competing ferroic orders - cornell...
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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 04
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2 Basic Training 2009– Lecture 04
Module Outline 1. Overview and Background
• Ferro ordering, the magnetoelectric effect
2. ME revisited, and basic oxide physics • ME effect revisited: Toroidal moments • Complex oxides basics: Types of insulators (i.e., ZSA classifications), Coordination chemistry
3. 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)
4. Magnetism • Basics, exchange interactions, superexchange, Dzyaloshinskii-Moria • How spins couple to the lattice! Phenomenology and microscopics (spin-phonon, spin-lattice, etc) • Competing magnetic orders • Systems: ZnCr2O4, EuTiO3, SeCuO4, TeCuO4
2
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3 Basic Training 2009– Lecture 04
Magnetism and how it couples to the lattice
Tuning AFM ⇔FM – The Goodenough-Kanamori rules
Spin-phonon coupling – ZnCr2O4 → spin-induced phonon anisotropy – EuTiO3 → magneto-capacitance
Spin-lattice coupling – Weak ferromagnetism Fe2O3 – Spin spiral Lifshitz invariant
3
Isotropic Exchange
Dzyaloshinskii-Moria Exchange
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4 Basic Training 2009– Lecture 04
Magnetic properties of localized systems
4
Eu2O3
Eu: 4f7 5d0 6s2 → Eu3+: 4f6 5d0 6s0
Magnetism arises from an incomplete shell
Question: is Eu2O3 magnetic? What do I mean by magnetic? For now I mean, Does a magnetic field couple to the fairly large S=3µB, NO! why? Well magnetic field couples to J=|S-L|
Hund’s Rules 1) Max S =3 2) Max L =3 3) J=|S-L| =0 less than half filled (J=|S+L| if more …)
But is it magnetic? Yes!
〈0|µ|0〉 = gJµB〈0|J |0〉 = 0
χ =N
V
(2µ2
B
∑
n
|〈0|Lz + gSz|0〉|2
En − E0− e2mu0
6me
Z∑
i=1
〈r2i 〉
)
Van Vleck Paramagnetism diamagnetism
Both terms, small and temperature independent (χVV ~ 1/Δ)
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5 Basic Training 2009– Lecture 04
Magnetic properties of localized systems
Currie Weiss Susceptibility
5
Temperature ΘCW = Tc
0
χ-1
TN -ΘCW
AFM PM FM
χ ∝ 1T − θCW
E = −∑
ij
JijSi · Sj + gµB
∑
i
Si · H
kBθCW =23
∑
n
JnznS(S + 1)
Magnetic exchange interactions Mean-field
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6 Basic Training 2009– Lecture 04
Properties of common Antiferromagnets
6
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7 Basic Training 2009– Lecture 04
Effect of strong magnetic field (AFM)
Susceptibility below TN and Spin flop – Hard vs easy axis
7
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8 Basic Training 2009– Lecture 04
Origin of easy/hard axis
Spin-orbit interaction
1. Single-ion anisotropy
2. Anisotropic exchange coupling
8
λ"L · "S
1. Crystal field couple to charge density 2. Charge density couple to spin E-field
overlap
Different energy
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9 Basic Training 2009– Lecture 04
General form of spin-spin coupling
9
+kxS1xS2x + kyS1yS2y + kzS1zS2z
Isotropic symmetric exchange (l=0)
Dzyaloshinskii-Moriya Antisymmetric exchange (l=1)
Anisotropic symmetric exchange (l=2) D and K both spin-orbit effects
D ∼ λJ ≈ (∆g/g)J
k ∼ λ2J ≈ (∆g/g)2J
H = J!S1 · !S2 + !D · (!S1 × !S2)
H = !S1 · J · !S2
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10 Basic Training 2009– Lecture 04
10
Dzyaloshinskii-Moriya
E = |J| 〈Si⋅Sj〉 + ∑ Dij⋅〈SixSj〉
Collinear AFM Spin canting
D = 0 M
D ≠ 0
Relativistic correction to Anderson’s superexchange
• Direction of canting determined by the sign of D
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11 Basic Training 2009– Lecture 04
Weak ferromagnetism vs ME effect
11
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
Cr2O3
L = M1-M2+M3-M4
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
L = M1+M2-M3-M4
Fe2O3
Neel temperature ~500K Morin temp ~260K
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12 Basic Training 2009– Lecture 04
Exchange: Background
12
Hund’s rule ➟ like FM
Covalent bond ➟ like AFM
bonding
Anti-bonding
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13 Basic Training 2009– Lecture 04
Exchange: Background
13
If they are the same orbital ➟ AFM
If they are the different orbital ➟ FM
Direct exchange
TM-d O TM-d
Superexchange 90˚ SE
90˚ Hund’s rule J_H coupling on the anion
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14 Basic Training 2009– Lecture 04
Exchange: Background
14
NiO
Competing interactions 1. Strong
AFM Ni-O-Ni 180
2. Weaker FM 90˚ SE
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15 Basic Training 2009– Lecture 04
Exchange: Background
15
Arbitrary angle
J =J90 sin2θ + J180 cos2θ
90˚ 180˚
Empirically 135˚
SeCuO3 FM TeCuO3 AFM
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16 Basic Training 2009– Lecture 04
ACr2X4: Cubic Spinel Structure
16
eg
t2g ⇒ S=3/2
Cr3+: 3d3 4s0
What is Moment? µ ~ S =3/2 Why -> orbital dof quenched
Superexchange pathways
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17 Basic Training 2009– Lecture 04
ACr2X4: Cubic Spinel Structure
Network of edge sharing octahedra
17
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18 Basic Training 2009– Lecture 04
Background: Spinels
18
ACr2X4
• A= Zn, Cd, Hg
• X= O, S, Se
eg
t2g ⇒ S=3/2
a (Å) Theory Exp.
