Blaubeuren 2006 Relaxation mechanisms in exchange coupled spin systems – I Line broadening and the...

Blaubeuren 2006Blaubeuren 2006

Relaxation mechanisms in exchange coupled spin systems – I

Line broadening and the Kubo-Tomito approach

Joachim Deisenhofer

Université de Genève

Blaubeuren6th September 2006


Outline

ESR in exchange-coupled system: definition of the problem

linear response and ESR:

the Kubo-Tomita approach

the effective spin Hamiltonian and the broadening mechanisms

summary


Motivation - ESR in transition-metal compounds

ESR locally probes the spin of interest

low-dimensional spin systems–S = 1/2 chains: LiCuVO4, CuGeO3, TiOCl - and ladders: NaV2O5 –chains with larger spin PbNi2V2O8 (S = 1), (NH4)2MnF5 (S =2)–2D honeycomb-lattice BaNi2V2O8

metal-insulator-transitions–heavy-fermion-properties in Gd1-xSrxTiO3 –spin-state transitions in GdBaCo2O5+

materials with colossal magnetoresistance effect–magnetic structure in thiospinels FeCr2S4, MnCr2S4

–orbital ordering in La1-xSrxMnO3


parameters: intensity:local spin susceptibility

resonance field:g = g - 2.0023local symmetry

linewidth H:spin-relaxation,anisotropic interactions

The absorption signal

resBHg


Why one Lorentzian only? “Exchange narrowing”

isotropic exchange interaction between neighbouring spins causes local field fluctuations

originally Gaussian lineshapes are therefore “narrowed“ in the middle and extendend in the wings

phenomenon similar to “motional narrowing“ in liquids as seen by NMR


~

What are we measuring? - Definition of the problem

usual case: Faraday configuration;microwave field perpendicular to static magnetic field couples to the spin

Microwave absorption in the cavity is given by

Problem: What is the (linear) response function of the spin system to the microwave perturbation?


ESR in Linear Response – ingredients

observed quantity:

time-dependent perturbation:

general spin susceptibility:

Heisenberg equation of motion:

Hamiltonian of the spin system:


The effective spin Hamiltonian

Hamiltonian for strongly correlated spin systems:

H'H i

iii

i Jg 1B SSSH

Zeeman energy

isotropicexchange

additionalinteractions

strong isotropic coupling averages local fields similar to

fast movements of the spins “exchange narrowing“ of the ESR signal

local fluctuating fields local, static resonance shift inhomogenous broadening of

the ESR signal

e.g. crystal fieldanisotropic exchangedipole-dipole interactionhyperfine interaction


final general expression:

with

lineshape is completely determined by

: resonance line infinitely sharp at ω=ω0 (δ –peak)

assuming a Lorentzian lineshape and

gives linewidth:

and resonance shift:

Where does linear response theory take us?

Oshikawa and Affleck, PRB 65, 134410 (2002)

Nagata and Tatsuke, J. Phys. Soc. Jpn. 32, 337 (1972)


The Kubo-Tomita formula for the linewidth

T ∞: high-temperature approximation

correlation function decays exponentially:

characteristic time is governed by isotropic exchange J:

equivalent expression:

R. Kubo and K. Tomita, J. Phys. Soc. Jpn. 9, 888 (1954)


Characteristics of the Kubo-Tomita approach - I

temperature dependence is governed by the static susceptibility

describes many 3D magnetic systems, e.g. manganites

Causa et al., PRB 58, 3233 (1998)


Characteristics of the Kubo-Tomita approach - II

anisotropies completely contained in the second Moment M2:

remaining tasks:

calculate the second moment for the different contributions to the spin Hamiltonian, find the dominating line-broadening mechanism (and check for the anisotropy)


The effective spin Hamiltonian

Hamiltonian for strongly correlated spin systems:

H'H i

iii

i Jg 1B SSSH

Zeeman energy

isotropicexchange

additionalinteractions

strong isotropic coupling averages local fields similar to

fast movements of the spins “exchange narrowing“ of the ESR signal

local fluctuating fields local, static resonance shift inhomogenous broadening of

the ESR signal

e.g. crystal fieldanisotropic exchangedipole-dipole interactionhyperfine interaction


