This dissertation investigates various aspects of helimagnets. Helimagnets are magnets with spins aligned in helical order at low temperatures. It exists in materials of crystal structure lacking the spatial inversion symmetry. The helical order is due to the Dzyaloshinskii-Moriya (DM) mechanism. Examples of helimagnets include MnSi, FeGe and Fe<sub>1-<italic>x</italic></sub>Co<sub><itaic>x</italic></sub>Si. A field theory appropriate for such magnets is used to derive the phase diagram in a mean-field approximation. The helical phase, the conical phase, the columnar phase and the non-Fermi-liquid (NFL) region in the paramagnetic phase are discussed. It is shown that the orientation of the helical vector along an external magnetic field within the conical phase occurs via two distinct phase transitions. The columnar phase, believed to be a Skyrmion lattice, is found to exist as Abrikosov Skyrmions near the helimagnetic phase boundary, and the core-to-core distance is estimated. The Goldstone modes that result from the long-range order in the various phases are determined, and their consequences for electronic properties, in particular, the specific heat, single-particle relaxation rate and the electrical conductivity, are derived. In addition, Skyrmion gases and lattices in helimagnets are studied, and the size of a Skyrmion in various phases is estimated. For isolated Skyrmions, the long distance tail is related to the magnetization correlation functions and exhibits power-law decay if the phase spontaneously breaks a continuous symmetry, but decays exponentially otherwise. The size of a Skyrmion is found to depend on a number of length scales. These length scales are related to the strength of DM interaction, the temperature, and the external magnetic field.

1. Properties of Metallic
Helimagnets
Kwan-yuet Ho
Institute for Physical Science and Technology &
Department of Physics
University of Maryland
1UMD
Apr 3rd, 2012

2. Outline
 Helimagnets (Introduction)
 Model and Energy Scales
 Phase Diagram
 Goldstone Modes
 Electronic Properties
 Columnar Phase and Skyrmions
 Conclusion
UMD 2

3. Helimagnets
 Helimagnets: materials
exhibiting
helimagnetism in one
of the phases.
 In helimagnetism,
there is ferromagnetic
order on each plane.
 The direction of the
spin rotates as one
goes along the helix.
 Low-temperature
system
 q-1 >> a
 Examples: MnSi,
FeGe, FexCo1-xSi
3UMD
For MnSi,
Lattice constant: 4.56Å
Helical wavelength: 180Å
Ordering temperature: 29.5K
Resistivity ~ 0.33μΩ cm
(T=0)
(kFl~6000)(Ishikawa, Tajima, Bloch, Roth
1976)
(Bauer et al
2010)
q
(Uchida, Onose,
Matsui, Tokura
2006)

4. Helimagnets
 Helimagnets
generally have richer
phase diagrams than
the other magnets.
 Helimagnets are
sensitive to a
change of pressure.
Helical order is
destroyed at high
pressures.
 Universal quantum
fluctuations lead to
the tricritical point
(TCP). UMD 4
(Thessieu et al 1997)
(Pfleiderer, Julian, Lonzarich
2001)
(Kirkpatrick, Belitz, Vojta
1997)Experimental Phase Diagrams of MnSi

5. Helimagnets
 An exotic columnar
phase (A phase) at
T≈Tc and
intermediate H.
 Six-fold symmetry
was found in neutron
scattering.
 It is a 2D hexagonal
columnar lattice,
confirmed by Lorentz
TEM images.
 Believed to be a
Skyrmion lattice.
UMD 5
(Mühlbauer et al 2009)
MnSi
(Yu et al 2010)Fe0.5Co0.5Si
(For MnSi, see Mühlbauer et al
2009; for FeGe, see Yu et al
2010; for Fe0.5Co0.5Si, see Yu et al

6. Helimagnets
 Helimagnets
generally have richer
phase diagrams than
the other magnets.
 Helimagnets are
sensitive to a
change of pressure.
Helical order is
destroyed at higher
pressures.
 Universal quantum
fluctuations lead to
the tricritical point
(TCP). UMD 6
(Thessieu et al 1997)
(Pfleiderer, Julian, Lonzarich
2001)
(Kirkpatrick, Belitz, Vojta
1997)Experimental Phase Diagrams of MnSi

7. Helimagnets
 The disordered phase at p>pc has a
non-Fermi-liquid (NFL) transport
properties that Δρ~T3/2.
UMD 7
(Pfleiderer, Julian,
Lonzarich 2001)

8. Helimagnets
 These properties are due to the huge
fluctuations.
 Like cholesteric liquid crystals, a pure
helimagnet has a Goldstone mode
(called helimagnon) with “transverse”
susceptibility χ┴
-1~k||
2+ck┴
4.
◦ True helimagnetic long-range order cannot
exist in d=3.
 Like columnar phases in liquid crystals,
columnar phase in helimagnets have a
fluctuation spectrum k┴
2+ckz
4.
UMD 8
(Belitz, Kirkpatrick, Rosch
2006)
(Kirkpatrick & Belitz 2010)

