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Page 1: Spin-wave calculations for Heisenberg magnets with reduced ...kreisel/Spinwaves_disputation.pdf · Spin-wave calculations for Heisenberg magnets with reduced symmetry Andreas Kreisel,

1

Spin-wave calculations for Heisenberg magnets with

reduced symmetryAndreas Kreisel, Frankfurt 14. Juli 2011

Institut für Theoretische PhysikGoethe Universität FrankfurtGermany

Page 2: Spin-wave calculations for Heisenberg magnets with reduced ...kreisel/Spinwaves_disputation.pdf · Spin-wave calculations for Heisenberg magnets with reduced symmetry Andreas Kreisel,

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1.1 Introduction: Magnetism

� Types of magnetism

� Diamagnetismresponse to magnetic field due to Lenz� rule

� Paramagnetismalignment of magnetic moment in a magnetic field

� Collective magnetismcorrelation effects of spin degrees of freedom

(phase transitions)

 =@ ~M

@ ~H

~L j0i = 0 ~S j0i = 0

Â! 1

 = ¡ e2

6m¹0nr

2aZa ¼ ¡10¡4 < 0

 » 1

T> 0

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1.2 Introduction: Collective magnetism

� Itinerant magnetism in metalsmagnetization from occupation of states

� Magnetism on mesoscopic scales

� Magnetism of localized spins in insulators

Heisenberg model

M » n" ¡ n#

Fe18 CsFe8

finite numbers of spins: nevertheless collective behaviour

method: exact diagonalisationWaldmann et al. PRL ('06)

W. Heisenberg Z. Phys. 49, 619 (1928)

H = J ~S1 ¢ ~S2 ! H =1

2

X

ij

Jij ~Si ¢ ~Sj

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0

1

2

−1

−2

1.3 Introduction:Spin wave theory (method)

� determine classical groundstate

� expand in terms of bosons� Holstein-Primakoff

transformation� Expansion in powers of 1/S

exact for , but� Interacting theory of bosons� Methods of quantum mechanics� Theory of many particle systems

H =X

~k

E~kby~kb~k +

X

~k1;~k2;~k3

¡3(~k1;~k2; ~k3)by~k1b~k2b~k3 +

X

1;2;3;4

¡4(1; 2; 3; 4)by1by2b3b4 + : : :

Holstein, PrimakoffPhys. Rev. ('40)

H=X

ij

Jij ~Si ¢~Sj

S ! 1 S = O(1)

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1.4 Introduction:Spin waves (general results)

� Ferromagnet(exchange interaction only)

� Quadratic excitation spectrum� Vanishing interaction at k=0

� Antiferromagnet� Linear spectrum

(Goldstone mode)� Two modes in magnetic field

(2 sublattices)� Divergent interaction vertices

¡4 »s

j~k1jj~k2jj~k3jj~k4j

Ã

1§~k1 ¢ ~k2j~k1jj~k2j

!

¡4 » ¡(~k1 ¢ ~k2 + ~k3 ¢ ~k4)

Hasselmann, Kopietz, EPL ('06)Chernychev, Zhitomirsky, PRB ('09)Veillette et al. PRB ('05)AK, et al. PRB ('08)

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2.1 Model system 1:Thin-film ferromagnet

� Motivation: Experiments on YIG (yttrium iron garnet) films

� Excitation and detection of spin waves with high energy resolution (parametric pumping, Brillouin light scattering spectroscopy)

� Low spin-wave damping in YIG� Good experimental control

103

104

105

2.5

3

3.5

4

4.5

Parametric pumping of magnons at high k-vectors creates magnetic excitations

Observation of the occupation number using microwave antennas or Brillouin Light Scattering (BLS)Sandweg, et al.

Rev. Sci. Instrum. ('10)

Condensation of magnons at room temperature!Demokritov et al. Nature ('06)

Open questions:Time evolution of magnons:Non-equilibrium physics of interacting quasiparticles?Coupling to other degrees of freedom, thermalization?Kloss, Kopietz PRB ('11)

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2.2 Model system 1:Microscopic model

Crystal structure of YIG Microscopic Hamiltonian

quantum spin S ferromagnet

dipole-dipoleinteractions

Zeeman term

Hmag=¡1

2

X

ij

JijS i ¢ Sj ¡ hX

i

Szi

¡ 1

2

X

i

X

j 6=i

¹2

jrij j3[3(S i ¢ rij)(Sj ¢ rij)¡ S i ¢ S j ]

reduced symmetry

Magnetic system:40 magnetic ions in elementary cell40 magnetic bands

Gilleo, Geller Phys. Rev. (�58)

space group: Ia3dY: 24(c) whiteFe: 24(d) greenFe: 16(a) brownO: 96(h) red

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2.3 Model system 1:Linear spin-wave theory

� Geometry (thin film)

� Numerical approach� Ewald summation technique� Diagonalization of 2N x 2N matrix

� Analytic approaches� Approximation for lowest mode� Bogoliubov transformation

� No dipolar interaction: known result

H2 =

ÃA ~k B ~kB ¤¡~k ¡A T

¡~k

!

