experimental and numerical study of the low temperature ... · 2) site 4 : (1 2;0 ; 3 4) 4. the...

Experimental and numerical study of the

low temperature behaviour of LiYbF4

Aurore Finco

Laboratory for Quantum Magnetism,École Polytechnique Fédérale de Lausanne

September 6, 2013

1

Outline

Introduction

The LiREF4 systemCrystalline and ordered structuresHamiltonian

Numerical simulationsMean �eld approximationMonte Carlo calculation

Classical model and algorithm

Results

ExperimentSetup

Susceptometer

Dilution refrigerator

Phase diagram

2

Introduction: LiREF4

I RE = Rare Earth ions

I magnetic : Yb, Ho, Er, Gd, TbI non magnetic : Y

I Dipolar coupled quantum magnets.

I Possible apparition of both ferromagnetic (LiHoF4) andantiferromagnetic (LiErF4, LiYbF4) order.

I Disordered systems with apparition of a spin glass phase:

I Dilution of the magnetic moments with non magnetic Y ionsI Mixing of several RE ions with di�erent anisotropies

(LiHo1−xErxF4 or LiHo1−xYbxF4 for instance)

3

Crystalline structure of LiREF4

site 1

site 2

site 3

site 4

Rare earth ions on the 4

sites in the unit cell

Rare earth ions (in the

adjacent cells)

Li ions

ab

c

Sites coordinates

site 1: (0, 0, 0)

site 2: (0, 12, 14

)

site 3: (12, 12, 12

)

site 4 : (12, 0, 3

4)

4

The Bi-Layered AntiFerroMagnetic (BLAFM) orderedstructure

Moments parallel to the a-axis Moments parallel to the b-axis

5

Hamiltonian of the system

H = Hsingle ion + Hinteractions

Hcf =∑i

∑lmBml O

ml (Ji)

Crystal �eld

HZ = −µB gL ~J.~B

Zeeman term

Hhyp = A ~J ·~IHyper�ne coupling

Hdip = −12

∑i ,jgLi

gLjµ2B

~JiDij~Jj

Dipolar interactions

Hex =∑

i ,j(n.n.)Jij ~Ji · ~Jj

Exchange interactions

Negligible !

6

The crystal �eld

I Electric �eld created by the charge distribution around the ion

I Responsible for the magnetic anisotropy

Hcf =∑i

∑lm

Bm

l Om

l (Ji)

LiYbF4 LiErF4 LiHoF4

103B02 663 ± 80 60.23 -60.0

103B04 12.5 ± 4.5 -0.12 0.350

103B44 (c) 102 ± 41 -4.33 3.60

105B06 -62 ± 73 -0.19 4.0

103B46 (c) -16.0 ± 1.7 -0.085 0.070

106B46 (s) 0 -22.7 9.8

7

The crystal �eld

I Electric �eld created by the charge distribution around the ion

I Responsible for the magnetic anisotropy

Hcf =∑i

∑lm

Bm

l Om

l (Ji)

LiYbF4 LiYbF4 (new) LiErF4 LiHoF4

103B02 663 ± 80 646.2 60.23 -60.0

103B04 12.5 ± 4.5 15.3 -0.12 0.350

103B44 (c) 102 ± 41 116.5 -4.33 3.60

105B06 -62 ± 73 -68.6 -0.19 4.0

103B46 (c) -16.0 ± 1.7 -15.2 -0.085 0.070

106B46 (s) 0 0 -22.7 9.8

7

Mean �eld approximation

I Fluctuations are neglected and e�ective mean �elds areintroduced :

~hie� =4∑

j=1

Dij〈µB gLj~Jj〉

I Hamiltonians decoupled for each site :

HMF

dip =4∑

i=1

gLiµB ~Ji ·~hie�

I Algorithm :

