencies of eigenmodes With the dispersion relation for spin waves
Thermadynamics using quantized spin waves: magnons
0: Bh g H
Derivation of spin waves in the classical limitFor simplicity let’s consider classical Heisenberg spin chain
JClassical spin vectors S of length
S S
nS 1nS 1nS
J
S
Ground state : all spins parallel with energy
20E NJS NhS
Deviations from ground state are spin wave excitations which can be pictured as
Torque changes angular momentum
Deriving the spin wave dispersion relation
Spin is an angular momentum
Classical mechanics dLT
dt
Here: nAn
dST S H
dt
1 1A n nH J S S Exchange field, exchange interaction with neighbors can effectively be considered as a magnetic field acting on spin at position n
1 1n n nS J S S nS 1nS 1nS
J J
1 1n
n n n n
dSJ S S S S
dt
1 1 1 1 1 1
xn
x y z x y zy
x y z x y znn n n n n n
x y z x y zz n n n n n nn
dS
dt e e e e e edS
J S S S S S Sdt
S S S S S SdS
dt
Takes care of the fact that spins are at discrete lattice positions xn=n a
Let’s write down the x-component the rest follows from cyclic permutation(be careful with the signs though!)
1 1 1 1 1 1
xn
x y z x y zy
x y z x y znn n n n n n
x y z x y zz n n n n n nn
dS
dt e e e e e edS
J S S S S S Sdt
S S S S S SdS
dt
1 1 1 1 1 1 1 1
xy z z y y z z y y z z z y ynn n n n n n n n n n n n n n
dSJ S S S S S S S S J S S S S S S
dt
We consider excitations with small amplitude
,,z x yn nS S S S
1 1 1 12 2x
y y y y y ynn n n n n n
dSJ S S S S S JS S S S
dt
1 12y
x x xnn n n
dSJS S S S
dt
0zndS
dt
Solution with plane wave ansatz:
i nka txn
i nka tyn
S uSe
S vSe
With
xi nka tn
yi nka tn
dSi uSe
dt
dSi vSe
dt
into1 12
xy y ynn n n
dSJS S S S
dt
1 12y
x x xnn n n
dSJS S S S
dt
i nka txn
i nka tyn
S uSe
S vSe
and
2
2
i nka t i nka t i nka t i nka tika ika
i nka t i nka t i nka t i nka tika ika
i uSe JS vSe vSe e vSe e
i vSe JS uSe uSe e uSe e
2 1 cos
2 1 cos
i u vJS ka
i v uJS ka
2 1 cos
02 1 cos
i JS ka u
JS ka i v
Non-trivial solution meaning other than u=v=0 for:
2 1 cos0
2 1 cos
i JS ka
JS ka i
2 1 cosJS ka
Magnon dispersion relation
Thermodynamics of magnons
Calculation of the internal energy:
1
2k kk
E n
1
2k kk
U E n
01k
k
k
Ee
in complete analogy to the photons and phonons
33
... ...(2 )k
Vd k
We consider the limit T->0:
Only low energy magnons near k=0 excited
2 22 1 cosJS ka JSa k
With
2 2
2 22
0 34
(2 ) 1JSa k
V JSa kU E k dk
e
With2
B B
JSa Dx k k
k T k T
and hence
B
Ddx dk
k T
2
2
24
0 3
5/ 23/ 2 40 2
4(2 ) 1
2 1
B B
x
B x
V k T D k TU E x dx
D De
V dxE D k T x
e
Just a number which becomes with integration to infinity
2
4
0
3 3(5 / 2) 1.3419
8 81x
dxxe
3/ 2
BV
V
U k TC
T D
Exponent different than for phonons due to difference in dispersion
1
1( )
sk
sk
Z. Physik 61, 206 (1930):
0 ( ) kk
U E k n
The internal energy
can alternatively be expressed as
2 2
0
1( )2
x yk k
k
U E k S SS
where nik rn k
k
S S e2 21
2x yk k kS S n
S
Intuitive/hand-waving interpretation:# of excitations in a mode <-> average of classical amplitude squared
2 2 21 1
2 2E m x m x
2 2 2 2 21 1
2 2E m x m x m x n
2x n
Magnetization and its deviation from full alignment in z-direction is determined as ( ) zB
nn
gM T S
V
2 22 x yB
n nn
gS S S
V
Magnetization and the celebrated T3/2 Bloch law:
Let’s closer inspect
2 22 x yn nS S S and
remember 2 2 2x y
n nS S S for T->0
2 2 2 2
2 222 2
1 12
x y x yn n n nx y
n n
S S S SS S S S S
S S
2 2 2 2
2( ) 1
2 2
x y x yn n n nB B
n n
S S S Sg gM T S NS
V S V S
with nik rn k
k
S S e
2 21( )
2x yBk k
k
gM T NS S S
V S
using( )
,ni k k r
k kn
e and k kS S
( ) Bk
k
gM T NS n
V
Intuitive interpretation:Excitation of spin waves (magnons) means spins point on average less in z-direction -> magnetization goes down
Now let’s calculate M(T) with magnon dispersion at T->0
( ) Bk
k
gM T NS n
V
3
3... ...
(2 )k
Vd k
2 2JSa k with and
2 2
23
( ) 4(2 ) 1
B
JSa k
g V dkM T NS k
V e
Again with2
B B
JSa Dx k k
k T k T
and hence
B
Ddx dk
k T
2
3/ 22
2( )
2 1B B
x
g V k T dxM T NS x
V D e
3/ 2
2
(3 / 2)( ) ( 0) 1
2 4BV k T
M T M TNS D
Bg NS
V
Felix Bloch(1905 - 1983)Nobel Prize in 1952 for NMR
Spin waves and phase transitions: Goldstone excitations A stability analysis against long wavelength fluctuations gives hints for the possible existence of a long range ordered phase
( , )n m
n m
H J S S Heisenberg Hamiltonian example for continuous rotational symmetry which can be spontaneously broken depending on the dimension, d
d=1
d=2Let’s have a look to spin wave approach for
( 0) ( )M M T M T in various spatial dimensions d
( ) Bk
k
gM T NS n
V
... ...
(2 )dk
ddLd k
From
and
1
2
ddk kM
k
2 2 2 2
01JSa k
ke JSa k
When a continuous symmetry is broken there must exist a Goldstone mode (boson) with 0 for k0
In low dimensions d=1 and d=2 integral diverges at the lower bound k=0
Unphysical result indicates absence of orderedlow temperature phase in d=1 and d=2