Line broadening mechanisms:

Zero-Field Splitting (crystal field)

Hyperfine interaction

Dipole-dipole interaction

Anisotropic Zeeman interaction

Anisotropic exchange interactions:

Symmetric anisotropic exchange

Antisymmetric anisotropic exchange(Dzyaloshinsky-Moriya interaction)


Crystal field – zero-field splitting

two independent elements for orthorhombic symmetry:

perturbation theory (spin-orbit coupling perturbs the crystal-field levels):

important source of line broadening and anisotropy (e.g. in manganites)

i

iyixiz SSEDS )( 222ZFSH



Different Cu sites give rise to superposition of resonance lines, e.g. in CuGeO3 (1D Heisenberg spin-chain)

exchange coupling along the b-directionyields then only one broadened resonance line

Second moment depends on the external field! good candidate for field-dependent linewidth

B. Pilawa, J. Phys.: Cond. Mat. 9, 3779 (1997)



Zero-Field Splitting (crystal field)

Hyperfine interaction

Dipole-dipole interaction


Anisotropic exchange interactions:

Symmetric anisotropic exchange

Antisymmetric anisotropic exchange(Dzyaloshinsky-Moriya interaction)


magnetic ions interact indirectly via an intermediate diamagnetic ion (O2-, F1-,..)

potential exchange: describes the self-energy of the charge distribution → ferromagnetic

kinetic exchange:electrons can hop,stabilization of the singletover the triplet state :→ antiferromagnetic

perturbationtreatment:

Reminder: Isotropic superexchange

2

ˆ ˆˆ, with h.c.

1 2 2 , 2 .ab a b ab

V VV t a b

tJ S S J

H H

H H


Mechanism of anisotropic exchange interaction

This effect adds to the isotropic exchange interaction an anisotropic part (dominant source of anisotropy for S=½ systems!)

free spin couples to the lattice via the spin-orbit interactionHLS=(l·s)

excited orbital states are involved in the exchange process

described as virtual hoppings of electrons via the excited orbital states (additional perturbation term – (LS)-coupling – acts on one site between the orbital levels)


Theoretical treatment - perturbation theory

4th order: describes 4 virtual electrons hoppings Isotropic superexchange

5th order: 4 hoppings + on-site (LS)-coupling Antisymmetric part of anisotropic exchange = Dzyaloshinsky-Moriya interaction

6th order: 4 hoppings + 2 times on-site (LS)-coupling Symmetric part of anisotropic exchange

= Pseudo-dipol interaction


Effective spin Hamiltonian of the antisymmetric exchange in form of a cross-product:

direction of D (Dzyaloshinsky-Moriya vector)

Perturbative result:

Antisymmetric part of anisotropic exchange

2j ja b

iD l J

S S

sa sb

ra rb

dVVVVVVVVVVVVVVab a bD r r

j = {x, y, z}, – orbital levels, – energy splitting,lj – operator of the LS-coupling,J – exchange integral.

jiij SSD

sssl baaa

DM

IsoSELSDM

H

HHH


Symmetric part of anisotropic exchange

Is

2

oSE

( )Г

8

AE LS LS a a a b a a

AE a ab

a b a b

aa

a

b

a b

l s s s l s

S S

lJ S S S S

S S

l

H H H H

H I >

I> I>

1 3 2

I I>

ba

, = {x, y, z};’ – orbital

levels.

Effective exchange constant of the pseudo-dipol interaction is a tensor of second rank and does not allow a simple graphical presentation:

Nonzero elements of can be determined by the product of the matrix elements of the (LS)-coupling and the hopping integrals.


Uffhh – How to sort out things?