9. Outline
 Helimagnets (Introduction)
 Model and Energy Scales
 Phase Diagram
 Goldstone Modes
 Electronic Properties
 Columnar Phase and Skyrmions
 Conclusion
UMD 9

10. Model and Energy Scales
 Ferromagnets can be described by the
LGW functional.
UMD 10
(Heisenberg 1930s)

11. Model and Energy Scales
 To stabilize a
helimagnet over a
ferromagnet,
Dzyaloshinski-
Moriya (DM)
interaction is
needed.
 Spin-orbit coupling
constant:
gso(dimensionless)
c=akFgso
 It exists in systems
with no inversion
symmetry. UMD 11
DM interaction
(Dzyaloshinski 1958, Moriya
1960)
B20 cubic crystal, P213
gso ≈ 0.05 for MnSi

12. Model and Energy Scales
 LGW functional
UMD 12
c~gso b, b1 ~gso
2 q=c/2a~gso
(Ho, Kirkpatrick, Sang, Belitz
2010)
gso <<1
Cubic
anisotropyPinning
(Bak & Jenson 1980)
Space group P213
(Belitz, Kirkpatrick,
Rosch 2006)
v ~gso
4
O(gSO
0)
O(gSO
2)
O(gSO
4)

13. Outline
 Helimagnets (Introduction)
 Model and Energy Scales
 Phase Diagram
 Goldstone Modes
 Electronic Properties
 Columnar Phase and Skyrmions
 Conclusion
UMD 13

14. Phase Diagram
 Mean-field theory
 H=0: pinned helical
phase, q in (1,1,1)
for b<0, |b|~gso
2
 M(x)=msp
[cos(q.x)e1+sin(q.x)e
2]
 0<H<Hc1: q rotates
from (1,1,1) to H
 M acquires a
homogeneous
component along H
 Elliptical conical
phase near Hc1
UMD 14
(Ho, Kirkpatrick, Sang, Belitz
2010)
(Ishikawa,
Tajima, Bloch,
Roth 1976)
MnSi

15. Phase Diagram
 Hc1 < H < Hc2:
conical phase
 q aligns with H
 M(x)=msp
[cos(qz)x+sin(qz)y] +
m//z
 When H increases,
msp decreases and
m// increases.
 msp vanishes at
H=Hc2
 Columnar phase: 2D
hexagonal lattice of
columns
UMD 15

16. Phase Diagram
UMD 16
(Thessieu et al 1997)MnSi
(Ishimoto et al 1995)
Fe0.8Co0.2Si

17. Outline
 Helimagnets (Introduction)
 Model and Energy Scales
 Phase Diagram
 Goldstone Modes
 Electronic Properties
 Columnar Phase and Skyrmions
 Conclusion
UMD 17

18. Goldstone Modes
 A pure helimagnet breaks the continuous
translational symmetry  Goldstone mode 
unusual electronic properties through
electron-Goldstone-mode coupling
 M(x)=msp (cos[qz+ϕ(x)], sin[qz+ϕ(x)],0)
 Energy fluctuations ~ ∫d3x [∇ϕ(x)]2  wrong,
because free energy is independent of the
direction of q. Perpendicular fluctuations
should not cost extra energy.
 The next available order of perpendicular
fluctuation ~ ∫d3x [∇⊥
2ϕ(x)]2
 Fluctuation energy ~ ∫d3x {[∂zu(x)]2 +
c[∇⊥
2u(x)]2}, leading to Goldstone mode
kz
2+ck┴
4
UMD 18
(Belitz, Kirkpatrick, Rosch
2006)

19. Goldstone Modes
 Magnetic field and small crystal field
effects (~gSO
4) make the Goldstone
mode less soft.
 For conical phase, χ┴
-
1~kz
2+H2k┴
2+ck┴
4
 For pinned helical phase, χ┴
-1~kz
2+|b|
k┴
2+ck┴
4
UMD 19
(Ho, Kirkpatrick, Sang, Belitz
2010)

20. Goldstone Modes
 There are two
Goldstone modes in
columnar phase, as
the translational
symmetry breaking
is on a plane.
 By similar argument,
the Goldstone
modes have the
spectrum χ┴
-
1~k┴
2+c’kz
4
 Disordered columnar
phase has one
Goldstone mode of
the same spectrum.
UMD 20
(Kirkpatrick & Belitz 2010)