E~k =

q[h+ ½ex~k2 +¢(1¡ f~k) sin

2£~k][h+ ½ex~k2 +¢f~k]

¢ = 4¼¹MS

¢ = 0E~k = h+ ½ex~k2

structure factor

Page 9: Spin-wave calculations for Heisenberg magnets with reduced ...kreisel/Spinwaves_disputation.pdf · Spin-wave calculations for Heisenberg magnets with reduced symmetry Andreas Kreisel,

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2.4 Model system 1:Results for spectra

E0 =ph(h+ 4¼¹Ms)

Hybridization: surface mode

Minimum for BECDemokritov et al.

Nature (�06)

102

103

104

105

106

2.5

3

3.5

4

4.5

5

5.5

kz

(cm−1

)

Ek

z

(GH

z)

k◦

E0

Θ = 0

102

103

104

105

106

4

4.5

5

5.5

ky

(cm−1

)

Ek

y

(GH

z)

E0

Θk = 90◦

� Parallel mode

� Perpendicular mode

d = 400a ¼ 0:5¹mN = 400He = 700 Oe

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2.5: Model system 1:Comparison to experiments

−4−2

02

4x 10

4

01

2x 105

2.5

3

3.5

4

4.5

ky

(cm−1

)

Ek

(GH

z)

kz

(cm−1

)

� Excitation and detectionof spinwaves usingBrillouin light scatteringspectroscopy (BLS)

� Outlook, future investigations� Condensation of magnons

(nonequilibrium, finite momentum)

� Interactions(magnon-magnon, magnon-phonon)

Hick et al. EPJB ('10)

£~k = 90±

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3.1 Model system 2:Triangular antiferromagnet

� Motivation: Frustrated magnets� Rich phase diagram� Strong quantum effects

due to large fluctuations� Exactly known microscopic model for the

frustrated antiferromagnet Cs2CuCl4

Coldea et al. PRL ('02)

ordered phaseat T<0.6 KColdea et al.PRB ('03)

J=0.37 meVJ'=0.13 meVD=0.02 meV

Hspin =1

2

X

ij

[JijS i ¢ Sj + D ij ¢ (S i £ Sj)]¡X

i

h ¢ S i ;

Jij = J(R i ¡ R j) =

½J if R i ¡ R j = §(±1 + ±2)J 0 if R i ¡ R j = §±1 or §±2

D ij = §Dez

spatially anisotropicexchange

Dzyaloshinsky-Moriyaanisotropy

magnetic field(perpendicular to plane)

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3.2 Model system 2:Spin-wave approach

� Classical groundstate:�cone state�

� Spin-wave spectra

� Interactions� Large for finite field

symmetric with respect to k

antisymmetric, but(Dzyaloshinsky-Moriaanisotropy)

/ k3

0 5 100

100

200

300

400

500

600

magnetic Field (T)

velo

city (

m/s

)

vx

vy

Goldstone mode: U(1) symmetry

Veillette et al. PRB ('05)

Projectionon the plane

¡bybyb3 (k1;k2;k3) ¼ ¡ h

hc

r1¡

³ hhc

´2p2Si

hcS

Ek =qv2xk

2x + v

2yk2y +O(k3)

Ek =q(A+k )

2 ¡B2k +A¡k 6= E¡k

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3.3 Model system 2:Coupling to lattice vibrations

� Include lattice vibrations

� Spin-phonon coupling via expansion of exchange integrals

phonon dispersion (acoustic)

exchange DM anisotropy

magnon phonon coupling

Chakraborty, TuckerPhysica A ('87)

bare exchange

H =1

2

X

ij

[JijSi ¢ Sj + Dij ¢(Si£Sj)]¡X

i

h ¢Si

+X

·P¡k¸Pk¸2M

+M

2!2k¸X¡k¸Xk¸

¸

Jij = J(R ij) + (X ij ¢rR )J(r)jr=rij + : : : = J(R ij) + X ij ¢ J (1)ij + : : :