Things are not as bad as they might look like:

Second moments of the broadening mechanisms are known

Geometry and order of magnitude estimations often allow to single out the dominant interactions

Within the KT-approach the temperature dependence is the same for all spin-spin interactions anisotropies are “very helpful”

But… keep in mind:

lots of questions for one single broad absorption line

justification of the KT- approach (dimension)?

possibility of spin-lattice (phonon) relaxation


summary and outlook

High-temperature KT-approach describes well 3D magnets

Lineshape completely determined by perturbation term

Dominating relaxation mechanism has to explain order of magnitude and anisotropy

Tomorrow:

Success of the KT-approach: the case of manganites

Limits of the KT approach - the case of linear spin chains:e.g. NaV2O5, LiCuVO4, CuGeO3, TiOCl


Relaxation mechanisms in exchange coupled spin systems – II

Two case files: the manganite system La1-xSrxMnO3 and ESR in spin chain compounds

Joachim Deisenhofer

Université de Genève

Blaubeuren7th September 2006


outline

Part 1: the manganite system La1-xSrxMnO3

Phase diagram of La1-xSrxMnO3

ESR results

spin relaxation in La0.95Sr0.05MnO3: orbital ordering

Part 2: ESR in 1D Heisenberg spin chains

Magnetic properties of spin S = 1/2 chains

temperature dependence of the ESR linewidth in the model systems NaV2O5, LiCuVO4, CuGeO3

the case of TiOCl


phase diagram of La1-xSrxMnO3

Hemberger et al., PRB 66, 094410 (2002).


Jahn-Teller distortion in LaMnO3

cooperative JT-distortions of MnO6 octahedra in the orthorhombic O'-phase up to TJT

750 K

A-type AFM (TN = 140 K):

FM coupling in ac-planes AFM coupling between ac-planes

Huang et al., PRB 55, 14987 (1997)


estimate for JT? (~0.1eV instead of ~1eV)

orbital excitations (orbitons) in LaMnO3 or multiphonon processes?

OO due to electron-phonon and/or electron-electron interaction?

What is the value of ? Suggestions: 90°, 102°, 106°, 120°

orbital physics in LaMnO3

OO

?

2222

2sin3

2cos yxrzg

xy

z

xy

z


parameters:intensity:local spin susceptibility

resonance field:g = g - 2.0023local symmetry

linewidth H:spin-relaxation,anisotropic interactions

the ESR Signal

resBHg

3.3 3.4 3.5 3.6

intensity

Hres

linewidth2 H

Abso

rptio

n P

H (kOe)

9.4 GHz LorentzianES

R s

igna

l dP/

dH


0.00 0.05 0.10 0.15 0.200

200

400

600

800

1000

O"/I / FM O / M

FM

R / MFM

R / MPM

O / I PM

O'/IPM

T (

K)

Sr concentration x

O'/I/CA/A-type

ESR in La1-xSrxMnO3 (0 x 0.2)

Ivanshin et al., PRB 61, 6213 (2000)

phase diagram

Paraskevopoulos et al., J. Phys.:Cond.Mat. 12, 3993 (2000).

linewidth

La0.95Sr0.05MnO3:

still an AFM insulator (TN = 140 K)reduced JT transition temperature TJT = 600 K effective treatment like LaMnO3


T-dependence of the linewidth

0 200 400 6001.0

1.5

2.0

2.5

ac-plane: FM coupling

b-axis : AFM coupling

O' O

H || a H || c H || b

La0.95

Sr0.05

MnO3

H

(kO

e)

T (K)



Observed linewidth: ~ 1-2 kOe

• Zero-Field Splitting (crystal field): ~1 kOe

• Anisotropic exchange interactions:

Symmetric anisotropic exchange ~1 Oe

Antisymmetric anisotropic exchange ~1 kOe(Dzyaloshinsky-Moriya interaction)

• Hyperfine interaction: ~10 Oe

• Dipole-dipole interaction ~1 Oe

• Anisotropic Zeeman interaction ~1 Oe


Crystal field – zero-field splitting

two independent elements for orthorhombic symmetry:

perturbation theory (spin-orbit coupling perturbs the crystal-field levels):

i

iyixiz SSEDS )( 222ZFSH


Effective spin Hamiltonian of the antisymmetric exchange in form of a cross-product:

direction of D (Dzyaloshinsky-Moriya vector)

