B - Institut für Physik

Persistent spin currents in mesoscopic
Heisenberg magnets
Marcus Kollar
Institut für Theoretische Physik, J.W.Goethe-Universität Frankfurt/Main
in collaboration with Florian Schütz and Peter Kopietz
PRL 91, 017205 (2003); PRB (in press); cond-mat/0301351, 0308230
supported in part by DFG-Forschergruppe 412 Spin- und Ladungsträgerkorrelationen
in niedrigdimensionalen metallorganischen Festkörpern
Karlsruhe, 24-Nov-03
Outline
1. Introduction: persistent currents in mesoscopic normal metal rings
2. Spin transport: itinerant vs. localized spins
3. Persistent spin currents in insulators: ferromagnets vs. antiferromagnets
4. Conclusion & outlook
1

1. Introduction: persistent currents
metal rings in Aharonov-Bohm geometry: ground-state currents
B
(Pauling 1937, London 1938, Hund 1939, . . . ; Imry, Introduction to mesoscopic physics, 1997)
∮
• magnetic flux φ = A · dr, A = ê ϕ Br/2
φ=πR 2 |B|
• simplest model: free 1d electrons (φ 0 = hc/e)
R
ϕ
s=Rϕ
2
2m ∗ (−i d ds + 2π L
φ
φ 0
) 2
ψ(s) = Eψ(s)
• periodic boundary conditions ψ(s + L) = ψ(s)
2

1. Introduction: persistent currents
metal rings in Aharonov-Bohm geometry: ground-state currents
B
(Pauling 1937, London 1938, Hund 1939, . . . ; Imry, Introduction to mesoscopic physics, 1997)
∮
• magnetic flux φ = A · dr, A = ê ϕ Br/2
φ=πR 2 |B|
• simplest model: free 1d electrons (φ 0 = hc/e)
R
ϕ
s=Rϕ
2
2m ∗ (−i d ds + 2π L
φ
φ 0
) 2
ψ(s) = Eψ(s)
• periodic boundary conditions ψ(s + L) = ψ(s)
( iφs
)
• gauge trafo: ˜ψ(s) = exp ψ(s) ⇒ twisted boundary conditions
φ 0 R
2
2m ∗ (−i d ds
) 2
˜ψ(s) = E ˜ψ(s) with ˜ψ(s + L) = exp
( 2πiφ
φ 0
)
˜ψ(s)
2

Ballistic regime
• eigenstates ˜ψ n (s) = e ikns / √ L with eigenenergies ɛ n = 2 kn
2
2m ∗
• quantized wavevectors: k n = 2π (n + φ )
, n = 0, ±1 . . . ⇒ k −n ≠ k n
L φ 0
ballistic regime, L ≪ l:
(l = mean free path)
• equilibrium current from thermodyn. potential
I(φ) = −c ∂Ω gc(φ)
∂φ
• velocities: v n = 1
∂ɛ n
∂k n
=
= −e
L
∑
n
v n
e (ɛ n−µ)/T + 1
m ∗ 2π
L (n + φ φ 0
) ≠ v −n
(Cheung et al. 1988)
3

Ballistic regime
• eigenstates ˜ψ n (s) = e ikns / √ L with eigenenergies ɛ n = 2 kn
2
2m ∗
• quantized wavevectors: k n = 2π (n + φ )
, n = 0, ±1 . . . ⇒ k −n ≠ k n
L φ 0
ballistic regime, L ≪ l:
(l = mean free path)
• equilibrium current from thermodyn. potential
I(φ) = −c ∂Ω gc(φ)
∂φ
• velocities: v n = 1
• I max = ev F
L
∂ɛ n
∂k n
=
= −e
L
∑
n
v n
e (ɛ n−µ)/T + 1
m ∗ 2π
L (n + φ φ 0
) ≠ v −n
≈ 4 nA ⇒ observed in GaAs loop
(Mailly et al. 1993)
(Cheung et al. 1988)
3