CdCr2S4 10.12 10.24 CdCr2Se4 10.63 10.74
HgCr2Se4 10.65 10.74
ZnCr2O4 8.26 8.31
CdCr2O4 8.54 8.59
Ferromagnetic Insulators
Tc ~100K,
TΘ ~200K
Anti-ferromagnetic Insulators
TN ~10K,
TΘ ~ 100K
Cr3+: 3d3 4s0
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19 Basic Training 2009– Lecture 04
Exchange interactions Direct Cr-Cr exchange → AFM 90° Cr-O-Cr SE → FM
19 oxygen
chromium
AFM → FM a) as lattice constant increases b) as electronegativity of anion decreases
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20 Basic Training 2009– Lecture 04
Huge spin-phonon coupling
20
3-fold T1u phonon mode
Singlet A2u
Doublet Eu
Temperature (K)
Tn= 12.5K
Symmetry lowering at Tn
Oh → D4h Cubic to tetragonal
Phonon splitting at Tn
T1u → A2u ⊕ Eu
AFM-ZnCr2O4
Exp: λ =6-10 cm-1
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21 Basic Training 2009– Lecture 04
21
Spin-phonon coupling
J(u)
J(u+δ)
Phonon modulated exchange interaction Baltensperger and Helman, Helvetica physica acta 1968.
e.g. can understand large spin-phonon coupling in ZnCr2O4 Fennie and Rabe, Phys. Rev Lett. May 2006
Eph = 1/2 ω02u2
E = E0 +Ephonon+ Espin
Esp = -∑Jij 〈Si⋅Sj〉
⇒ ω2∝ ω02 - ∂2J/∂u2 〈Si⋅Sj〉
renormalized bare magnetic contribution phonon phonon
J(u) ≈ J(0) + 1/2 ∂2J/∂u2 〈Si⋅Sj〉 u2
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22 Basic Training 2009– Lecture 04
22
Spin-phonon coupling
J(fu)
J(fu+δ)
Phonon modulated exchange interaction Baltensperger and Helman, Helvetica physica acta 1968.
e.g. can understand large spin-phonon coupling in ZnCr2O4 Fennie and Rabe, Phys. Rev Lett. May 2006 to lowest order in Si Sj
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23 Basic Training 2009– Lecture 04
Origin of large anisotropy
23
spin up
spin down
Cr ions
Direct-exchange length scale, α-1
Exp: α=8.9 Å-1
Theory: α=9.05 Å-1
f3 has significant anisotropy in force constant matrix
x y z
f3: One set of T1u Partner functions
z-mode x-mode (or equivalently y-mode)
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24 Basic Training 2009– Lecture 04
24
TN
Bulk Eu2+Ti4+O3: Ground state antiferromagnetic paraelectric
• r(Eu2+) ~ r(Sr2+); Cubic perovskite • Eu2+ → J=S=7/2; Tn ~ 5.5K, G-type AFM
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25 Basic Training 2009– Lecture 04
25
Perovskites and the Period Table
Substitutions on A, B or both (A1-xA’x)(B1-yB’y)O3 Random distribution or ordered
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26 Basic Training 2009– Lecture 04
Phonon anomaly at Tc
FM- CdCr2S4
26
Symmetry lowering at Tc
Oh → Oh
Cubic to cubic (ignoring LS)
No phonon splitting at Tc
T1u → T1u