Dzyaloshinsky-Moriya interaction

jiij SSD

DMH


200 400 6000.5

1.0

1.5

2.0

2.5

TJT

= 605 K

T(K)

9.35 GHz

O O'

H || a H || b H || c

La0.95

Sr0.05

MnO3

H (

kOe)

fitting the T-dependence of the linewidth

(E/D)

6(E/D)1)(

),(),(

2

),(

div

N

NCFDregCF

JTDM

CW fTT

Tf

T

T

T

TTTH

3(+1) fit parameters:ZFS = 0.57 kOe

DM = 1.0 kOe

= 0.16 (critical JT-exponent)

+

ZFS-ratio E/D=0.37

Kochelaev et al., Mod. Phys. Lett. B 17, 459 (2003).

JD et al., PRB 68, 214427 (2003).


consistent ESR fitparameters

ZFS ratio E/D=0.37was obtained from fitting

theg-factor anisotropy

Deisenhofer et al., PRB 68, 214427 (2003). .

100 200 300

1.9

2.0

2.11

2

3

La0.95

Sr0.05

MnO3

gef

f

34 GHz

T (K)

H || a H || b H || c

H

(kO

e)

200 400 6000.5

1.0

1.5

2.0

2.5

TJT

= 605 K

T(K)

9.35 GHz

O O'

H || a H || b H || c

La0.95

Sr0.05

MnO3

H (

kOe)


zero-field splitting and orbital order

/3tan

sin)/(3-

cos)/-3(2

2

DE

E

D

perturbation theory:

ESR=106

°

ESR

ND=106°

21

221

222

23JT

/2/tan

3

cc

yxcrzc

TQTQg

g

iixiizi

H

neutron diffraction

[Rodriguez-Carvajal et al., PRB 57, R3189 (1998)]

excellentagreement!


summary - part I

KT-approach allows to describe the temperature dependence and the anisotropy of the linewidth

Dominant contribution are the ZFS and the DM interaction in agreement with neutron diffraction data

type of orbital ordering in La0.95Sr0.05MnO3 has been derived from the analysis of the ESR g-factor and linewidth (ZFS)


outline

Part 1: the manganite system La1-xSrxMnO3

Phase diagram of La1-xSrxMnO3

ESR results

spin relaxation in La0.95Sr0.05MnO3: orbital ordering

Part 2: ESR in 1D Heisenberg spin chains

Magnetic properties of spin S = 1/2 chains

temperature dependence of the ESR linewidth in the model systems NaV2O5, LiCuVO4, CuGeO3

the case of TiOCl


Susceptibility of free spins

0 100 200 3000

20

40

60

80

Curie law = M/H ~ 1/T

M = gBS

i

susceptibility

spin Si

magnetic field H

magnetization

(

10-4 e

mu/

mol

)

temperature T(K)


Interacting spins (3D)

0 100 200 3000

20

40

60

80

-J SiS

j

TN

~1/T

Heisenberg exchange

~ 1/(T - )

Curie-Weiss ~ J

antiferromagnet < 0

order T < TN ~ J

(

10-4 e

mu/

mol

)

temperature T(K)


Interacting spins (1D)

0 100 200 3000

20

40

susceptibility (S=1/2)Bonner-Fisher(numerical)

Curie-Weiss for T>Jmaximum at T

MAX~J

no magnetic order !

Si

J

TMAX

~(T-)-1

antiferromagnetic chain

(10-4

em

u/m

ol)

temperature T(K)


Real spin chains

• Only in ideal 1D antiferromagnets no phase transition

• In real systems:

– weak inter-chain coupling not negligible 3D antiferromagnetic order at T << J

– electron-phonon interaction Spin-Peierls transition into dimerized ground state


Spin-Peierls transition

0 1000

20

= (T )

Si+S

i+1 = 0

T < TSP

dimerization

TSP

(T<TSP

) ~ exp(-/T )

Si

J

TMAX

~J

antiferromagnetic chain

(10-4

em

u/m

ol)

temperature T(K)


