Spin Fluctuation Theory of Itinerant Electron Magnetism

190 Pages·2013·2.857 MB·English

by  Yoshinori Takahashi (auth.)

Springer Tracts in Modern Physics 253
Yoshinori Takahashi
Spin Fluctuation
Theory of
Itinerant Electron
Magnetism
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Yoshinori Takahashi
Spin Fluctuation
Theory of Itinerant
Electron Magnetism
123
YoshinoriTakahashi
Graduate School ofMaterial Science
Universityof Hyogo
Hyogo
Japan
ISSN 0081-3869 ISSN 1615-0430 (electronic)
ISBN 978-3-642-36665-9 ISBN 978-3-642-36666-6 (eBook)
DOI 10.1007/978-3-642-36666-6
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Preface
My research career in the field of itinerant electron magnetism started in 1977
whenIgotajobinInstituteforSolidStatePhysics,theUniversityofTokyo.Since
thenasamemberofProfessorT.Moriya’slaboratory,Ihavebeeninvolvedinthe
development of the spin fluctuation theory of itinerant electron magnetism, well
known now as the SCR spin fluctuation theory. The finite temperature Stoner-
Wohlfarth theory has been improved by these studies as reviewed in the book,
‘‘Spin Fluctuations in Itinerant Electron Magnetism’’, by Moriya published in
1985. From the beginning, I was interested in a few difficulties still unresolved
around 1980 and several assumptions of the theory. These have become the
motivationformysubsequentstudies.MostofthemhavebeendoneafterImoved
to Faculty of Science, Himeji Institute of Technology.
The aim of this book is to review the new theoretical development of the spin
fluctuation theory that began from around the middle of 1980s. It is based on the
verysimpleideas,i.e,theassumptionofthetotalspinamplitudeconservationand
theexplicit accounttheeffectofzero-pointspinfluctuations,etc.Theseallowsus
to deal with wide variety of phenomena ranging from the ground state to the
paramagnetic phase at high temperatures. Various interesting predictions on the
properties of itinerant electron ferromagnets have been also derived in agreement
with a number of experimental observations.
The book does not attempt to cover a wide area of magnetism. What I would
liketoemphasize isthatmagneticandthermalpropertiesofitinerantmagnetsare
even more determined under the influence of magnetic fluctuations than we have
thought.Subjects are mainly confined inthermodynamic properties. It is valuable
to capture the current status of the spin fluctuation theories for graduate students
and researchers in the field of magnetism and is also helpful for the analyses of
experimental data.
TheauthorisgratefultoDr.ToruMoriyaforinvitingmeinthisfieldofresearch
and various useful advice and instructions. I would like to thank Dr. Kazuyoshi
Yoshimura,Dr.TakeshiKanomata,Dr.HironoriNishihara,Dr.KazuakiShimizu,
Dr. Yuichi Dazuke, Dr. Masayuki Shiga, Dr. Tsuneaki Goto, Dr. Hideji Yamada,
v
vi Preface
Dr. Keiichi Koyama, Dr. HiroyukiNakamura, Dr.Kazuaki Fukamichi,Dr. Asaya
Fujita,Dr.Deguchi,andDr.TakehideKoyamaforcollaborationsandmanyfruitful
suggestions and discussions. I also want to thank my staff and students in Himeji
Instituteof Technology, now renamed Universityof Hyogo.
Himeji, December 2012 Yoshinori Takahashi
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Historical Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Localized Heisenberg Magnets . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Mean Field Approximation . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Phase Transitions of Heisenberg Magnets
and Spin Amplitude Conservation. . . . . . . . . . . . . . . . . 6
1.3 Band Theoretical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Hartree–Fock Approximation . . . . . . . . . . . . . . . . . . . . 12
1.3.2 Free Energy of Stoner-Wohlfarth Theory. . . . . . . . . . . . 13
1.4 Magnetic Properties Derived from the SW Theory . . . . . . . . . . 15
1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Fluctuations and Magnetism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Fluctuations and Responses . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Kramers-Kronig Relation. . . . . . . . . . . . . . . . . . . . . . . 26
2.3 SCR Spin Fluctuation Theory. . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.1 Free Energy of the SCR Theory. . . . . . . . . . . . . . . . . . 27
2.3.2 Curie–Weiss Law of Magnetic Susceptibility. . . . . . . . . 31
2.4 Discontinuous Change of Magnetization. . . . . . . . . . . . . . . . . . 33
2.4.1 Temperature Dependence of Magnetization . . . . . . . . . . 33
2.4.2 Origin of the Discontinuity. . . . . . . . . . . . . . . . . . . . . . 35
2.5 Thermal and Zero-Point Spin Fluctuation Amplitudes . . . . . . . . 36
2.5.1 Spectral Properties of Spin Fluctuation Amplitudes. . . . . 38
2.5.2 Thermal Spin Fluctuation Amplitude. . . . . . . . . . . . . . . 39
2.5.3 Zero-Point Spin Fluctuation Amplitude . . . . . . . . . . . . . 41
2.6 Spin Amplitude Conservation . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6.1 Neutron Scattering Experiment on MnSi . . . . . . . . . . . . 43
2.6.2 Theoretical Explanation for Experiments on MnSi . . . . . 46
2.6.3 Giant Magnetic Fluctuations Observed in (Y,Sc)Mn . . . 47
2
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
vii
viii Contents
3 Effects of Spin Fluctuations on Magnetic Properties . . . . . . . . . . . 49