Effect of disorder
• mesoscopic persistent current exists also in disordered conductor
(Büttiker, Imry, Landauer 1983)
• I typ ≈ ev F
L
in experiments on single Au rings (Chandrasekhar et al. 1991)
• 〈I〉 ≈ 3 · 10 −3 ev F
in experiments on 10 7 Cu rings (Lévy et al. 1990)
L
• 〈I〉 ≈ ev F 1
for M-channel ring in canonical ensemble ⇒ too small
L M
• 〈I〉 ≈ ev F
L
l
L (2g 4 − g 2 )
(Schmid 1990, v. Oppen & Riedel 1990, Altshuler et al. 1990)
from Coulomb interaction ⇒ too small
(Ambegaokar & Eckern 1990)
unsolved problem:
experimentally observed currents much larger than theoretical predictions
4

Effect of disorder
• mesoscopic persistent current exists also in disordered conductor
(Büttiker, Imry, Landauer 1983)
• I typ ≈ ev F
L
in experiments on single Au rings (Chandrasekhar et al. 1991)
• 〈I〉 ≈ 3 · 10 −3 ev F
in experiments on 10 7 Cu rings (Lévy et al. 1990)
L
• 〈I〉 ≈ ev F 1
for M-channel ring in canonical ensemble ⇒ too small
L M
• 〈I〉 ≈ ev F
L
l
L (2g 4 − g 2 )
(Schmid 1990, v. Oppen & Riedel 1990, Altshuler et al. 1990)
from Coulomb interaction ⇒ too small
(Ambegaokar & Eckern 1990)
unsolved problem:
experimentally observed currents much larger than theoretical predictions
today’s topic:
spin analogue of persistent currents
for single ring in ballistic regime
4

2. Spin transport: itinerant vs. localized spins
spin transport with itinerant electrons:
• conducting rings in inhomogeneous fields
(Loss et al. 1990, Stern 1992, Balatsky & Altshuler 1993, Gao & Qian 1993)
• conducting rings with spin-orbit coupling
(Frustaglia et al. 2001, Mal’shukov et al. 2002, Splettstoesser et al. 2003, Rashba 2003)
• thin film ferromagnets with spiral states, helimagnets
(König et al. 2001, Heurich et al. 2003)
Rashba spin-orbit coupling in semiconductors (Rashba 1960)
Ĥ = p2 x + p 2 y
2m ∗
eigenvalues: E ± = 2 k 2
+ α (
R
(σ ψ1
)
x ˆp y − σ y ˆp x ), Ψ =
ψ 2
√
2m ∗ ± α R k, k = kx 2 + ky
2
E
k
5

Spin-based electronic devices
spintronics: spin-based electronics for information processing
(Awschalom, Loss, Samarth, Semiconductor spintronics and quantum computation, 2003)
• spin transistor (Datta & Das 1990)
V g
FM 2DES FM
spin-dependent transmission
controlled by gate voltage
difficulties: spin injection, propagation, precession, collection
6

Spin-based electronic devices
spintronics: spin-based electronics for information processing
(Awschalom, Loss, Samarth, Semiconductor spintronics and quantum computation, 2003)
• spin transistor (Datta & Das 1990)
V g
FM 2DES FM
spin-dependent transmission
controlled by gate voltage
difficulties: spin injection, propagation, precession, collection
• spin filtering with quantum wires (Governale et al. 2002)
shift E ± with
V g and B
⇒
select
tunneling
events
6

Magnetic insulators
localized electrons: S i = 1 2
[ c
+
i↑
c + i↓
] T
σ
[
ci
↑
c i ↓
]
, S z i = 1 2 (n i↑ −n i↓ ) etc.
• motion of electrons frozen ⇒ Heisenberg exchange interactions
Ĥ = 1 2
∑
∑
J ij S i · S j − gµ B B i · S i ,
i,j
i
[S x i , S y j ] = iδ ijS z i
• S 2 i
= S(S + 1), S = 1 2
, 1, . . . ⇒ Holstein-Primakoff representation
S +
i
= √ 2S − n i b i
◮ bosons: [b
S −
i
= b † √ i
, b † i ] = 1, n i = b † i b i
i 2S − ni ◮ only n i ≤ 2S is allowed
Si z = S − n i ◮ S ≫ 1 2
⇒ expand in 1/S
• elementary excitations: spin waves (“magnons”)
7

