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Introduction to theoretical methods to describe caloric effects in ferroic materials Peter Entel, Sanjubala Sahoo, Mario Siewert, Markus E. Gruner, Heike C. Herper Faculty of Physics, University of Duisburg-Essen, 47048 Duisburg, Germany 1.2 Ni

slide-1
SLIDE 1

150 180 210 240 270 300 330 Temperature (K)

  • 0.8
  • 0.4

0.0 0.4 0.8 1.2 ∆Tad (K)

Experiment Model

Ni50Mn34In16

∆H = 1 T

Introduction to theoretical methods to describe caloric effects in ferroic materials

Peter Entel, Sanjubala Sahoo, Mario Siewert, Markus E. Gruner, Heike C. Herper Faculty of Physics, University of Duisburg-Essen, 47048 Duisburg, Germany JPD:AP 44, 064012 (2011)

IIMEC-2012 MCE – p.1/30

slide-2
SLIDE 2

Motivation: Functional properties of Heuslers

Interplay of magnetism and structural transformation: – Exchange bias (EB) effect: Shift of magnetic hysteresis curve – Magnetocaloric effect (MCE): Conventional (heating) and inverse (cooling) effect – Magnetic shape memory effect (MSME): Huge strain effect in the martensitic phase in an external magnetic field Origin: Competing ferro- and antiferromagnetic interactions

IIMEC-2012 MCE – p.2/30

slide-3
SLIDE 3

Introduction

Solid state refrigeration can reduce the worldwide CO2 emission Cooling requires to control the entropy of the refrigeration medium: (possible at phase transitions) Solid state refrigeration requires diffusionless transformation because diffusion is too slow

IIMEC-2012 MCE – p.3/30

slide-4
SLIDE 4

Introduction

Solid state refrigeration can reduce the worldwide CO2 emission Cooling requires to control the entropy of the refrigeration medium: (possible at phase transitions) Solid state refrigeration requires diffusionless transformation because diffusion is too slow

  • S. Fähler et al.,
  • Adv. Eng. Mater. 13, 1 (2011)
  • L. Mañosa et al., Nat, Mater. 9, 478 (2010)

IIMEC-2012 MCE – p.3/30

slide-5
SLIDE 5

Introduction

Solid state refrigeration can reduce the worldwide CO2 emission Cooling requires to control the entropy of the refrigeration medium: (possible at phase transitions) Solid state refrigeration requires diffusionless transformation because diffusion is too slow

  • S. Fähler et al.,
  • Adv. Eng. Mater. 13, 1 (2011)
  • L. Mañosa et al., Nat, Mater. 9, 478 (2010)

Barocaloric cooling cycle: Adiabatic compression of austenite in- duces martensite / twins and increases T Heat is releasesed to external reservoir Adiabatic decompression induces austen- ite and decreases T System is connected to cold reservoir be- coming colder

IIMEC-2012 MCE – p.3/30

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SLIDE 6

Electrocaloric effect cooling cycle

Cycle involving two constant-entropy transitions and two at constant field E

IIMEC-2012 MCE – p.4/30

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SLIDE 7

Electrocaloric effect cooling cycle

Cycle involving two constant-entropy transitions and two at constant field E

  • J. F

. Scott., Annu. Rev. Res. 41, 229 (2011)

  • Z. Zhao et al., Nat, Mater. 5, 8233 (2006)

IIMEC-2012 MCE – p.4/30

slide-8
SLIDE 8

Electrocaloric effect cooling cycle

Cycle involving two constant-entropy transitions and two at constant field E

  • J. F

. Scott., Annu. Rev. Res. 41, 229 (2011)

  • Z. Zhao et al., Nat, Mater. 5, 8233 (2006)

Electrocaloric cooling cycle: (a) Initial state T - E is rapidly applied bringing the crystal to lower entropy (b) At higher T - subsequently the crystal is allowed to cool at constant E lowering entropy to (c) (c) E is reduced to 0 and further cooling by adiabatic depolarization (d) System warms up to initial state absorbing heat from the local