21. Outline
 Helimagnets (Introduction)
 Model and Energy Scales
 Phase Diagram
 Goldstone Modes
 Electronic Properties
 Columnar Phase and Skyrmions
 Conclusion
UMD 21

22. Electronic Properties
 Specific heat:
 Helimagnon: C(T) ~ T2
 Columnar phase: C(T) ~ T5/2
UMD 22
(Belitz, Kirkpatrick, Rosch 2006)
(Kirkpatrick & Belitz 2010)
(Ho, Kirkpatrick, Sang, Belitz
2010)

23. Electronic Properties
 Single-particle relaxation rate:
 Helimagnons (k||~T, k┴~T1/2): τ-1~ T3/2
(clean); τ-1~ T (ballistic disorder)
 Columnar phase (k||~T1/2, k┴~T): τ-1~
T2 (clean); τ-1~ T3/2 (ballistic disorder)
UMD 23
(Belitz, Kirkpatrick, Rosch 2006)
(Ho, Kirkpatrick, Sang, Belitz
2010)
(Belitz, Kirkpatrick, Rosch 2006)
(Kirkpatrick & Belitz 2010)

24. Electronic Properties
 Transport relaxation rate:
 Helimagnons (k||~T, k┴~T1/2): τ-1~ T5/2
(clean); τ-1~ T (ballistic disorder)
 Columnar phase (k||~T1/2, k┴~T): τ-1~
T3 (clean); τ-1~ T3/2 (ballistic disorder)
UMD 24
(Belitz, Kirkpatrick, Rosch 2006)
(Belitz, Kirkpatrick, Rosch 2006)
(Ho, Kirkpatrick, Sang, Belitz
2010)
(Kirkpatrick & Belitz 2010)

25. Electronic Properties
 In ballistic disorder, both relaxation
rates of the columnar phase show
T3/2-dependence.
 The NFL phase that has electrical
resistivity T3/2 might be a liquid of
columns.
UMD 25

26. Outline
 Helimagnets (Introduction)
 Model and Energy Scales
 Phase Diagram
 Goldstone Modes
 Electronic Properties
 Columnar Phase and Skyrmions
 Conclusion
UMD 26

27. Columnar Phase and
Skyrmions
 Columnar phase:
believed to be a
Skyrmion lattice
 Skyrmion: a
topological object
 Winding number:
W=(1/4π)∫d2x
(n.∂xn×∂yn), where
n=M/|M|.
 n = (-2yl, 2xl, (x2+y2-
l2))/(x2+y2+l2), in σ
model.
 Algebraic decay at
large distances
 Size: believed to be
-1
UMD 27
W=-1
(Pfleiderer & Rosch 2010)
(Abanov & Prokrovsky
1998; Belavin & Polyakov
1975)
(Skyrme 1961)

28. Columnar Phase and
Skyrmions
 n(x)=-sinθ(ρ)φ
+cosθ(ρ) z
 Core size R, defined
by core behavior
θ(ρ) = π (1-ρ/R)
 Tail length lT, defined
by the long-range tail
exponential decay
length exp(-ρ/lT).
 Matching length L as
the size.
UMD 28
(Ho, Kirkpatrick, Belitz 2011)
lT
(Röβler, Leonov, Bogdanov 2011) θ: polar angle between the
spin direction and the
ferromagnet

29. Columnar Phase and
Skyrmions
 The Skyrmion size is
not always q-1, but it is
the result of the
competition of different
length scales, e.g.,
correlation lengths,
magnetic length, q-1
UMD 29
(Ho, Kirkpatrick, Belitz 2011)

30. Columnar Phase and
Skyrmions
 T~Tc, intermediate H:
columnar phase (A
phase)
 Believed to be 2D
hexagonal columnar
Skyrmion lattice
 With some efforts, it
can be derived from
LGW functional
 Core-to-core
distance in a
Skyrmion lattice ~
qξp
2 ~ (c/a) (a/r) ~ c/r
UMD 30
(Ho, Kirkpatrick, Belitz 2011)
(Mühlbauer et al 2009)
(Han, Zang, Yang,
Park, Nagaosa,
2010)

31. Outline
 Helimagnets (Introduction)
 Model and Energy Scales
 Phase Diagram
 Goldstone Modes
 Electronic Properties
 Columnar Phase and Skyrmions
 Conclusion
UMD 31

32. Conclusion
 Helimagnets are more favored than
ferromagnetism through DM
interaction.
 Helimagnets have a richer phase
diagram than other magnets in
general.
 Helimagnets have softer Goldstone
modes, resulting in huge fluctuations
and special electronic properties.
 The size of Skyrmions in helimagnets
is the result of competition of various
UMD 32

33. Acknowledgments
 Theodore Kirkpatrick (UMD)
 Dietrich Belitz (UOregon)
 Yan Sang (UOregon)
UMD 33