3.1 Basic Idea of the Spin Fluctuation Theory . . . . . . . . . . . . . . . . 49
3.2 Magnetic Isotherm in the Ground State . . . . . . . . . . . . . . . . . . 50
3.2.1 Magnetic Isotherm for Ferromagnets. . . . . . . . . . . . . . . 51
3.2.2 Magnetic Isotherm for Exchange Enhanced
Paramagnets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3 Magnetic Properties in the Paramagnetic Phase. . . . . . . . . . . . . 55
3.3.1 Temperature Dependence of Magnetic Susceptibility. . . . 55
3.3.2 Magnetic Susceptibility in the Low Temperature
Limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.3 Magnetic Susceptibility Around the Critical Point. . . . . . 57
3.3.4 Curie–Weiss Law of Magnetic Susceptibility. . . . . . . . . 58
3.3.5 Magnetic Isotherm in the Paramagnetic Phase . . . . . . . . 61
3.4 Critical Magnetic Behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4.1 Critical Magnetic Isotherm. . . . . . . . . . . . . . . . . . . . . . 64
3.4.2 Scaling Law Relations Among Critical Indexes . . . . . . . 68
3.5 Crossover Behavior Around the Quantum Critical Point. . . . . . . 69
3.5.1 Scaling Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5.2 Temperature Induced Crossover Between Classical
and Quantum Critical Phenomena. . . . . . . . . . . . . . . . . 71
3.5.3 Temperature Dependence of Magnetic Susceptibility
of Paramagnets Near the QC Point . . . . . . . . . . . . . . . . 71
3.5.4 Magnetic Susceptibility of Paramagnets Around
the Quantum Critical Point. . . . . . . . . . . . . . . . . . . . . . 75
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4 Magnetic Properties in the Ordered Phase . . . . . . . . . . . . . . . . . . 79
4.1 Initial Value Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.1.1 Analytic Property of Thermal Amplitude. . . . . . . . . . . . 80
4.1.2 Effect of the Presence of Spin Waves . . . . . . . . . . . . . . 81
4.2 Temperature Dependence of Spontaneous Moment . . . . . . . . . . 85
4.2.1 Magnetic Properties at Low Temperatures. . . . . . . . . . . 85
4.2.2 Comparison with Experiments at Low Temperatures. . . . 87
4.2.3 Magnetic Properties Around the Critical Temperature. . . 92
4.2.4 Numerical Results of Temperature Dependence . . . . . . . 93
4.2.5 Magnetization Curve. . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5 Thermal Properties of Itinerant Magnets . . . . . . . . . . . . . . . . . . . 99
5.1 Difficulties Involved in the Spin Fluctuation Theory
of Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Contents ix
5.2 Free Energy of Spin Fluctuations . . . . . . . . . . . . . . . . . . . . . . 100
5.2.1 Free Energy in the Presence of Magnetic Moment . . . . . 100
5.2.2 Stability Conditions of the Free Energy. . . . . . . . . . . . . 101
5.2.3 Free Energy Corrections . . . . . . . . . . . . . . . . . . . . . . . 102
5.3 Temperature Dependence of Entropy and Specific Heat. . . . . . . 104
5.3.1 Temperature Dependence of Paramagnetic Entropy. . . . . 105
5.3.2 Temperature Dependence of the Specific Heat. . . . . . . . 106
5.3.3 Temperature Dependence of the Entropy
and the Specific Heat in the Ordered Phase . . . . . . . . . . 109
5.4 Specific Heat Under the External Magnetic Field . . . . . . . . . . . 114
5.4.1 Maxwell Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.4.2 Temperature Derivatives in the Static External
Magnetic Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.4.3 Entropy and Specific Heat in the Paramagnetic Phase. . . 118
5.4.4 External Field Effect in the Ordered Phase . . . . . . . . . . 122
5.4.5 Numerical Estimate. . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.4.6 External Field Effect on Paramagnets Near the QCP. . . . 128
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6 Magneto Volume Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.1.1 Thermal Expansion Due to Lattice Vibrations . . . . . . . . 132
6.2 Stoner-Edwards-Wohlfarth Theory and its Correction . . . . . . . . 133
6.2.1 SEW Theory of Magneto-Volume Effect. . . . . . . . . . . . 133
6.2.2 Correction of the Free Energy of Spin Fluctuations. . . . . 135
6.3 Volume Dependence of the Free Energy . . . . . . . . . . . . . . . . . 138
6.3.1 Magnetic Grüneisen Parameters . . . . . . . . . . . . . . . . . . 141
6.3.2 Forced Magneto-Striction and Maxwell Relation . . . . . . 143
6.4 Volume Magneto-Striction for Ferromagnets. . . . . . . . . . . . . . . 144
6.4.1 Magneto-Volume Effect in the Ground State . . . . . . . . . 144
6.4.2 Ferromagnets at Finite Temperatures. . . . . . . . . . . . . . . 145
6.5 Magneto-Volume Effect in Some Temperature Ranges . . . . . . . 151
6.5.1 Magneto-Volume Effect at Low Temperature
and Grüneisen Relation . . . . . . . . . . . . . . . . . . . . . . . . 152
6.5.2 Around the Critical Point. . . . . . . . . . . . . . . . . . . . . . . 154
6.5.3 In the Paramagnetic Phase. . . . . . . . . . . . . . . . . . . . . . 158
6.5.4 Numerical Results on Volume-Strictions . . . . . . . . . . . . 161
6.6 Magneto-Volume Effect for Paramagnets. . . . . . . . . . . . . . . . . 162
6.6.1 Spontaneous Magneto-Striction for Paramagnets. . . . . . . 164
6.6.2 Forced Magneto-Striction for Paramagnets. . . . . . . . . . . 166
6.7 Pressure Effects on Spontaneous Magnetic Moment
and the Critical Temperature. . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.7.1 Effect of Pressure on the Critical Temperature. . . . . . . . 167