IIMEC-2012 MCE – p.4/30

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SLIDE 9

Electrocaloric entropy change

Calculated entropy S(T) of BaTiO3:

IIMEC-2012 MCE – p.5/30

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SLIDE 10

Electrocaloric entropy change

Calculated entropy S(T) of BaTiO3: H.-X. Cao et al., J. Appl. Phys. 106, 094104 (2009) ∆T (model calculation) is of the order of 10 K

IIMEC-2012 MCE – p.5/30

slide-11
SLIDE 11

Electrocaloric entropy change

Calculated entropy S(T) of BaTiO3: H.-X. Cao et al., J. Appl. Phys. 106, 094104 (2009) ∆T (model calculation) is of the order of 10 K

IIMEC-2012 MCE – p.5/30

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SLIDE 12

Caloric effects in ferroic materials

Magnetocaloric effect (MCE): Magnetic materials change their thermodynamic properties like entropy and specific heat under the influence of a control parmeter: S(T, V, H, x, . . . ), C(T, V, H, x, . . . ) Effect known since 1880: E. Warburg, Ann. Phys. 13, 131 (1881) Last decade: Materials which work at ambient temperature MCE: ∆S(T, H) ≈ 10 J/(kg K), ∆Tad(T, H) ≈ 1 − 10 K Challenge: How can one improve “systematically” the caloric effect? Issue: Strong interaction of experimental and theoretical groups Reviews: A.M. Tishin & Y.I. Spichkin (IOP , Bristol, 2003) The Magnetocaloric Effect and its Applications N.A. de Oliveira & P .J. von Ranke, Phys. Rep. 489, 89 (2010) Theoretical aspects of the magnetocaloric effect V.D. Buchelnikov & V.V. Sokolovskii, Phys. Met. Metallogr. 112, 633 (2011) Magnetocaloric effect in Ni-Mn-X (X = Ga, In, Sn, Sb) Heusler alloys

IIMEC-2012 MCE – p.6/30

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SLIDE 13

Giant MCE Materials 1990 FeRh Nikitin et al. 1997 Gd5(Ge1−xSix)4 Pecharsky & Gschneidner 1998 RCo2 Foldeaki et al. 2000-2002 La(Fe, Si)13 Hu et al., Fukamichi et al. 2001 MnAs1−xSbx Wada et al. 2002 MnFe(P , As) Tegus et al. 2003 Co(S1−xSex)2 Yamada & Goto 2005 Ni2Mn1+xIn1−x Krenke et al. 2009 MnCoGeB Trung et al. Complex crystalline and magnetic structures & “magnetostructural” phase transformations

IIMEC-2012 MCE – p.7/30

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SLIDE 14

Example of magnetostructural transition

0.0 0.1 0.2 0.3 0.4

Ni excess (x)

100 200 300 400 500 600 7.50 7.55 7.60 7.65 7.70 7.75 7.80

Valence electron number/atom (e/a)

100 200 300 400 500 600

Temperature (K)

TC MS

Ni2+xMn1-xGa

TI

PM L21 FM L21 FM 5M, 7M (c/a > 1) (c/a < 1) FM non-modulated tetragonal martensite PM martensite FM L21 FM L10

MCE MSME

0.0 0.1 0.2 0.3 0.4

Ni excess (x)

100 200 300 400 500 600 7.50 7.55 7.60 7.65 7.70 7.75 7.80

Valence electron number/atom (e/a)

100 200 300 400 500 600

Temperature (K)

TC MS

Ni2+xMn1-xGa

TI

PM L21 FM L21 FM 5M, 7M (c/a > 1) (c/a < 1) FM non-modulated tetragonal martensite PM martensite FM L21 FM L10

20 40 60 80 100 120

Compressive stress σ || [001]P (MPa)

200 210 220 230 240 250 260 270

Temperature (K)

Ni2MnGa P (L21) X I 5M

continuous discontinuous discontinuous multicritical point Phonon softening (TA2) of P phase: precursor to the X-phase?

