Second harmonic generation on multiferroics
Optical spectroscopy seminar
2013 spring
Orbán Ágnes, Szaller Dávid
2013. 04. 04.
Outline
•Introduction to nonlinear optics
• Theoretical background – selection rules
•The basic idea of the instrumental setup
• The structure of multiferroic materials
• Examples:
• Cr2O3 – domain structure, spin orientation in spin flop phase
• Hexagonal manganites – spin orientation
•YMnO3 – multiferro domains
Introduction to nonlinear optics
• Goal: determination/analysis of magnetic and/or electric
ordering in crystal strucutres
• Ususal method: x-ray, electron, neutron diffraction →
microscopic structure
• Nonlinear optics:
• spatial resolution: 1- 100 µm → visible domain structure
• surface sensitivity
• symmetry considerations → resolution of structural
ambiguities
• temporal resolution: 1-10 fs → spin dynamics
• faster, cheaper
How fast?
Introduction to nonlinear optics
• Hamiltonina for light-matter interaction, linear terms:
0
1
0
0
δH  ε0 pi  Ei   ε0Θij   i E j   μBmi  Bi   ...
3
• Induced polarozation:
ΔP
1
1
ω
Θ
ω
m
ω

ε
χ
E

ε
χ

E

μ
χ
B





0
ij
j
0
ijk
j
k
B
ij
j 0  ...
ω
0
0
3
• For strong EM fields
ΔP  ε0 χ 1ij E j ω  ε0 χ 2ijkEjωEkω  ε0 χ 3ijkmE j ωEkωEmω  ...
• The Kubo-formula:
Types of ferro orderings
+q
-q
M
-M
-P
P
-q
inversion
r → -r
+q
j
time
reversal
-j
t → -t
• spontaneous symmetry-breaking
• multiferroic: ferroic order of more than
one degree of freedom
• extended definition: materials having
multiferroic sublattices
 includes antiferromagnetic and
ferrimagnetic order
Basic symmetry arguments
• Paramagnetic or nonmagnetic, centro-symmetric material
– Spatial inversion and time reversal symmetry, 11
𝑃2𝜔 ~𝜒 𝑒𝑒𝑒 𝐸 𝜔 𝐸 𝜔
Basic symmetry arguments
• Paramagnetic or nonmagnetic, centro-symmetric material
– Spatial inversion and time reversal symmetry, 11
𝑃2𝜔 ~𝜒 𝑒𝑒𝑒 𝐸 𝜔 𝐸 𝜔
𝑀2𝜔 ~𝜒 𝑚𝑒𝑒 𝐸 𝜔 𝐸 𝜔
𝜒 𝑒𝑒𝑒 = 0
Basic symmetry arguments
• Paramagnetic or nonmagnetic, centro-symmetric material
– Spatial inversion and time reversal symmetry, 11
𝑃2𝜔 ~𝜒 𝑒𝑒𝑒 𝐸 𝜔 𝐸 𝜔
𝜒 𝑒𝑒𝑒 = 0
𝑀2𝜔 ~𝜒 𝑚𝑒𝑒 𝐸 𝜔 𝐸 𝜔
𝜒 𝑚𝑒𝑒 (𝑖)
• Magnetic phase, 1
Basic symmetry arguments
• Paramagnetic or nonmagnetic, centro-symmetric material
– Spatial inversion and time reversal symmetry, 11
𝑃2𝜔 ~𝜒 𝑒𝑒𝑒 𝐸 𝜔 𝐸 𝜔
𝜒 𝑒𝑒𝑒 = 0
𝑀2𝜔 ~𝜒 𝑚𝑒𝑒 𝐸 𝜔 𝐸 𝜔
𝜒 𝑚𝑒𝑒 (𝑖)
• Magnetic phase, 1
𝜒 𝑒𝑒𝑒 (𝑐)
Introduction to nonlinear optics
• Transformation properties of magnetic crystals:
32 crystallographic pont groups  time-reversal operation (T) → 122 magnetic
point groups
• Nonmagnetic crystals: reciprocal susceptibilities, χijk  i 
magnetic crystals: additional, nonreciprocal susceptibilities, linear dependence
on the relevant order parameter, vanishes above Tc, χ  c 