I Computation of Dij

I Initialization of the moments 〈Ji 〉I Computation of hi

e�for the current con�guration

I Diagonalization of HMFi and update of 〈Ji 〉I Evaluation of ∆ =

∑i

|〈Ji 〉new − 〈Ji 〉old |8

Monte Carlo calculation: the quantum e�ective model

Ho Er Yb

11.5 K

25.7 K

334.8 K

E�ective model for H0 = Hcf +HZ

I Diagonalization of H0

I Projection of J and H0 on thesubspace of the doublet

I Diagonalization of Jz

I Obtention of a basis {|+〉 , |−〉}

State:

|α, β〉 = cos(α) |+〉+ e iβ sin(α) |−〉

where α ∈ [0, π2

] and β ∈ [0, 2π].

9

The classical e�ective model

I 2 parameters: α and β

I Classical moment: ~Je� = 〈α, β|~J |α, β〉

x

y

z

~Je�

•ϕ

θ

Distribution of θ

−π2

0 π2

0

1

2

θ

Occurrences·10−4

LiYbF4

−π2

0 π2

0

400

800

θ

Occurrences

LiErF4

−π2

0 π2

0

1

2

θOccurrences·10−4

LiHoF4

10

The classical e�ective model

I 2 parameters: α and β

I Classical moment: ~Je� = 〈α, β|~J |α, β〉

x

y

z

~Je�

•ϕ

θ

Distribution of θ

−π2

0 π2

0

0.05

0.1

θ

Occurrences·10−4

LiYbF4

−π2

0 π2

0

400

800

θ

Occurrences

LiErF4

−π2

0 π2

0

1

2

θOccurrences·10−4

LiHoF4

10

Critical exponents

C ∝ |T − TN |−α

Jaltxy ∝ |T − TN |β

χ ∝ |T − TN |−γ

Exponent α β γ

Mean-�eld 0 0.5 1

3D Ising -0.11 0.32 1.24XY 0.01 0.35 1.32

Heisenberg 0.12 0.36 1.39

2D Ising 0 0.125 1.75XY/h4 0.1-0.25

LiErF4 (at TN) 0.28 ± 0.04 0.15 ± 0.02 0.82 ± 0.04LiErF4 (at Hc) 0.31 ± 0.02 1.44 ± 0.2

11

Monte Carlo simulation of LiErF4

0 50 100 150 200

0

1

2

3

· 10−4

127

Temperature (mK)

Standarddeviation

(meV)

Energy per ion

• TN = 127 mK Exp: 373 mK

12


0 5 10 15 20 25

0

2

4

0.5 20.8 22.5

External �eld (T)

Quantum phase transition

Jz

Jaltxy • TN = 127 mK Exp: 373 mK• HC = 0.5 T Exp: 0.4 T

12


0 50 100 150 200

0

1

2

3· 10−4

122

Temperature (mK)

Speci�c heat (FDT)

• TN = 127 mK Exp: 373 mK• HC = 0.5 T Exp: 0.4 T• αFDT = 0.33 Exp: 0.28

12


0 50 100 150 200−0.5

0

0.5

1

124

Temperature (mK)

Speci�c heat

• TN = 127 mK Exp: 373 mK• HC = 0.5 T Exp: 0.4 T• αFDT = 0.33 Exp: 0.28• α = 0.27 Exp: 0.28

12


0 50 100 150 2000

1

2

3

124.8

Temperature (mK)

Order parameter

• TN = 127 mK Exp: 373 mK• HC = 0.5 T Exp: 0.4 T• αFDT = 0.33 Exp: 0.28• α = 0.27 Exp: 0.28• β = 0.15 Exp: 0.15

12


0 50 100 150 200

0

1,000

2,000

3,000

126.4

Temperature (mK)

Susceptibility

• TN = 127 mK Exp: 373 mK• HC = 0.5 T Exp: 0.4 T• αFDT = 0.33 Exp: 0.28• α = 0.27 Exp: 0.28• β = 0.15 Exp: 0.15• γ = 0.91 Exp: 0.82

12

Monte Carlo simulation of LiYbF4

Wrong CF parameters !