Spin transport in magnetic insulators
Heisenberg magnets in spatially varyingtransition
magnetic fields:
• magnetic field difference at ends ofregion
spin system (Meier & Loss 2003)
−L / 2 L / 2 x
B(x)
ε
B+ ∆B/2
B+ ∆B/2
B− ∆B/2
B− ∆B/2
reservoir
R1
B(x)
reservoir
R1
spin chain
B+ ∆B/2
B− ∆B/2
reservoir R1 R2
R2
I m
transition
region −L / 2 L / 2 x
−L / 2 L / 2 x
ε
⇒ magnetization transport
semiclassical picture of spin transport:
n
B( ε)
time−L evolution / 2 of L / 2M(r, x t) ⇒ magnetization transport
R1 R2
g µ
B∆
B
I m
reservoir
R2
g µ B∆
B
n
B( ε)
carried by magnons
• magnetic field B i ⇒ magnetic moments m i (t) = gµ B 〈ψ(t)|S i |ψ(t)〉
• magnetization M(r, t) = ∑ i δ(r − r i)m i (t)
8

Charge current vs. magnetization current
current of electric monopoles q i :
current of magnetic dipoles m i :
q
i
v i
da
da
m i
v i
• current density: vector
j µ (r) = ∑ i q i(v i ) µ δ(r − r i )
• current through
∫
surface A:
I(A) = da · j(r)
A
• magnetization current density:
j α µ (r) = ∑ i (m i) α (v i ) µ δ(r − r i )
2nd rank tensor, α = x, y, z
• current through
∫
surface A:
I α (A) = da µ jµ α (r)
A
9

¡
¢
¡
¡
¡
Electrodynamics of magnetization currents
stationary charge currents
⇒ static magnetic fields
stationary magnetization currents
⇒ static electric fields
(Hirsch 1999; Meier & Loss 2003)
m
B
p
E
v
q
v
m
• Biot-Savart ∫ law:
d 3 r ′
B(r) =
c j(r′ ) × (r − r′ )
|r − r ′ | 3
• far zone: A = ×
| |
∫ 3
d 3 r ′
m =
2c r′ × j(r ′ )
magnetic dipole moment
• Biot-Savart-type law:
∫ d 3 r ′
φ(r) = −
c [M(r′ )×v(r ′ )]· (r − r′ )
|r − r ′ | 3
·
• far zone: φ = , E = −∇φ
| |
∫ 3 d 3 r ′
p = − M(r ′ ) × v(r ′ )
c
electric dipole moment
10

3. Persistent spin currents in insulators
ferromagnetic Heisenberg ring in a crown-shaped magnetic field:
• Hamiltonian:
Ĥ = 1 ∑
∑
J ij S i·S j −gµ B B i·S i
2
ij
• magnetic moments
m i = gµ B 〈S i 〉 = m i ˆm i
• Heisenberg equation of motion:
∂S i
∂t + h i × S i + ∑ j
i
J ij S i × S j = 0 where h i ≡ gµ B B i
^e z
m^
i
B i
• spin current operator: I i→j = J ij (S i × S j ) (Chandra, Coleman, Larkin 1990)
• longitudinal spin current: I s = ˆm i · 〈I i→j 〉
∂m i (t)
∂t
+ ∑ j
ˆm i · 〈I i→j 〉 = 0
continuity equation
11

Classical ground state
• classically S i = S ˆm i ⇒ stability: ˆm i × (h i − ∑ j J ijS ˆm j ) = 0
^e z
ϑ
m^
i
ϑ m
B i
FM ring (J < 0) with B(r) ∝ ˆr
⇒ sin(ϑ−ϑ m ) = |J|S
h
( 2π
⇒ assume h JS
N
[
1 − cos 2π
N
]
sin 2ϑ
) 2
≡ ∆
12

Classical ground state
• classically S i = S ˆm i ⇒ stability: ˆm i × (h i − ∑ j J ijS ˆm j ) = 0
^e z
ϑ
m^
i
ϑ m
B i
FM ring (J < 0) with B(r) ∝ ˆr
⇒ sin(ϑ−ϑ m ) = |J|S
h
( 2π
⇒ assume h JS
N
[
1 − cos 2π
N
) 2
≡ ∆
]
sin 2ϑ
• quantum spins: S i = S ‖ i ˆm i + S ⊥ i ⇒ longitudinal part S ‖ i = S − b† i b i
• decomposition of Hamiltonian: Ĥ = Ĥ‖ + Ĥ⊥ + O(S 1/2 )
◮ longitudinal: Ĥ ‖ = 1 ∑
2 i,j J ij ˆm i · ˆm j S ‖ i S‖ j − ∑ i h i · ˆm i S ‖ i = O(S2 )
◮ transverse: Ĥ ⊥ = 1 ∑
2 i,j J ijSi ⊥ · Sj ⊥ = O(S)
12