IIMEC-2012 MCE – p.8/30

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SLIDE 15

Example of magnetostructural transition

0.0 0.1 0.2 0.3 0.4

Ni excess (x)

100 200 300 400 500 600 7.50 7.55 7.60 7.65 7.70 7.75 7.80

Valence electron number/atom (e/a)

100 200 300 400 500 600

Temperature (K)

TC MS

Ni2+xMn1-xGa

TI

PM L21 FM L21 FM 5M, 7M (c/a > 1) (c/a < 1) FM non-modulated tetragonal martensite PM martensite FM L21 FM L10

MCE MSME

0.0 0.1 0.2 0.3 0.4

Ni excess (x)

100 200 300 400 500 600 7.50 7.55 7.60 7.65 7.70 7.75 7.80

Valence electron number/atom (e/a)

100 200 300 400 500 600

Temperature (K)

TC MS

Ni2+xMn1-xGa

TI

PM L21 FM L21 FM 5M, 7M (c/a > 1) (c/a < 1) FM non-modulated tetragonal martensite PM martensite FM L21 FM L10

20 40 60 80 100 120

Compressive stress σ || [001]P (MPa)

200 210 220 230 240 250 260 270

Temperature (K)

Ni2MnGa P (L21) X I 5M

continuous discontinuous discontinuous multicritical point Phonon softening (TA2) of P phase: precursor to the X-phase?

26 28 30 32 34 Mn (at. %) 19 20 21 22 23 Ga (at. %)

Tetragonal Mixed Orthorhombic Cubic / 5M 370 K 300 K MS = 280 K (e/a = 7.61)

50% Ni 48% Ni 46% Ni 44% Ni

(e/a = 7.64) (e/a = 7.74)

Ni2-xMn1+x+yGa1-y

Ni49Mn32Ga19 TC = 368, MS = 353 K

IIMEC-2012 MCE – p.8/30

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SLIDE 16

Example of magnetostructural transition

0.0 0.1 0.2 0.3 0.4

Ni excess (x)

100 200 300 400 500 600 7.50 7.55 7.60 7.65 7.70 7.75 7.80

Valence electron number/atom (e/a)

100 200 300 400 500 600

Temperature (K)

TC MS

Ni2+xMn1-xGa

TI

PM L21 FM L21 FM 5M, 7M (c/a > 1) (c/a < 1) FM non-modulated tetragonal martensite PM martensite FM L21 FM L10

MCE MSME

0.0 0.1 0.2 0.3 0.4

Ni excess (x)

100 200 300 400 500 600 7.50 7.55 7.60 7.65 7.70 7.75 7.80

Valence electron number/atom (e/a)

100 200 300 400 500 600

Temperature (K)

TC MS

Ni2+xMn1-xGa

TI

PM L21 FM L21 FM 5M, 7M (c/a > 1) (c/a < 1) FM non-modulated tetragonal martensite PM martensite FM L21 FM L10

20 40 60 80 100 120

Compressive stress σ || [001]P (MPa)

200 210 220 230 240 250 260 270

Temperature (K)

Ni2MnGa P (L21) X I 5M

continuous discontinuous discontinuous multicritical point Phonon softening (TA2) of P phase: precursor to the X-phase?