!k¸ = c¸(k)jkj

Dij ¼ D(R ij)

Page 14: Spin-wave calculations for Heisenberg magnets with reduced ...kreisel/Spinwaves_disputation.pdf · Spin-wave calculations for Heisenberg magnets with reduced symmetry Andreas Kreisel,

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3.4 Model system 2:Calculation of observables

� Shift of the phonon velocity� Integrate out magnetic degrees of freedom

� Ultrasound attenuation rate� Diagrammatic perturbation theory (1/S expansion)

e¡S2pho

eff[X ] =

ZD[¯; ¹]e¡S2pho[X ]¡S2mag[ ¹;¯]¡S

1pho1mag[X ; ¹;¯]

(¢c¸)totc¸

= limjkj!0

³s

1¡ §pho0 (k; ¸)

!2k¸¡ 1 +

¯¡X¯k ¢ ek¸

¯2

2M!3k¸

´

°k¸ = ¡ Im§pho2 (!k¸ + i0+;k; ¸)

2!k¸

Two contributions:1. Classical spin background2. Hybridization tomagnetoelastic waves

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3.5 Model system 2:Comparison to experiments

� Model

� Shift of ultrasound velocityfor c22 mode (P. T. Cong):

� Fix parameters:� Project result for c33 mode

without adjustable parameters� Attenuation rate

calculate from parameters

0 2 4 6 8

−15

−10

−5

0

x 10−4

B (T)

∆c/c

0 1 2−8

−6

−4

−2

0

2x 10

−5

B (T)

∆c/c

°k¸ ¼ ¼2

64

µk2

2M

¶µS2c2¸k

2

VBZvxvy

¶hfX¯1 (k) ¢ ek¸

i2

(1¡ h=hc)2

·¼ 15 ·0 ¼ 51

J(x) = J(b)e¡·(x¡b)=b

J 0(r) = J 0(d)e¡·0(r¡d)=d

0 2 4 6 80

0.2

0.4

0.6

0.8

B (T)

(db

/cm

)

T ¼ 50 mK

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4 Summary

� Theoretical investigations on Heisenberg magnets with reduced symmetry

� 3 different model systems (2 presented in detail)

� Application and further development of concepts using the spin-wave approach

� Connection to recent experimental research and comparison of corresponding results

Page 17: Spin-wave calculations for Heisenberg magnets with reduced ...kreisel/Spinwaves_disputation.pdf · Spin-wave calculations for Heisenberg magnets with reduced symmetry Andreas Kreisel,

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5.1 Experimental detailsBLS spectroscopy

� Wavevector resolvedBLS setup(C. Sandweg,TU Kaiserslautern)

� large wavevector range

� high resolution

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5.2: Experimental DetailsBLS spectroscopy

� Scattering of light ongrating created fromspin waves

cut along line

wavenumber from geometryenergy from interferometer

£~k = 90±

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5.3 Experimental detailsTechniques: Thin-film magnets

� Inverse spin-hall effect (ISHE):spin polarized current

C. Sandweg, et al., Appl. Phys. Lett, '10

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5.4 Experimental detailsMaterial parameters for YIG

� Heisenberg magnet withdipole-dipole interactions

material parameters

Gilleo et al. �58

Cherepanov et al. �93

Tittmann �73

a = 12:376 ºA

4¼Ms = 1750 G

Tupitsyn et al. �08

alternatively

½ex¹= 5:17¢10¡13Oe m2

J=1:29 KS=14:2 ¹ :=2¹B

Hmag=¡12

X

ij

JijS i ¢ Sj ¡ hX

i

Szi

¡ 1

2

X

i

X

j 6=i

¹2

jrij j3[3(S i ¢ rij)(S j ¢ rij)¡ S i ¢ S j]

Crystal structure:space group: Ia3dY: 24(c) whiteFe: 24(d) greenFe: 16(a) brownO: 96(h) red

Gilleo et al. �58

Magnetic system:40 magnetic ions in elementary cell40 magnetic bands

Elastic system:160 atoms inelementary cell3x160 phonon bands

Druzinin et. al

Sov. Phys. Solid. ('81)

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5.5 Experimental detailsUltrasonic technique

� pulse echo method (phase-sensitive detection technique), P. T. Cong (Uni FFM)