Transverse basis: local U(1) gauge freedom
• classical spin orientation ˆm i ⇒ local quantization axis (fixed)
• but local triad {ê 1 i , ê2 i , ˆm i} can be arbitrarily rotated around ˆm i
• transverse Hamiltonian: (e ± i
= ê 1 i ± iê2 i , S± i
= e ± i
· Si ⊥)
Ĥ ⊥ = 1 8
∑
i,j
J ij
∑
p,p ′ =±
(e p i · ep′
j ) S−p i
S −p′
j
• local U(1) gauge transformation:
e ± i
→ ẽ ± i e±iα i
and S ± i
→ S ± i e±iα i
, α i arbitrary
• spin-wave expansion: S + i
= √ 2S b i
[1 + O( 1 S )] 13

Transverse basis: local U(1) gauge freedom
• classical spin orientation ˆm i ⇒ local quantization axis (fixed)
• but local triad {ê 1 i , ê2 i , ˆm i} can be arbitrarily rotated around ˆm i
• transverse Hamiltonian: (e ± i
= ê 1 i ± iê2 i , S± i
= e ± i
· Si ⊥)
Ĥ ⊥ = 1 8
∑
i,j
J ij
∑
p,p ′ =±
(e p i · ep′
j ) S−p i
S −p′
j
• local U(1) gauge transformation:
e ± i
→ ẽ ± i e±iα i
and S ± i
→ S ± i e±iα i
, α i arbitrary
• spin-wave expansion: S + i
= √ 2S b i
[1 + O( 1 S )]
useful reference: parallel-transported basis
• for neighbors i and j let {ẽ 1 i , ẽ2 i , ˆm i} with ẽ 2 i = ˆm i × ˆm j
• express e ± i
= ẽ ± i
e ±iω i→j
via angles ω i→j
13

Parallel transport on surface of unit sphere
given:
• closed curve C on unit sphere,
parametrized by ˆm(s)
• transverse vector defined
by angular velocity ω(s):
de(s)
ds
parallel transport:
= ω(s) × e(s)
(M.V.Berry, in: Shapere & Wilczek,
Geometric Phases in Physics, 1989)
• ω(s) = ˆm(s) × d ˆm(s)
ds
• ˆm(s) · ω(s) = 0 ⇒ transverse basis does not twist around ˆm(s)
• e initial ≠ e final (“anholonomy”)
⇒ defect angle α(C) = ∠(e initial , e final ) = solid angle subtended by C
14

Gauge invariance and current conservation
transverse Hamiltonian:
Ĥ ⊥ = 1 8
∑
i,j
J ij
[(1 + ˆm i · ˆm j ) e i(ω i→j−ω j→i ) S − i S+ j
− (1 − ˆm i · ˆm j ) e i(ω i→j+ω j→i ) S − i S− j + h.c. ]
• ferromagnet: ˆm i · ˆm j = 1 + O( 1 N
• antiferromagnet: ˆm i · ˆm j = −1 + O( 1 N
) ⇒ FM magnons
) ⇒ AF magnons
gauge invariance: ω i→j → ω i→j + α i ,
S p i → Sp i eipα i
• conserved spin current: (Noether’s theorem)
0 = 〈 ∂Ĥ⊥
∂α i
〉 = 〈 ∑ j
∂Ĥ ⊥
∂ω i→j
〉 = − ∑ j
〈 ˆm i · J ij (S ⊥ i × S ⊥ j )〉
} {{ }
= ˆm i · 〈I i→j 〉
• agrees with Heisenberg equation of motion
15

Gauge invariant persistent current
• ferromagnetic next-neighbor coupling J < 0 ⇒ ˆm i · ˆm i+1 ≈ 1
• gauge phases away ⇒ twisted boundary conditions S ± i+N = exp(±iΩ) S± i
where
N∑
Ω = (ω i→i+1 − ω i→i−1 ) = gauge invariant geometric flux
i=1
= total defect angle of ˆm i on the unit sphere
= 2π(1 − cos ϑ m ) for crown-shaped configuration
16