ijk
• Total SH intensity:
• Interference term: linear dependence on χ(c) → magnetic order parameter
• For fixed polarizations χ(i) constant, χ(c) depends on the domain structure
Hexagonal manganites
• RMnO3 with R=Sc, Y, In, Ho, Er, Tm, Yb, Lu
• simultaneous ferroelectric and frustrated
triangular antiferromagnetic ordering
• P63cm in the ferroelectric paramagnetic
phase
𝑃63 𝑐𝑚
𝑃63 𝑐𝑚
M. Fiebig, R.V. Pisarev, JMMM, 272 e1607 (2004)
Hexagonal manganites
• RMnO3 with R=Sc, Y, In, Ho, Er, Tm, Yb, Lu
• simultaneous ferroelectric and frustrated
triangular antiferromagnetic ordering
• P63cm in the ferroelectric paramagnetic
phase
Phys. Rev. Lett. 84, 5620 (2000)
𝑃63 𝑐𝑚
𝑃63 𝑐𝑚
M. Fiebig, R.V. Pisarev, JMMM, 272 e1607 (2004)
Hexagonal manganites
• RMnO3 with R=Sc, Y, In, Ho, Er, Tm, Yb, Lu
• simultaneous ferroelectric and frustrated
triangular antiferromagnetic ordering
• P63cm in the ferroelectric paramagnetic
phase
Phys. Rev. Lett. 84, 5620 (2000)
M. Fiebig, R.V. Pisarev, JMMM, 272 e1607 (2004)
Cr2O3 : magnetic spectroscopy and domain topography
• Crystal and magnetic structure:
• Cr3+ in distorted
octahedral errengement of
O2- , chains along the z axis
• AF magnetic order
• order parameter: L vector
• above TN: centrosym.
point group 3̅m
• below TN: 3̅m
m
m
m
m
• experiment: k ǁ z → χm  i   χ yyy  i    χ yxx  i    χ xyx  i    χ xxy  i 
m
m
m
m
χm  c   χ yyy
 c    χyxx
 c    χ xyx
 c    χ xxy
c 
Cr2O3 : magnetic spectroscopy and domain topography
• cirkuláris bázisban: E = E+e+ + E-e- + Ezez, where e±=± (-1)* (1/√2)(ex±iey)
• incoming left cirkularly pol. light → right circularly pol. SH
incoming right cirkularly pol. light → left circularly pol. SH
Cr2O3 : magnetic spectroscopy and domain topography
•
, where C
is constant and the second term is the interference
• change of interference term: reversing the circular
polarization or AFM vector
• same spectra, but with reversed dependence of σ
T = 295 K (<TN), exposure time
35 min, σ+ polarized light
T = 295 K (<TN), exposure time
35 min, σ- polarized light
T = 325 K (>TN), exposure time
15 min, σ+/- polarized light
Cr2O3 : spin-flop phase
• Crystal and magnetic structure:
• Cr3+ in distorted octahedral errengement
of O2- , chains along the z axis
• AF magnetic order
• order parameter: L vector
• above TN: centrosym. point group 3̅m
• below TN: 3̅m
• spin-flop phase: below TN, B=5.8 T ǁ z, AFM
order
• 3-fold rotation is lost, six possible domains
• two possibilities:
•Lǁy, where y is the twofold axis, 2/m
• Lǁx, where x is the glid plane, 2/m
• experiment: Ey polarization
Magnetic dipole SHG
𝑀2𝜔 ~𝜒 𝑚𝑒𝑒 𝐸 𝜔 𝐸 𝜔
M. Fiebig, et. al. PRL, 87 137202 (2001)