26 28 30 32 34 Mn (at. %) 19 20 21 22 23 Ga (at. %)

Tetragonal Mixed Orthorhombic Cubic / 5M 370 K 300 K MS = 280 K (e/a = 7.61)

50% Ni 48% Ni 46% Ni 44% Ni

(e/a = 7.64) (e/a = 7.74)

Ni2-xMn1+x+yGa1-y

Ni49Mn32Ga19 TC = 368, MS = 353 K

Top:

  • V. V. Khovaylo et al., PRB 72, 224408 (2005)
  • H. Kushida et al., Scripta Mater. 60, 96 (2009)

Left:

  • M. Richard et al., Scripta Mater. 54, 1797 (2006)

IIMEC-2012 MCE – p.8/30

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SLIDE 17

S-T diagrams: Carnot, Ericsson, Brayton cycles

Temperature Total entropy

S(T, H1, p) S(T, H2, p)

A B C D

Carnot cycle

Material absorbs heat Material releases heat

Temperature Total entropy

S(T, H1, p) S(T, H2, p)

A B C D

Ericsson cycle

Material absorbs heat Material releases heat

  • Z. Yan & J. Chen, JAP 72, 1 (1992)

Temperature Total entropy

S(T, H1, p) S(T, H2, p)

A B C D

Brayton cycle

Optimal T-S curves for C.R. Cross et al., in: Advances in Cryogenic Engineering, Vol. 33 magnetic refrigeration: (Plenum, New York, 1988), p. 767 IIMEC-2012 MCE – p.9/30

slide-18
SLIDE 18

Experimental S-T curves of Ni49.26Mn36.08In14.66

IIMEC-2012 MCE – p.10/30

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SLIDE 19

Experimental S-T curves of Ni49.26Mn36.08In14.66 Barocaloric effect (top)

  • L. Mañosa et al.,
  • Nat. Mater. 9. 478 (2010)

Inverse Barocaloric effect

  • L. Mañosa et al.,
  • Nat. Commun. 2. 595 (2011)

Magnetocaloric effect (bottom)

  • T. Krenke et al., Nat. Mater. 4, 450

(2008)

IIMEC-2012 MCE – p.10/30

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SLIDE 20

Quantities of interest: Entropy and specific heat

∆Smag(T, H) = Smag(T, H) − Smag(T, 0) (0) ∆Tad(T, H) = −T ∆Smag(T, H) C(T, H) (0) ∆Smag(T, H) = µ0

H

Z dH′ „ ∂M ∂T «

H′

(0) ∆Tad(T, H) = −µ0

H

Z dH′ T C(T, H′) „∂M ∂T «

H′

(0) ∆C(T, H) = 1 kbT 2 h˙ H2¸ − H

2i

In general, ∆S(T, Y ) = Z

∆Y

dY ′ „∂X ∂T «

Y ′

X = {Xi} Y = {Yi} X = M, Y = H (MagnetoCE) X = V , Y = −p (BaroCE) X = ε, Y = σ (ElastoCE) X = P, Y = E (ElectroCE)

IIMEC-2012 MCE – p.11/30

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SLIDE 21

Modelling magnetostructural coupling

H = Hm + Hlat + Hint Hm = − X

ij

Jm(i, j) δSi,Sj − gµBHext X

i

δSi,Sg Hlat = − J X

ij

σiσj − K X

ij

(1 − σ2

i )(1 − σ2 j ) − kBT ln(p)

X

i

(1 − σ2

i )

− K1gµBHext X

i

δσi,σg X

ij

σiσj Hint = 2U X

ij

δSi,Sj ` 1

2 − σ2 i

´ ` 1

2 − σ2 j

´ − 1

2U

X

ij

δSi,Sj Mapping ab initio results onto q-state Potts model for Mn and Ni extended to include σ = 0, ±1 for cubic and tetragonal distorted structures (BEG model)

  • T. Castán et al., PRB 60, 7071 (1999)
  • V. D. Buchelnikov et al., PRB 78, 184427 (2008)
  • V. D. Buchelnikov et al., PRB 81, 094411 (2010)

IIMEC-2012 MCE – p.12/30

slide-22
SLIDE 22

Modelling magnetostructural coupling

H = Hm + Hlat + Hint Hm = − X

ij

Jm(i, j) δSi,Sj − gµBHext X

i

δSi,Sg Hlat = − J X

ij

σiσj − K X

ij

(1 − σ2

i )(1 − σ2 j ) − kBT ln(p)