Model systems

LiCuVO4

Cu2+

S = 1/2 chain

J = 40 K

TN = 2.1 K

antiferromagnetic

order

NaV2O5

S = 1/2 per 2 V4.5+

¼-filled ladder

J = 570 K

TCO = 34 K

dimerization

via

charge order

CuGeO3

Cu2+

S = 1/2 chain

J = 120 K

TSP = 14 K

dimerized,

spin-Peierls S = 0

ground state


Temperature dependence of the ESR linewidth

LiCuVO4 CuGeO3 NaV2O5

0 100 200 3000.0

0.5

1.0

1.5

2.0

TN

H || a H || b H || c

LiCuVO4

H (

kOe

)

T (K)0 100 200 300

0.0

0.5

1.0

1.5

TSP

CuGeO3

H || a H || b H || c

T(K)0 200 400 600

0.0

0.2

0.4

TCO

NaV2O

5

H || a H || b H || c

T(K)


Universal temperature law

0 100 200 3000.0

0.5

1.0

1.5

2.0

C1 = 60 (5) K

C2 = 15 (5) K

TN

H || a H || b H || c

LiCuVO4

H (

kO

e)

T (K)0 100 200 300

0.0

0.5

1.0

1.5

C1 = 235 (5) K

C2 = 40 (2) K

TSP

CuGeO3

H || a H || b H || c

T(K)0 200 400 600

0.0

0.2

0.4

C1 = 420 (20) K

C2 = 80 (10) K

TCO

NaV2O

5

H || a H || b H || c

T(K)

1

2

( ) ( ) expC

H T HT C


Limits of the KT-approach

High-temperature approximation fails for T < J (!)

Field theoretical approach (M. Oshikawa and I. Affleck, Phys. Rev. B 65, 134410, 2002)

For temperatures T << J :

H (T ) ~ T for symmetric anisotropic exchange

H (T ) ~ 1/T 2 for antisymmetric DM interaction

in LiCuVO4, CuGeO3 and NaV2O5 symmetric anisotropic

exchange is the dominant relaxation process


Universal behavior of the linewidth

low temperatures T << J : H (T) ~ T for symmetric anisotropic exchange H (T) ~ 1/T 2 for antisymmetric DM interaction

0 100 200 3000.0

0.5

1.0

1.5

2.0

TN

H || a H || b H || c

LiCuVO4

H (

kO

e)

T (K)0 100 200 300

0.0

0.5

1.0

1.5

TSP

CuGeO3

H || a H || b H || c

T(K)0 200 400 600

0.0

0.2

0.4

TCO

NaV2O

5

H || a H || b H || c

T(K)

What about the antisymmetric interaction? Observation of a low-temperature 1/T2 divergence due to this interaction?


The system TiOCl

There is no center of inversion between the ions in the Ti-O-layers

antisymmetric anisotropic exchange

[A. Seidel et al., Phys. Rev. B 67, 020405(R) (2003)]

Isotropic exchange constant J = 660 K


Analysis of the anisotropic exchange mechanisms Dzyaloshinsky-Moriya interaction Pseudo-dipol interaction

D is almost parallel to the b-direction

Dominant component of the tensor of the pseudo-dipol interaction is (aa)


Temperature dependence of H

60 90 120 1500

100

200

300

400

500TiOCl

a-axis b-axis c-axis

H (

Oe)

T (K)

Tc1

Tc2

2

1

2

( ) ( ) ( ) expDM AE

CJH T K K

T T C

[Oe] KAE (∞) KDM (∞)

H || a 1429 1.397H || b 765 2.319H || c 930 1.344

The temperature and angular dependence of H can be described as a competition of the symmetric and the antisymmetric exchange interactions!

[Zakharov et al., PRB 73, 094452 (2006)]


Summary

Anisotropic exchange dominates the ESR line broadening in low dimensional S=1/2 transition-metal oxides in contrast to estimations based on the KT-approach

Universal temperature dependence of the ESR linewidth in spin chains with dominant symmetric anisotropic exchange

Interplay of antisymmetric Dzyaloshinsky-Moriya and symmetric anisotropic exchange in TiOCl

Blaubeuren 2006 Relaxation mechanisms in exchange coupled spin systems – I Line broadening and the...

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