Gauge invariant persistent current
• ferromagnetic next-neighbor coupling J < 0 ⇒ ˆm i · ˆm i+1 ≈ 1
• gauge phases away ⇒ twisted boundary conditions S ± i+N = exp(±iΩ) S± i
where
N∑
Ω = (ω i→i+1 − ω i→i−1 ) = gauge invariant geometric flux
i=1
= total defect angle of ˆm i on the unit sphere
= 2π(1 − cos ϑ m ) for crown-shaped configuration
gauge invariant persistent spin current:
I s (Ω) = −
compare with persistent charge current:
I(φ) = −c ∂Ω gc(φ)
∂φ
∂F (Ω)
∂Ω = J〈 ˆm i · (S ⊥ i × S ⊥ i+1) 〉
, φ = magnetic flux
16

Persistent spin current carried by magnons
• FM spin waves:
Ĥ sw = ∑ k
(ɛ k + h) b † k b k with ɛ k = 2JS(1 − cos(ka)) ∝ k 2
• quantized wavevectors: k = 2π L
(
n + Ω )
, n = 0, ±1, . . . , ± N
2π
2
17

Persistent spin current carried by magnons
• FM spin waves:
Ĥ sw = ∑ (ɛ k + h) b † k b k with ɛ k = 2JS(1 − cos(ka)) ∝ k 2
k
• quantized wavevectors: k = 2π L
(
n + Ω )
, n = 0, ±1, . . . , ± N
2π
2
magnetization current: I m = gµ B
I s = − gµ B
I m (Ω) = − gµ ∑
B v k
L e (ɛk+h)/T − 1
≈ gµ BT
k
sin Ω
cos Ω − cosh(2π √ h/∆)
∂F sw
∂Ω
where v k = 1
∂ɛ k
∂k ∝ k
for ∆ ≡ JS( 2π N )2 ≪ T ≪ J
17

Persistent spin current carried by magnons
• FM spin waves:
Ĥ sw = ∑ (ɛ k + h) b † k b k with ɛ k = 2JS(1 − cos(ka)) ∝ k 2
k
• quantized wavevectors: k = 2π L
(
n + Ω )
, n = 0, ±1, . . . , ± N
2π
2
magnetization current: I m = gµ B
I s = − gµ B
I m (Ω) = − gµ ∑
B v k
L e (ɛk+h)/T − 1
≈ gµ BT
k
sin Ω
cos Ω − cosh(2π √ h/∆)
∂F sw
∂Ω
where v k = 1
∂ɛ k
∂k ∝ k
for ∆ ≡ JS( 2π N )2 ≪ T ≪ J
• bosonic analogue of charge current: I(φ) = −e ∑
L
k
• thermal fluctuations: I m ∝ T
v k
e (ɛ k−µ)/T + 1
• mesoscopic quantum effect: I m vanishes in bulk limit or classical limit
17

Electric field generated by spin current
lines of constant electric potential for ϑ m = 30 o
4
15
10
2
m^
m^
5
z / R
0
z / R
0
-2
-5
-4
-10
-4 -2 0 2 4
x / R
-15 -10 -5 0 5 10 15
generalized Biot-Savart law:
far zone: φ(r) = p · r/|r| 3
φ(r) = I ∮
m
[dr ′ × ˆm(r ′ )] · (r − r′ ) with electric dipole moment
c
|r − r ′ | 3 I m
p = −ê z
c sin ϑ m
x / R
18

Experimental parameters
• desired:
measurement of voltage difference ∆U
at distance ≈ L above and below Heisenberg ring
• optimal parameters:
◮ solid angle Ω = 2π(1 − cos ϑ m ) = π/2 ⇒ ϑ m ≈ 41 o
◮ magnetic field: gµ B B ≈ ∆ = JS(2π/N) 2
◮ temperature: ∆ ≪ T ≪ J
• T = 50 K, L = 100 nm, N = 100, J = 100 K ⇒ ∆U ≈ 0.2nV, B ≈ 0.1T
• requirements:
◮ well-characterized rings with large Heisenberg coupling J
◮ submicron B inhomogeneity
◮ nanovolt sensitivity
• detection may be difficult due to screening
• spin rings in inhomogeneous electric fields: Ahoronov-Casher effect
19