X

i

(1 − σ2

i )

− K1gµBHext X

i

δσi,σg X

ij

σiσj Hint = 2U X

ij

δSi,Sj ` 1

2 − σ2 i

´ ` 1

2 − σ2 j

´ − 1

2U

X

ij

δSi,Sj Mapping ab initio results onto q-state Potts model for Mn and Ni extended to include σ = 0, ±1 for cubic and tetragonal distorted structures (BEG model)

  • T. Castán et al., PRB 60, 7071 (1999)
  • V. D. Buchelnikov et al., PRB 78, 184427 (2008)
  • V. D. Buchelnikov et al., PRB 81, 094411 (2010)

100 200 300 400

Temperature (K)

0.0 1.0 2.0 3.0 4.0

Magnetization (µB/Mn)

m (Hext = 0 T) m (Hext = 5 T) 0.0 0.2 0.4 0.6 0.8 1.0

Tetragonal distortion (ε)

ε (Hext = 0 T) ε (Hext = 5 T)

Ni50Mn34In16 ε m

IIMEC-2012 MCE – p.12/30

slide-23
SLIDE 23

Example 1: Ni-Mn-In

100 200 300

Temperature (K)

1 2 3

Magnetization (Am

2kg

  • 1)

FC FH ZFC TC

M

Ms TC

A

µ0H = 5 mT

Ni50Mn34In16 Experiment

0.5 1.0 1.5 2.0 2.5 3.0 d/a

  • 2.0

0.0 2.0 4.0 6.0 Jij (meV)

Mn-Mn Mn-In Mn-Ni

(a) Ni50Mn25In25 (c/a = 1)

0.5 1.0 1.5 2.0 2.5 3.0 d/a

  • 6.0
  • 4.0
  • 2.0

0.0 2.0 4.0 Jij (meV)

Mn1-Mn1 Mn1-Mn2 Mn2-Mn2 Mn1-Ni Mn2-Ni

(b) Ni50Mn34In16 (c/a = 1)

0.5 1.0 1.5 2.0 2.5 3.0 d/a

  • 6.0
  • 4.0
  • 2.0

0.0 2.0 4.0 Jij (meV)

Mn1-Mn1 Mn1-Mn2 Mn2-Mn2 Mn1-Ni Mn2-Ni

0.5 1.0

  • 20
  • 15
  • 10
  • 5

Mn1-Mn1 Mn1-Mn2 Mn2-Mn2 Mn1-Ni Mn2-Ni

(c) Ni50Mn34In16 (c/a = 0.94)

  • T. Krenke et al., PRB 73, 174413 (2006)
  • V. D. Buchelnikov et al., JPD:AP 44, 064012 (2011)

IIMEC-2012 MCE – p.13/30

slide-24
SLIDE 24

Heusler structures X2YZ

0.8 1 1.2 1.4 1.6

c/a ratio

100 200 300

E-E0 (meV/f.u.)

Ni2MnGa Ni2CoGa Ni2CoZn Co2NiGa Fe2CoGa PtNiMnGa

0.8 1 1.2 1.4

c/a ratio

  • 50

50

Energy (meV/atom)

0% 25 % 50 %, para 50 %, ortho 75 %

Ni2Mn1+xIn1-x

IIMEC-2012 MCE – p.14/30

slide-25
SLIDE 25

For comparison: Jij of Ni-Mn-Sn

IIMEC-2012 MCE – p.15/30

slide-26
SLIDE 26

For comparison: Jij of Ni-Mn-Sn

  • M. Acet et al., Handbook of Magnetic Ma-

terials, vol. 19, 231 (2011)

IIMEC-2012 MCE – p.15/30

slide-27
SLIDE 27

For comparison: Jij of Ni-Mn-Sn

  • M. Acet et al., Handbook of Magnetic Ma-

terials, vol. 19, 231 (2011)