0 20 40 60 80

· 10−2

0

1

2

3

· 10−5

34

Temperature (mK)

Standarddeviation

(meV)

Energy per ion

• TN = 34.4 mK Exp: 128.6 mK

13



0 10 20

0

0.5

1

1.5

External �eld (T)

Quantum phase transition

Jz

Jaltxy

• TN = 34.4 mK Exp: 128.6 mK• HC = 25 T Exp: 0.48 T

13



0 20 40 60 80

· 10−2

0

0.5

1

1.5

· 10−4

35.2

Temperature (mK)

Speci�c heat (FDT)

• TN = 34.4 mK Exp: 128.6 mK• HC = 25 T Exp: 0.48 T• αFDT = 0.25

13



0 20 40 60 80

· 10−2

0

0.1

0.2

0.3

33.6

Temperature (mK)

Speci�c heat

• TN = 34.4 mK Exp: 128.6 mK• HC = 25 T Exp: 0.48 T• αFDT = 0.25• α = 0.28

13



0 20 40 60 80

· 10−2

0

0.5

1

1.5

34.4

Temperature (mK)

Order parameter

• TN = 34.4 mK Exp: 128.6 mK• HC = 25 T Exp: 0.48 T• αFDT = 0.25• α = 0.28• β = 0.18

13



0 20 40 60 80

· 10−2

0

1,000

2,000

3,000

4,000

34.4

Temperature (mK)

Susceptibility

• TN = 34.4 mK Exp: 128.6 mK• HC = 25 T Exp: 0.48 T• αFDT = 0.25• α = 0.28• β = 0.18• γ = 0.90

13

AC susceptibility measurements

χ′ =ω

µ0Ha0π

2πω∫

0

〈B〉 cos(ωt)dt − 1

χ′′ =ω

µ0Ha0π

2πω∫

0

〈B〉 sin(ωt)dt

Primary coilSecondary coils

Sample

coils1 cm

sample

thermalization wires

carbon �ber

14

The dilution refrigerator

Mixing Chamber

Still

Pump

1K Pot

Heat exchangers

Susceptometer

4He Condenser

Concentra

tedphase

Concentrated phase

Dilute phase

Dilute

phase

3Hegas

mixing

chamber

heat

exchanger

still

1 K pot

15

Measurement of the temperature

I Thermometer: thick �lm of RuOx

I Calibration with 13 superconducting references and aparamagnetic salt.

Resistance

Superconductingreferences

Susceptibility ofthe paramagneticsalt

16

Critical �eld and temperature

0 50 100 150 200 250

6

7

· 10−6

128.6

Temperature (mK)

χ′(a.u.)

0 0.5 1

0

0.5

1· 10−5

Field (T)

χ′

0 0.5 1

1

1.2

1.4

· 10−7

χ′′

17

Field scans

0.1 0.2 0.3 0.4 0.50.2

0.4

0.6

0.8

1

· 10−5

Field (T)

χ′(a.u.)

18

Phase diagram

0 20 40 60 80 100 120

0

0.5

1

Temperature (mK)

Field

(T)

ExperimentMF without hyper�necoupling (T rescaled)

MF with hyper�necoupling (T rescaled)

TN (mK) exp.Experiment 128 0.32MF no hyp 186 0.24MF hyp 182

HC (T) exp.Experiment 0.48 0.54MF no hyp 0.46 0.52MF hyp 1.2 0.5

19

Future work

I Monte Carlo simulation for LiYbF4 with the new parameters

I Study of the disordered compounds LiHo1−xYbxF4I Monte Carlo simulationI Measurement of the TC -x phase diagram

20

experimental and numerical study of the low temperature ... · 2) site 4 : (1 2;0 ; 3 4) 4. the...

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