Antiferromagnetic spin ring
• antiferromagnetic next-neighbor coupling J > 0
crown-shaped field B(r) ∝ ˆr
⇒ spins flip for h > JS ≡ h c
^z e
^m i
B i
stability of Néel-like state:
⇒ sin 2( ¯ϑ − ϑ) = ( JS h )2 sin 2 ( 2π N
) sin(2 ¯ϑ)
ϑ
⇒ assume h JS/N (≪ h c )
• two sublattices with ˆm i ≈ (−1) i ˆn i and ˆn i · h i ≈ 0
20

Antiferromagnetic spin ring
• antiferromagnetic next-neighbor coupling J > 0
crown-shaped field B(r) ∝ ˆr
⇒ spins flip for h > JS ≡ h c
^z e
^m i
B i
stability of Néel-like state:
⇒ sin 2( ¯ϑ − ϑ) = ( JS h )2 sin 2 ( 2π N
) sin(2 ¯ϑ)
ϑ
⇒ assume h JS/N (≪ h c )
• two sublattices with ˆm i ≈ (−1) i ˆn i and ˆn i · h i ≈ 0
• twisted boundary conditions b i+N
= e ±iΩ b i
with Ω = ∑ N
i=1 (−1)i (ω i→i+1 + ω i→i−1 ) = total defect angle of ˆn i
AF spin waves: quantum fluctations ⇒ new magnon vacuum
Ĥ sw = ∑ k≥0
ɛ k (α † k α k + β † k β k − 1), ɛ k = 2JS sin(ka) ∝ k
20

Persistent spin current carried by AF magnons
• spin current: I s (Ω) = J〈ˆn i · (S ⊥ i
• magnetization current:
I m (Ω) = − 2gµ ∑
B
L
T = 0
−→
k
I m = I 0 m
(
c k n k + 1 )
2
(
1 − Ω π
× S ⊥ i+1)〉 = −∂F sw /∂Ω
)
n k =
1
e ɛ k/T − 1 ,
I 0 m = gµ Bc 0
L
= gµ B
c k = 1
JS
N
∂ɛ k
∂k
21

Persistent spin current carried by AF magnons
• spin current: I s (Ω) = J〈ˆn i · (S ⊥ i
• magnetization current:
I m (Ω) = − 2gµ ∑
B
L
T = 0
−→
k
I m = I 0 m
(
c k n k + 1 )
2
(
1 − Ω π
× S ⊥ i+1)〉 = −∂F sw /∂Ω
)
n k =
1
e ɛ k/T − 1 ,
I 0 m = gµ Bc 0
L
= gµ B
c k = 1
JS
N
∂ɛ k
∂k
integer spin: Haldane gap = 2JS∆ H ∝ e −πS ∝ a/ξ , ∆ H (S =1) ≈ 0.2
I m / I 0 m
1
0
-1
N=100
∆ H =0
∆ H =0.1/N
∆ H =1/N
∆ H =2/N
0 1 2
Ω / π
“modified” spin-wave theory: suppress
Neél order on average ∑ i 〈S i · m i 〉 = 0
via staggered field
∆ H ≫ 1 N ⇒ I m
I 0 m
(Takahashi 1989, Hirsch & Tang 1989)
∝
√
L
ξ e−L/ξ sin Ω
21

4. Conclusion & outlook
persistent spin currents in Heisenberg magnets:
• mesoscopic quantum interference effect
• bosonic analogue of charge currents in metal rings:
• magnetization transported by magnons due to
◮ thermal fluctuations (ferromagnets)
◮ quantum fluctuations (antiferromagnets)
Ω
2π ↔ φ φ 0
• spin current induces dipole-like electric field
• experimental requirements:
◮ well-defined Heisenberg rings
◮ submicron field inhomogeneities
◮ detection of nanovoltages
open questions:
• diffusive regime of disordered magnets
22

Metallasiloxanolates
current activities in DFG-Forschergruppe FOR 412: (Molodtsova et al. 2003)
C 68 H 172 Cu 10 N 16 O 56 Si 20
23