0.5 1.0 1.5 2.0 2.5 3.0 d/a

  • 120.0
  • 100.0
  • 80.0
  • 60.0
  • 40.0
  • 20.0

0.0 20.0 Jij (meV)

Mn1-Mn1 Mn1-Mn2 Mn2-Mn2 Mn1-Ni Mn2-Ni

c/a = 0.91

Ni8Mn7Sn (GGA)

0.5 1.0 1.5 2.0 2.5 3.0 d/a

  • 120.0
  • 100.0
  • 80.0
  • 60.0
  • 40.0
  • 20.0

0.0 20.0 Jij (meV)

Mn1-Mn1 Mn1-Mn2 Mn2-Mn2 Mn1-Ni Mn2-Ni

c/a = 1.0

Ni8Mn7Sn (GGA)

0.5 1.0 1.5 2.0 2.5 3.0 d/a

  • 120.0
  • 100.0
  • 80.0
  • 60.0
  • 40.0
  • 20.0

0.0 20.0 Jij (meV)

Mn1-Mn1 Mn1-Mn2 Mn2-Mn2 Mn1-Ni Mn2-Ni

c/a = 1.35

Ni8Mn7Sn (LDA)

  • S. Sahoo et al., unpublished

IIMEC-2012 MCE – p.15/30

slide-28
SLIDE 28

Monte Carlo simulations of Ni50Mn34In16

100 200 300

Temperature (K)

1 2 3

Magnetization (Am

2kg

  • 1)

FC FH ZFC TC

M

Ms TC

A

µ0H = 5 mT

Ni50Mn34In16 Experiment

50 100 150 200 250 300 350

Temperature (K)

200 400 600 800 1000

C (J/kgK)

Model Experiment Hext = 0 T

Ni50Mn34In16

(a)

1 2 3 4 5

Magnetic field (T)

  • 8.0
  • 6.0
  • 4.0
  • 2.0

0.0 2.0 4.0 6.0 8.0 10.0

∆Smag (J/kgK)

Positive MCE Negative MCE

Ni16Mn34In16

(b)

(Theory)

  • V. D. Buchelnikov et al., JPD:AP 44, 064012 (2011)

IIMEC-2012 MCE – p.16/30

slide-29
SLIDE 29

Calculated ∆Smag(T) and ∆Tad(T) (Ni-Mn-In)

150 200 250 300 350 Temperature (K)

  • 5

5 10 ∆Smag (J/kgK)

Model Experiment

(a) Ni50Mn34In16

∆ Hext= 5 T

150 180 210 240 270 300 330 Temperature (K)

  • 0.8
  • 0.4

0.0 0.4 0.8 1.2 ∆Tad (K)

Experiment Model

Ni50Mn34In16

∆H = 1 T

MCE: V.D. Buchelnikov et al., JPD:AP 44, 064012 (2011) BCE: L. Mañosa et al., Nat. Mat. 9, 478 (2010)

IIMEC-2012 MCE – p.17/30

slide-30
SLIDE 30

How to get a Larger MCE: Scale the Jij

100 200 300 400 500 Temperature (K) 0.2 0.4 0.6 0.8 1 Normalized magnetization

n = 0.50 n = 0.75 n = 1.00 n = 1.50 n = 2.00

Ni50Mn34In16

180 210 240 270 300 330 Temperature (K)

  • 2.0
  • 1.0

0.0 1.0 ∆Tad (K)

Experiment Model (n = 1.0) Model (n = 0.5) Model (n = 1.5)

Ni50Mn34In16

∆ Hext= 1 T

Effect of scaling on the MCE (unpublished): n = Jnew

ij

/Jold

ij

Yields a larger MCE and, simultaneously, a smaller “hysteresis”

IIMEC-2012 MCE – p.18/30

slide-31
SLIDE 31

With 5 at.% transition metals

0.5 1.0 1.5 2.0 2.5 3.0 d/a

  • 30.0
  • 20.0
  • 10.0

0.0 10.0 Jij (meV)

