Spin waves and magnons
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Transcript Spin waves and magnons
Spin waves and magnons
Consider an almost perfectly ordered ferromagnet at low temperatures T<<Tc
At
z
rn
y
x
J
rn
magnetic moment characterized by spin:
S n S nx , S ny , S nz
Interaction between spins and magnetic field given by Hamiltonian
H J S n S m g B 0 H S n
( n,m)
Spin waves:
n
Thermal properties of ferromagnet determined at T<<Tc by low
energy excitations, quantized spin waves
Strategy similar to photons and phonons
Phonons: 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
Derivation of spin waves in the classical limit
For simplicity let’s consider classical Heisenberg spin chain
S n1 S n S n1
J
J
Classical spin vectors S of length
S S
Ground state : all spins parallel with energy
h : g B 0 H
E0 NJS 2 NhS
Deviations from ground state are spin wave excitations which can be pictured as
S
Deriving the spin wave dispersion relation
Spin is an angular momentum
Classical mechanics
Here:
dL
T
dt
Torque changes angular momentum
dSn
T S n H A S n J S n1 S n1
dt
H A J S n1 S n1
S n1 S n S n1
J
J
Exchange field, exchange interaction with neighbors can effectively be considered
as a magnetic field acting on spin at position n
dSn
J S n S n 1 S n S n 1
dt
dSnx
dt
ex e y ez
ex e y ez
x
dSny
y
z
x
y
z
Sn Sn Sn
dt J Sn Sn Sn
x
y
z
x
y
z
z
S
S
S
S
S
S
n 1
n 1
n 1
n 1
n 1
n 1
dSn
dt
dSnx
dt
dSny
dt
z
dSn
dt
ex e y ez
ex e y ez
x
y
z
x
y
z
J
S
S
S
S
S
S
n
n
n
n
n
n
x
y
z
x
y
z
S
S
S
S
S
S
n
1
n
1
n
1
n
1
n
1
n
1
Let’s write down the x-component
the rest follows from cyclic permutation
(be careful with the signs though!)
dSnx
J Sny Snz1 Snz Sny1 Sny Snz1 Snz Sny1 J Sny Snz1 Snz1 Snz Sny1 Sny1
dt
We consider excitations with small amplitude
Snz S , Snx, y S
dSnx
J Sny 2S S Sny1 Sny1 JS 2Sny Sny1 Sny1
dt
dSny
JS 2Snx Snx1 Snx1
dt
dSnz
0
Takes care of the fact that spins are at discrete lattice positions x =n a
dt
i nka t
S nx uSe
n
Solution with plane wave ansatz:
S ny vSe
i nka t
With
S nx uSe
i nka t
S ny vSe
i nka t
i nka t
x
n
dS
i nka t
iuSe
dt
into
and
dSny
i nka t
i vSe
dt
dSnx
JS 2Sny Sny1 Sny1
dt
dSny
JS 2Snx Snx1 Snx1
dt
iuSe
i nka t
i nka t ika
i nka t ika
JS 2vSe
vSe
e vSe
e
ivSe
i nka t
i nka t ika
i nka t ika
JS 2uSe
uSe
e uSe
e
i nka t
iu 2vJS 1 cos ka
iv 2uJS 1 cos ka
i
2 JS 1 cos ka u
0
i
2 JS 1 cos ka
v
Non-trivial solution meaning other than u=v=0 for:
i
2 JS 1 cos ka
0
2 JS 1 cos ka
i
2JS 1 cos ka
Magnon dispersion relation
Thermodynamics of magnons
Calculation of the internal energy:
E
k
1
k nk
2
in complete analogy to the photons
and phonons
U E
k
k
1 E
k nk
0
k
e
1
2
k
We consider the limit T->0:
Only low energy magnons near k=0 excited
2JS 1 cos ka JSa2k 2
With
...
k
V
3
d
k ...
3
(2 )
V
JSa 2 k 2
2
U E0
4 k JSa2k 2
dk
3
(2 )
e
1
With
xk
JSa 2
D
k
k BT
k BT
and hence
dx dk
D
k BT
2
V
D
k BT
4 k BT
U E0
4
x
dx
x2
3
(2 )
D e 1 D
E0
V
dx
5/ 2
3/ 2
4
D
k
T
x
B x2
2
2
e 1
Just a number which becomes with integration to infinity
4
x
0
dx
e 1
x2
3
3
(5 / 2)
1.3419
8
8
1
s
k 1 k
( s)
Exponent different than for phonons due to difference in dispersion
U
k BT
CV
T V D
3/ 2
Magnetization and the celebrated
T3/2
Bloch law:
Z. Physik 61, 206 (1930):
The internal energy
U E0 (k ) nk
k
can alternatively be expressed as
U E0 (k )
k
where S n
2
1
Skx Sky
2S
ik r n
S
e
k
Intuitive/hand-waving interpretation:
# of excitations in a mode <-> average of classical amplitude squared
k
2
1
Skx S ky
2S
2
2
1
1
m x2 m 2 x2
2 1
2
1
E m x 2 m 2 x 2 m 2 x 2 n
2
2
E
nk
x2 n
Magnetization and its deviation from full alignment in z-direction is determined as
g B
M (T )
V
n
S
z
n
g B
V
n
S 2 Snx Sny
2
2
S 2 Snx Sny
Let’s closer inspect
S 2 Snx Sny
M (T )
with
2
g B
V
n
Sn Sk e
2
2
Snx Sny
2
1
and remember
S2
2
Snx Sny S 2 for T->0
2
using
2
S x 2 S y 2
n
n
S 1
2
2S
S x 2 S y 2
n
n
g B NS
S 1
2S 2
V
n
ik r n
k
S
2
Snx Sny
2
2S
i ( k k ) r n
e
k ,k and S k S k
2
n
g B
1
x 2
M (T )
Sk Sky
NS
V
k 2S
g B
M (T )
NS nk
V
k
2
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
g B
M (T )
NS nk with
V
k
M (T )
Again with
...
k
V
3
d
k ... and
3
(2 )
JSa 2 k 2
g B
V
dk
2
NS
4
k
JSa2 k 2
V
(2 )3
e
1
xk
g B
M (T )
V
JSa 2
D
k
k BT
k BT
and hence
dx dk
D
k BT
3/ 2
V k BT
dx
2
NS 2
x x2
2
D
e 1
3/ 2
V k BT
(3 / 2)
M (T ) M (T 0) 1 2
2 NS D
4
g B NS
V
Felix Bloch
(1905 - 1983)
Nobel Prize in 1952 for NMR
Modern research example:
Bloch’s T3/2-law widely applicable also in exotic systems
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
H J S n S m
( n,m)
Heisenberg Hamiltonian example for continuous rotational symmetry
which can be spontaneously broken depending on the dimension, d
When a continuous symmetry is broken there must exist
a Goldstone mode (boson) with 0 for k0
Let’s have a look to spin wave approach for
d=1
d=2
M M (T 0) M (T )
in various spatial dimensions d
From
g B
M (T )
NS nk
V
k
dk k d 1
M
k2
e
JSa 2 k 2
and
Ld
d
...
d
k (2 )d k ...
In low dimensions d=1 and d=2 integral diverges at the
lower bound k=0
1 JSa 2 k 2
k 0
Unphysical result indicates absence of ordered
low temperature phase in d=1 and d=2