� sound velocity� change of sound velocity� (relative) attenuation rate

c¸ = L=t0

¢® / 1

Lln(A1=A0)

df

f=dc¸c¸

Page 22: Spin-wave calculations for Heisenberg magnets with reduced ...kreisel/Spinwaves_disputation.pdf · Spin-wave calculations for Heisenberg magnets with reduced symmetry Andreas Kreisel,

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6.1 Theoretical detailsQAF in a magnetic field

� model Hamiltonian

� spin-wave modes

� spin-wave velocities

H =1

2

X

ij

JijS i ¢ Sj ¡ hX

i

Szi

E2~k+ = h2 + c2+~k2;

E2~k¡ = c2¡~k2;

c2+ = c20(1¡3

¢20h2)

c2¡ = c20(1¡1

¢20h2)

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6.2 Theoretical detailsQAF in a magnetic field

� Hermitian parametrization (compare harmonic oscillator)

� physical meaning for

uniform spin fluctuations

staggered spin fluctuations� effective action

X~k¾ :

P~k¾ :

k ¿ 1

ª~k¾ = p¾

"rº~k¾2X~k¾ +

ip2º~k¾

P~k¾

# [X~k¾; P~k0¾0 ] = i±~k;¡~k0±¾;¾0

Se® [X¾] = S0 +¯

2

X

E2k¾ + !2

¢k¾X¡K¾XK¾

r2

N

X

K1K2K3

±K1+K2+K3;0

h 13!¡(3)¡¡¡(K1;K2;K3)XK1¡XK2¡XK3¡

+1

2!¡(3)¡++(K1;K2; K3)XK1¡XK2+XK3+

i

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6.3 Theoretical detailsQAF in a magnetic field

� diagrammatic perturbation theory

� perturbation theory: 1/S corrections to self energy

G¾(K) =¢k¾

E2k¾ + !2

S inte® [X¾] = ¯q

2N

Ph13!V(3)¡ X¡X¡X¡ +

12!V(3)+ X¡X+X+

i

no frequency dependence, negligible

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6.4 Theoretical detailsQAF in a magnetic field

� results� spin-wave velocity

� magnon damping

~h =h

¢0

nonanalytic in h²

¡~k¡ / 1

S

µh

hc

¶2µq6 ¹A¡

¶D¡3aD+1j~kj2D¡1

¡~k ¼ 0 ¡~k / k2D¡1

c2¡c20

¼ 1¡ 6p3 ~h2

¼2Sln

µ2~h

¶D = 3

c2¡c20

¼ 1¡ 2~h

¼SD = 2

Page 26: Spin-wave calculations for Heisenberg magnets with reduced ...kreisel/Spinwaves_disputation.pdf · Spin-wave calculations for Heisenberg magnets with reduced symmetry Andreas Kreisel,

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6.5 Theoretical detailsSpin-wave theory for thin films

� spectrum for realistic samples

101

102

103

104

105

106

2

2.5

3

3.5

4

4.5

kz

[cm−1

]

Ek

z

[GH

z]

Θk = 0◦

E0

101

102

103

104

105

106

3.6

3.7

3.8

3.9

4

4.1

4.2

4.3

4.4

4.5

ky

[cm−1

]

Ek

y

[GH

z]

Θk = 90◦

E0

Es

d = 4040a He = 700 Oe

Page 27: Spin-wave calculations for Heisenberg magnets with reduced ...kreisel/Spinwaves_disputation.pdf · Spin-wave calculations for Heisenberg magnets with reduced symmetry Andreas Kreisel,

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� Comparison� analytical result

� uniform mode approximation

� eigenmode approximation

� numerical result

¢ = 4¼¹MS

0 5 10 15 200

0.2

0.4

0.6

0.8

1

f k

dk

6.6: Theoretical detailsSpin-wave theory for thin films

E~k =

q[h+ ½ex~k2 +¢(1¡ f~k) sin

2£~k][h+ ½ex~k2 +¢f~k]

f~k =1¡ e¡j~kjd

j~kjd

f~k = 1¡j~kdj j~kdj3+j~kdj¼2+2¼2(1+e¡j~kdj)

(~k2d2 + ¼2)2

102

103

104

105

106

−0.02

0

0.02

0.04

0.06

0.08

kz

[cm−1

]

∆E

kz

[GH

z]

Θk = 0◦

101

102

103

104

105

106

0

0.02

0.04

0.06

0.08

kz

[cm−1

]

∆E

kz

[GH

z]

Θk = 0◦