Mn2-Mn2 Mn1-Mn2 Mn2-Ni Mn2-Nb Mn1-Mn1 Mn1-Ni Mn1-Nb Ni-Ni

Ni50Mn34In16 with 5% Nb

c/a = 1

0.5 1.0 1.5 2.0 2.5 3.0 d/a

  • 30.0
  • 20.0
  • 10.0

0.0 10.0 Jij (meV)

Mn1-Mn1 Mn2-Ni Mn1-Ni Mn2-Mn2 Mn1-Mn2 Ni-Ni

c/a = 1

Ni50Mn34In16 with 5% Cu

Effect of nonmagnetic TM on magnetic exchange parameters (unpublished)

IIMEC-2012 MCE – p.19/30

slide-32
SLIDE 32

Example 2: Gd-Ge-Si

First-order bond breaking magnetostructural transition responsible for the giant MCE

  • W. Choe et al., PRL 84, 4617 (2000)
  • D. Haskel et al., PRL 98, 247205 (2007)

IIMEC-2012 MCE – p.20/30

slide-33
SLIDE 33

Survey of theoretical methods Landau and spin models with coupling to the lattice Molecular field approximation Monte Carlo simulations Ab initio magnetic exchange parameters as input First-principles DFT methods, fully relativistic Including external magnetic field: M(T)H curves Finite T calculations: Phonons, magnons Phase diagrams, MCE, BCE and ECE

IIMEC-2012 MCE – p.21/30

slide-34
SLIDE 34

Theoretical tools (I) (a) MCE (RE):

IIMEC-2012 MCE – p.22/30

slide-35
SLIDE 35

Theoretical tools (I) (a) MCE (RE):

H = Hspd

el

+ H4f

mag + Hlat

Complex parts: Crystal field Hamiltonian, Clat and Slat depend on magnetic order Gd5Si2Ge2 (solid lines) and Gd (dashed lines) upon magnetic field variation from 0 to 5T N.A. de Oliveira & P .J. von Ranke, Phys. Rep. 489, 89 (2010)

IIMEC-2012 MCE – p.22/30

slide-36
SLIDE 36

Theoretical tools (II) (b) MCE (TM):

IIMEC-2012 MCE – p.23/30

slide-37
SLIDE 37

Theoretical tools (II) (b) MCE (TM):

H = Hsp

el + Hd mag + Hlat + Hel−lat

Complex parts: Hubbard Hamiltonian, elctron-phonon and phonon-magnon interactions MnAs upon magnetic field variation from 0 to 2 T (solid lines) and from 0 to 5 T (dashed lines) compared to experiment (symbols) N.A. de Oliveira & P .J. von Ranke, Phys. Rep. 489, 89 (2010)

IIMEC-2012 MCE – p.23/30

slide-38
SLIDE 38

Theoretical tools (III) (c) MCE (Heusler alloys):

IIMEC-2012 MCE – p.24/30

slide-39
SLIDE 39

Theoretical tools (III) (c) MCE (Heusler alloys):

H = Hd

mag + HBEG + Hlat

Complex parts: Magnetic part coupled to the martensitic transformation

180 210 240 270 300 330

Temperature (K)

20 40 60 80

Magnetization (Am

2kg

  • 1)

Ni50Mn34In16 Experiment

µ0H = 1.3 T TC

A

Magnetostructural transition

150 200 250 300 350

Temperature (K)

1 2 3 4

Magnetization (µB/Mn)

0.2 0.4 0.6 0.8 1

Tetreganal distortion (ε)

m ε

Ni50Mn34In16

5 T 0 T 5 T µ0H = 0 T

Theory

Magnetization near the magnetostructural transition in external field from Monte Carlo simulations compared to experiment V.D. Buchelnikov et al., JPD:AP 44, 064012 (2011)

