Lecture VI: An Introduction to Modern Approach of Polarization Theory

South China University of Technology

Yu-Jun Zhao

South China University of Technology

背景－多铁体

多铁体（multiferroic）

Spaldin and Fiebig, Science 309, 391 (2005)

铁性分类

Multi：多，ferroic：含铁的，源自拉丁词汇ferrum(铁 )

Multiferroic: 具有多重铁性的材料多种极性序参量

—Hans Schmid, Ferroelectrics 162, 317 (1994)

Van Aken, Rivera, Schmid，Fiebig. Nature 449, 702 (2007)

铁弹性

铁涡性

背景－磁电效应

电磁场耦合方程

磁电效应

电磁场内稟性质

电场(磁场) 对物质磁性(介电)性质的影响

P = αH, M =αE

应用领域

电场(磁场)传感（探测）器,转换器

电场控制的磁纪录存储

Eerenstein, Mathur and Scott, Nature 442, 759 (2006)

Wood和Austin ：15种用途Int. J. Magn. 5，303（1973）

多铁体和磁电体的范畴

J. C. Maxwell

6/13/1831~11/5/1879

Phil. Trans. R. Soc. Lond. 155, 459–512 (1865)

magnetoelectric effect: P. Debye 1926 Z. Phys. 36 300

磁电耦合:表征介质磁学性质和介电性质的序参量，即磁化强度M和电极化强度P之间的耦合

磁电体自身性质

背景－历史回顾

早期 (1888-1958)

小高潮（1958-1993)

0

50

100

150

200

250

300

350

400

1993 1995 1997 1999 2001 2003 2005 2007

ISI Statistics: keyword magnetoelectric

and multiferroic

Magnetoelectric

Multiferroic

1888年，Röntgen ：处于电场中的运动介质会被磁化 （Ann. Phys. 35, 264 ）1905年, H.A. Wilson,逆效应：处于磁场中的运动介质会被电极化 （Phil.Trans. R. Soc. A 204, 129）1894年，P. Curie（对称性原理）：用磁场使介质电极化或者用电场使介质磁极化 J. Physique 3，393

实验努力大多失败Debye, Condon, van Vleck

1958年，Landau和Lifshitz ：时间反演对称性

1959年，Dzyaloshinskii：Cr203

1960年，Astrov实验证实（JETP 11， 708）

1958年，Smolensky和Ioffe：铁电性铁磁体 Pb(Fe0.5Nb0.5)O3

复兴（1993-Present)

至1973年共有80多种磁电体被发现，随后研究进入低谷

2000年，N.A. Hill, J. Phys. Chem. B 104, 6694

1994年，C. W. Nan Phys. Rev. B 50，6082

2005年，M. Fiebig, Phys. D: Appl. Phys. 38, R123

2003年，J. Wang等：BiFeO3，Science 299, 1719

2005年以来，平均每年5篇review papers多家杂志作了专题报道Science:2008年七大科技热点

复兴原因

理论发展，实验进步，社会需求

Electrical Polarization theory

Electric dipole

Water Molecule

x

y

Polarization in bulk

NaCl 离子晶体CM 模型：极矩可用局域的电荷中心来描述

Si

共价晶体

（阴影部分是外电场引起的负电荷集聚区域）

无法用电荷中心来描述（有很大一部分是键的贡献）

如果一定要用CM模型来描述的话，无论怎样取单位原胞和电荷中心，得到的高频介电常数仅为实验值(12)的~1/10.

Dielectric Constants of Bulks

Ferroelectric Crystals

Issue on Polarization in bulk

2.

In order to apply it, we need to assume a macroscopic

but finite crystal. Not a bulk description.

1.

However, this approach is also flawed, because the

result depends on the shape and location of the unit cell.

From textbook of Aschcroft and Mermin

Issue on Polarization in bulk

3. As a third approach, one might imagine defining P as the

cell average of a microscopic polarization Pmicro defined via

However, the above equation does not uniquely define Pmicro(r).

(E.g., shift a constant for Pmicro)

The periodic electronic charge distribution in a polarized

crystalline solid cannot, even in principle, be used to construct

a meaningful definition of bulk polarization.

However, this important message has not received the wide

appreciation it deserves, nor has it reached the most popular

textbooks.

Issue Polarization in bulk

From textbook of Aschcroft and Mermin

P is a bulk property or surface property?

From Kittel’s textbook (8th ed).

Issue Polarization in bulk

The conclusion to be drawn from the above discussion is that a

knowledge of the periodic electronic charge distribution in a

polarized crystalline solid cannot, even in principle, be used to

construct a meaningful definition of bulk polarization.

A modern approach has been developed since 1970s (significant

works were done in 90s).

Modern Approach

How is Induced Polarization Measured?

(介电常数）

How is Induced Polarization Measured?

以压电效应的测量为例：

How is Ferroelectric Polarization Measured?

Prescription for a theory of Polarization

Prescription for a theory of Polarization

Berry Phase Theory

Berry Phase Theory

Berry Phase Theory

Berry phase

Calculation of polarization

Berry phase

虚部

Calculation of polarization

The Quantum of Polarization

Berry phase 满足

所以形式电极矩（formal polarization)

Formal polarization vs normal polarization

（形式极矩 vs 常规极矩）

A useful way to think about the presence of this “modulus”

is to regard the formal polarization as a multivalued vector

quantity, rather than a conventional single-valued one.

Formal polarization vs normal polarization

（形式极矩 vs 常规极矩）

References

1. Physics of ferroelectrics: a modern perspective

By Karin Maria Rabe, Charles H. Ahn, Jean-Marc Triscone

Chapters by Resta and Vanderbilt

Available at springerlink.com

2. Macroscopic polarization in crystalline dielectrics: the

geometric phase approach

R. Resta, Rev. Mod. Phys. 66, 899(1994)