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8/17/2019 Spin waves and magnons unit 20.pptx
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Spin waves and magnons
Consider an almost perfectly ordered ferromagnet at low temperatures T
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0: Bh g H µ µ =
Derivation of spin waves in the classical limit&or simplicity let's consider classical Heisenberg spin chain
J
Classical spin vectors of length S S =
nS 1nS +1nS −
J
(round state : all spins parallel with energy 20 E NJS NhS = − −)eviations from ground state are spin wave e!citations which can be pictured as
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Tor#ue changes angular momentum
Deriving the spin wave dispersion relation
pin is an angular momentumClassical mechanics d L T
dt =
Here: n And S
T S H
dt
= = ×
( )1 1 A n n H J S S − += +*!change field" e!change interaction with neighbors can effectively be consideredas 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 nd S
J S S S S dt − +
= × + ×
1 1 1 1 1 1
xn
x y z x y z y
x y z x y z nn n n n n n
x y z x y z z n n n n n n
n
dS dt e e e e e e
dS J S S S S S S
dt S S S S S S dS
dt
− − − + + +
÷ ÷ ÷= + ÷ ÷
÷ ÷
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Ta+es care of the fact that spins are at discrete lattice positions ! n,n a
-et's write down the !.componentthe rest follows from cyclic permutation/be careful with the signs though01
1 1 1 1 1 1
xn
x y z x y z y
x y z x y z nn n n n n n
x y z x y z z n n n n n nn
dS dt e e e e e e
dS J S S S S S S
dt S S S S S S dS
dt
− − − + + +
÷
÷ ÷= + ÷ ÷ ÷
÷
( ) ( )1 1 1 1 1 1 1 1 x
y 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
dS J S S S S S S S S J S S S S S S
dt − − + + − + − + = − + − = + − +
%e consider e!citations with small amplitude ,
, z x yn nS S S S ≈
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%ith
( )
( )
xi nka t n
yi nka t n
dS i uSe
dt dS
i vSedt
ω
ω
ω
ω
−
−
= −
= −into
1 12 x
y y ynn n n
dS JS S S S
dt − + = − −
1 12 y
x x xnn n n
dS JS S S S
dt − + = − − −
( )
( )
i nka t xn
i nka t yn
S uSe
S vSe
ω
ω
−
−
==
and
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
2
2
i nka t i nka t i nka t i nka t ika ika
i nka t i nka t i nka t i nka t ika ika
i uSe JS vSe vSe e vSe e
i vSe JS uSe uSe e uSe e
ω ω ω ω
ω ω ω ω
ω
ω
− − − −− +
− − − −− +
− = − − − = − − −
( )( )
2 1 cos
2 1 cos
i u vJS ka
i v uJS ka
ω
ω
− = −− = − −
( )( )
2 1 cos0
2 1 cos
i JS ka u
JS ka i v
ω ω
− = ÷ ÷− −
Non-trivial solution meaning other than u=v=0 for:
( )( )
2 1 cos0
2 1 cos
i JS ka
JS ka i
ω ω
− =− − ( )2 1 cos JS kaω = −
Magnon dispersion relation
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Thermodynamics of magnons
Calculation of the internal energy:12k k k
E nω = + ÷ ∑h 1
2k k k U E nω = = + ÷ ∑h 0 1k
k
k
E eβ ω
ω = +−∑ h
h
in complete analogy to the photonsand phonons
3
3... ...(2 )k
V
d k π =∑ ∫
%e consider the limit T.23:
4nly low energy magnons near +,3 e!cited ( ) 2 22 1 cos JS ka JSa k ω = − ≈%ith
2 2
2 22
0 3 4(2 ) 1 JSa k V JSa k
U E k dk eβ
π π
= +−∫ h
h
%ith
2
B B
JSa D x k k k T k T = =
h
and hence B
Ddx dk k T =
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( )
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 T U E x dx
D De
V dx E D k T x
e
π π
π −
= + ÷ −
= +−
∫
∫
Just a number which becomes with integration to infinity
2
4
0
3 3(5 / 2) 1.3419
8 81 xdx
xe
π π ς
∞
= ≈ ×−∫
3/ 2
BV
V
U k T C
T D∂ = µ ÷ ÷∂
*!ponent different than for phonons due to difference in dispersion
1
1( ) s
k
sk
ς ∞
==∑
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56 $hysi+ ! " 738 /9 ;31:
0 ( ) k k
U E k nω = +∑hThe internal energy
can alternatively be e!pressed as2 2
01
( )2
x yk k
k
U E k S S S
ω = + +∑hwhere nik r n k
k
S S e=∑2 21
2 x yk k k S S nS
+ =
Intuitive hand.waving interpretation:= of e!citations in a mode = < > + < >= < >& nω ↔ < >h
2 x n< > ↔ < >
>agnetization and its deviation from full alignment in z.direction is determined as
( ) z B
nn
g M T S V
µ
= ∑ ( ) ( )( )2 22 x y B
n nn
g
S S S V
µ
= − +∑
Magneti"ation and the cele#rated T $%& 'loch law:
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-et's closer inspect ( ) ( )( )2 22 x yn nS S S − + and remember ( ) ( )2 2 2 x yn nS S S +
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Now let(s calculate M)T* with magnon dispersion at T-+0
( ) B
k k
g M T NS nV
µ = − ÷ ∑
3
3... ...(2 )k
V
d k π =∑ ∫ 2 2
JSa k ω ≈with and
2 2
23( ) 4(2 ) 1
B JSa k
g V dk M T NS k
V eβ µ
π π
= − ÷− ∫ h
Again with2
B B
JSa D x k k
k T k T = =
hand hence
B
Ddx dk
k T =
2
3/ 22
2( ) 2 1 B B
x
g V k T dx M T NS xV D e
µ π
= − ÷ ÷ ÷ − ∫
3/ 2
2
(3/2)( ) ( 0) 1
2 4 BV k T M T M T
NS Dπ ς
π
= = − ÷ ÷ ÷ B
g NS V
µ Felix Bloch(1905 - 1983)Nobel Prize in 1952 for
8/17/2019 Spin waves and magnons unit 20.pptx
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Modern research e,ample:'loch(s T $%&-law widely applica#le also in e,otic systems
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Spin waves and phase transitions: oldstone e,citations
. sta#ility analysis against long wavelength fluctuations giveshints for the possi#le e,istence of a long range ordered phase
( , )n m
n m
H J S S = − ∑% Heisenberg Hamiltonian e!ample for continuous rotational symmetrywhich can be spontaneously bro+en depending on the dimension" dd,9
d,7-et's have a loo+ to spin wave approach for
( 0) ( ) M M T M T ∆ = = −in various spatial dimensions d
( ) B k k
g M T NS n
V µ = − ÷
∑ ... ...(2 ) d k d
d L d k π
=∑ ∫ &rom and1
2
d dk k M
k
−=
∆ µ∫ 2 2 2 2
01 JSa k
k e JSa k β β
→− ≈h h
%hen a continuous symmetry is bro+en there must e!ista (oldstone mode /boson1 with ω→3 for + →3
In low dimensions d,9 and d,7 integral diverges at thelower bound +,3
/nphysical result indicates a#sence of orderedlow temperature phase in d=! and d=&