IIMEC-2012 MCE – p.24/30

slide-40
SLIDE 40

Theoretical tools (IV) (d) MCE (Heuslers with DFT):

IIMEC-2012 MCE – p.25/30

slide-41
SLIDE 41

Theoretical tools (IV) (d) MCE (Heuslers with DFT):

F(V, T) = Fel(V, T) + Fmag(V, T) + Fph(V, T) Complex parts: Ab initio evaluation of phonons and magnons and their coupling at finite T

100 200 300 400 Temperature (K)

  • 10
  • 5

Free energy (meV/atom) FMartensite FAustenite FPremartensite Ni2MnGa

Ms = 150 K TI = 240 K TC = 376 K

Austenite-martensite transition in Ni2MnGa

  • M. A. Uijttewaal et al., PRL 102, 035702 (2009)

IIMEC-2012 MCE – p.25/30

slide-42
SLIDE 42

Summary Large MCE near magnetostructural transitions with competing ferro- and antiferromagnetic interactions ∆Tad = −T ∆S(mag)(T, H, p, x, . . .) C(T, H, p, x, . . .) Mostly model calculations with magnetic and lattice degrees of freedom and parameter fit to experiment /ab initio data First-principles DFT calculations (T = 0 K) Needed: T > 0 K calculations for MCE, BCE . . .

IIMEC-2012 MCE – p.26/30

slide-43
SLIDE 43

Collaboration

Collaborators: Sanjubala Sahoo Markus E. Gruner Mario Siewert Heike C. Herper Cooperations: Raymundo Arróyave (Texas A&M University, USA) Ibrahim Karaman (Texas A&M University, USA) Navdeep Singh (Texas A&M University) Sebastian Fähler (IFW Dresden, Germany) Mehmet Acet (Uni Duisburg, Germany) Manfred Wuttig Uni Maryland, USA) Volodymyr V. Chernenko (Uni Bilbao, Spain) Sudipta R. Barman (UGC-DAE, India) Hubert Ebert (Uni München, Germany) Tilmann Hickel (MPIE, Germany) Jörg Neugebauer (MPIE, Germany) Hisazumi Akai (Osaka University, Japan) Aparna Chakrabarti (RRCAT, India) Vasiliy D. Buchelnikov (Uni Chelyabinsk, Russia)

IIMEC-2012 MCE – p.27/30

slide-44
SLIDE 44

Appendix: Specific heat of Ni2.18Mn0.82Ga

Buchelnikov et al., PRB 81, 094411 (2010)

IIMEC-2012 MCE – p.28/30

slide-45
SLIDE 45

Appendix: Qualitative S-T diagrams

Temperature T Entropy S

S0, T0 T1, H1 S1, H1

H1 > H0 = 0

S(H0) S(H1) Smag(H0) Smag(H1) ∆Tad Slat ∆Smag

  • V. K. Pecharsky et al., JMMM 200, 44 (1999)
  • V. D. Buchelnikov et al., PRB 81, 094411

(2010)

IIMEC-2012 MCE – p.29/30

slide-46
SLIDE 46

Appendix: Summary of caloric effects

Y → -p (pressure) Barocaloric effect x → V (Volume) Mechanical refrigeration

Nature Mater. 9 (2010) 478

Electrocaloric effect Y → E (Electric field) x → P (Polarization) Electric refrigeration

  • A. S. Mischenko et al.

∆S ∆T

Magnetocaloric effect Y → H (Magnetic field) x → M (Magnetization) Magnetic refrigeration

  • V. K. Pecharsky et al.

Elastocaloric effect Y → σ (Stress) x → ε (Strain) Elastic refrigeration

  • E. Bonnot et al.
  • Phys. Rev. Lett. 78 (1997) 4494
  • Phys. Rev. Lett. 100 (2008) 125901

REFRIGERATION IN SOLID STATE

Science 311 (2006) 1270

  • L. Manosa et al.

~

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