basic training 2009– lecture 01 competing ferroic orders

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Cornell University

“I would found an institution where any person can find instruction in any study.” − Ezra Cornell, 1868

Competing Ferroic Orders The magnetoelectric effect

Basic Training in Condensed Matter Theory 2009

Craig J. Fennie School of Applied and Engineering Physics [email protected]

Basic Training 2009– Lecture 01

2 Basic Training 2009– Lecture 01

Module Outline (Extremely tentative) 1.  Overview and Background

•  Ferro ordering, the magnetoelectric effect •  Complex oxides basics: Types of insulators (i.e., ZSA classifications), Coordination chemistry

2.  Structure and Ferroelectricity •  Basics of space groups •  Soft mode theory, lattice dynamics, group theoretic methods •  Competing lattice instabilities •  microscopic mechanisms, improper FE •  Modern theory of polarization (Berry Phase)

3.  Magnetism •  Basics, exchange interactions, superexchange, Dzyaloshinskii-Moria •  How spins couple to the lattice! Phenomenology and microscopics (spin-phonon, spin-lattice, etc) •  Competing magnetic orders

4.  Lets start putting things together •  Phase competition: magnetic and polar orders and colossal magnetoelectric responses •  Magnetic order induced ferroelectricity •  Ferroelectric induced ferromagnetism, switching magnetism 180˚ with an electric field aren't these

forbidden by some symmetry? NO!)

5.  Finish up loose ends and recent papers I wish I understood better

•  Toroidal moments

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Ab initio

 What does my title mean? “ferro” derived from the Latin “ferrum”→ Iron

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•  Iron is the sixth most abundant element in the Universe

•  While it makes up about 5% of the Earth's crust, the Earth's core is believed to consist largely of an iron-nickel alloy constituting 35% of the mass of the Earth as a whole.

•  Iron is consequently the most abundant element on Earth, but only the fourth most abundant element in the Earth's crust.

•  Most of the iron in the crust is found combined with oxygen as iron oxide minerals such as hematite and magnetite.

http://en.wikipedia.org/wiki/Iron

Yes a play on what I do.


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Ferromagnetism

Magnetization

Energy

F(M) = αM2 + βM4 - MH Spontaneous time-reversal, R, symmetry breaking

M(t)

j(t)

Ordering of spins → spontaneous magnetization

M(-t)

j(-t)

M ≡∑

i

〈Si〉 $= 0

Symmetry properties M(-t) = -M(t)

Time inversion R t → -t

M(t)

j(t)

Space inversion I r → -r M and H are Axial vectors

j(t)

M(t)

M(t)

j(t)

j(t)

M(t)

Mirror reflection m

M(t)

j(t)

M(-t)

j(-t)

Unit cell

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Centrosymmetric lattice

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Ferroelectricity

u ∝ polarization

Energy

F(u) = α u2 + β u4 – Z*⋅u⋅E or

F(P) = αP2 + βP4 - PE

Spontaneous space-inversion, ,I, symmetry breaking

Symmetry properties P(-x) = -P(x)

Time inversion R t → -t Space inversion I r → -r P and E are Polar vectors

P(x,y,z)

P(x,y,-z)

Mirror reflection m

P(x,y,z) P(-x,y,z)

Unit cell

Pα ≈∑

i

Z∗α,β;i〈uβ;i〉 $= 0

Noncentrosymmetric lattice - + -

- + -

+ -

+ -

P(x) P(x)

+ -

P(x)

+ - P(-x)

+ -

+ - +

- + -

x-y

y-z

Spontaneous polarization ⇒ Dipole moment per unit volume

Ordering of polar mode in many cases, this turns out to be a lattice mode


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AntiFerromagnetism

L

Energy

F(M) = αL2 + βL4 - LHq

q≠ 0, Ordering of spins

Symmetry properties L(-t) = -L(t) ; L(-x) = ± L(x)

Unit cell

M ≡∑

i

〈Si〉 = 0

L = S1 − S2

e.g., two sublattice spin system

Depends on details of crystallographic structure!

Cant ignore structure!! i.e., LATTICE + BASIS (don’t just pick your favorite sublattice and ignore the rest!!!)

Does the AFM ordering of spins spontaneously break time-reversal, R, symmetry breaking ? First, what does the question mean? Does the point group of the space group contain R?

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AntiFerromagnetism Does the AFM ordering of spins spontaneously break time-reversal, R, symmetry breaking ? Symmetry properties L(-t) = -L(t) ; L(-x) = ± L(x)

Unit cell L = S1 − S2

e.g., two sublattice spin system (in the ordered phase)

Example 1: two spins sit in the same unit cell in the paramagnetic phase, Yes! R is not a symmetry element. Crystallographic

Unit cell

Magnetic Unit cell

Example 2: two spins sit in different unit cells in the paramagnetic phase, No! the space group contains {R|τ} ⇒ the point group, τ→ 0 contains {R|0}.

n n+{0|τ}


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Ab initio

Def: Linear magnetoelectric effect Landau and Lifshitz, “Electrodynamics of continuous media” Dzyaloshinskii, JETP 1957; Astrov JETP 1960 -> Cr2O3 Hans Schmid, Geneva; Smolenskii USSR

Requirements: 1) broken space-inversion symmetry 2) broken time-reversal symmetry

γ ≡ linear magnetoelectric coefficient

 What does my title mean?

Yes a play on what I do.

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Crystal structure of Cr2O3

!

!"

!"

From Max Mostovoy


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Cr2O3

!"!

#$"

#$"

%

#

!

&

4321 MMMML −+−=

'"

0≠zL

+!!Ez

++!Lz

+!+Hz

3z2xI

zzzzz HEHEL ||αλ =

()**+,-.+(/01/23-3*345+,.6/273(+

( )yyxxz HEHEL +'583-.35,(9

zL∝⊥αα ,||

:;</0-=+-/23-3*+,+-/ >? @/!ABC

m3

z

z

m 3),(3,1

3),2(3,1

±

±

⊥

⊥

point group

From Max Mostovoy

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Cr2O3 m′′3

!!"

#$$%

&

y

x

H

H

!!"

#$$%

&

y

x

E

E

zE

zH

I ′x2 z3

!!"

#$$%

&

−

−

10

01!!"

#$$%

&

−10

01

!!"

#$$%

&

−−

−=

13

31

2

13/2πR

3/2πR

1− 1− 1+

!!"

#$$%

&

−

−

10

01!!"

#$$%

&

−10

013/2πR

1− 1− 1+

ITI =′

!"#$%&'(")*(+,'"$-).'/0)/'+$)%$#$%&12

345(6%(/1/'("

+17"$/'*)8('"/)7%(98

!"#1%'1"/&: ( )yyxxzz HEHEHEF +−−= ⊥αα ||me

;<++$/%'$&)(=)2(.6>)801&$:/zz m 3),(3,1,3),2(3,1 // ±± ⊥⊥

From Max Mostovoy


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Linear magnetoelectric effect

!"#$%

+= EP eχ Hα

=M Eα Hmχ+

!"#$"#%&'()*+,-.+/-- 0$12#&' 345#67898:;#%"#<"#=+>?*@;#0$12#&& AB5#6783B:

C"1"#D(E* 2DF#&% GG9#6783H:

From Max Mostovoy

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Next time

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ME revisited, and basic oxide physics •  ME effect revisited: Toroidal moments •  Complex oxides basics: Types of insulators (i.e., ZSA classifications), Coordination

chemistry

basic training 2009– lecture 01 competing ferroic orders

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