Topological Crystals: as aNew Paradigm
Satoshi Tanda
Dept. of Appl. Phys.,
Center of topological Science&tech.
SapporoHokkaido Univ.,,Japanhttp://exp-ap.eng.hokudai.ac.jp/index.html
MRS, Boston, 26 Nov 2012Outline(1) Introduction(2) Crystal Growth Covalent wires(3) Topological Crystals
Ring, Möbius, Figure8, Hopf-link Crystals
Mechanism(4) Classification of Topological Crystals(5) Electronic properties: AB effect, Frolich supercon.
(6) Summary

Introduction: Escher’s ants
Ants return to the original point
after two laps on Möbius Strips

No distinction between theObverse and the Reverse

An exotic one-sided world !!
from Artists to Scientists

Möbius Strips II (1963)
by M. C. Escher

Ants

Electrons,Spins
Photons,CDWs,Super

Strips

Lattice,Crystal
Is it Possible ?
What happens ?

Our Goal

1. Topological Crystallography (Ladder)
2. Quantum topological effects in the
topological crystals (Ants)
Macroscopic quantum coherence
in nontrivial topological space

Topological gauge fields

Strategy Is it Possible ?
For synthesize of topological crystals
Requirements★ One-dimensional wire

Organic Protein
polymerDNAToo flexible(CH)x(SN)x★ Rigidity

Inorganicpolymer

MX3,MX2C

Inorganic Covalent wires
Requirement for twisting and bending
materials such as Mobius crystals

chain

Semi-flexible !

Covalent Intrachain…

…

Typical materials
NbS3 NbSe3
TaS3 TaSe3
TiS3 ZrTe3
MX3

Van der waals force: Inter-chain

MX3 : CDW and Superconductors

SapporoResearches in Hokkaido Univ. 35 Years
Discovery of NbSe3 and TaS3: Yamaya, Sambongi,Tsutsumi (1977)
Memory Effect of CDW in NbSe3: Ido Oda, Okajima,Sambong (1986)
Ring Crystals in NbSe3: Kawamoto, Okajima, Yamaya, Tanda(1999)
Möbius Crystals in NbSe3 Tanda Tsuneta Inagaki Okajima YamayaHatakenaka

Hopf-link crystals

Nature 417 397 (2002)

Matsuura, Yamanaka, Hatakenaka, Matsuyama, Tanda

PRB (2006)

Topological quantum Effects in MX3: Tsuboa, Matsuura, Kumagai, Tanda

PRB (09,10,11,12)
Chiral CDW in TiSe2: Ishioka, Oda, Ichimura, Tanda PRL (2010,2012)
MX2 Nanotubes (03,04), Topology change(09,10,11,12)

Ring Crystals

NbSe3

TaSe3

Seamless!!

TaS31999

Synthesize and Condition
Chemical Vapor
Transportation

1. heating

Se Atmosphere

Closed Quartz Tube
<10-7Torr(initial vacuum)
10 days
Nb,Ta (99.999%),
Se (99.9999%)
FurnaceTemperature
600℃~800℃

Starting materials (Nb,Se) is sealed in an
evacuated ampoule.

Nb

Quartz Tube

2.transportation
3.final stage

After a few weeks of heating, products aretransported to the lowest-temperature end
where vapor pressure is also the lowest.

Nonequilibrium condition
Low Temp.

HighTemp.

Se Atmosphere

Quartz Tube

Nb

Large Temperature
gradient （more than150℃）

Se circulates throughVapor , mist , liquid droplet
Earth system makeLife polymer
Analogous method
These droplet need for formation of ring crystals.

Formation of Ring Crystals
The ribbonshaped NbSe3
crystals grownin the viscous
Se droplet are
bent due to Se
surface tension.

a growing crystal caneat its own tail.

NbSe3 fiber
circulate on theequator of Se
droplets during
growthSeamless ring

Disks and Tubes Crystals

Tubes

Disk with corn

Disk with hole

UFO ??

Hole radius
is less than
1 μｍ !!

Tube in tube

This system is ideal matter in observation ofInterference effect or AB effect of CDW

CDWCorrelation
length is 5 μm ,which is larger
than holeradius!

Figure-8 Crystals (2π-twist)
Knot with Framingin Math. term＋2π-Twist -Band－2π-Twist -Band
No evidence of breakingof Chiral Symmetry

Growing
Self-crossing
Maximum Crystals

2002 Nature

Mechanism of formation ofFigure 8 Crystals (2π twist)=

Möbius Crystals (π twist)

Copyright ©
NatureNbSe3

Copyright © Nature

NbSe3

TaSe3

Copyright ©
NatureBending to Twisting

formation mechanism of Mobius crystals
shear
Difficulties → twisting
Question is how the twisting is
introduced without actual shear forces

AnswerCrystal symmetry

The bending-twisting conversion

BendingTwistingComplianceΘTwist Angle： ΘMonoclinic , triclinic
TwistBentCross SectionComplianceＺ

Elasticity ( Expansion of Hook’s law )X＝(1/K)×F
F=kX

ＹＸ

Ｓ１１

Extensionandcompression

Ｓ１２
Ｓ１６
Ｓ４６

Ｓab:ComplianceCubic
・ : zero●: nonzero

Stress:σ(tensor) Strain:εTetragonal

Orthornbic

Shear

Ｓ６６

Monoclinic

Triclinic

Lower symmetry

Evidence of Twistingyxz

Ｓ ３５

Monoclinic NbSe3

・: zero●: nonzero●: Shear term for Monoclinic

twisting is introduced bybending through this mechanism
without actual shear forces
during the growth processes
along surface

Formation of Möbius Crystals

To form a Möbius strip (1π) ,a twist is needed to be
introduced during spooling.

Can we introduce the concept of
topology into crystals ?

Originally, What’s Crystal ?

Bragg reflection !
Definition of International Unionof Crystallography (1991)

Definition of Crystals
Snow crystal

Crystals show Bragg reflection
Bragg reflection of electron beam/X-ray

Mathematically, Fourier transform from realspace (lattice) to k-space (Bragg spots)

Reciprocal
spacespots

Quasicrystal

Crystals and quasicrystal
show sharp Bragg spots

Topological crystal?
Copyright © 2002 Nature

X-ray Diffraction: Ring
Imaging Plate

Ring

X-ray

X-ray

Bragg RingLine is homogeneous

Ring
Single Crystals !!

Texture pattern in 8 crystal

Short strong arcs are
diffracted fromsemi-straight partscK-space

Long weak arcs
are diffracted fromhair-pin curvesbReal space

Classification of Topological Crystals
ω: Wedge Disclinations
ω* : TwistConic

ω= 1/2π
ω*= 0πTopologicalCrystals

ω= 2πω*= 0π

New Definition !
Knot Theory
MöbiusRing

ω*, ω is independent,respectively,
due to the topological defect theoryof the crystals, as yet. (by Frank)

Figure-8NbSe3
ω= 2π
ω*= 0π
LLkk == 10Copyright on
Natureω= 2π
ω*= 1π
LLkk == 3/2
1/2

Lk= （ω＋ω*）/ 2π-1
Linking Number

ω= 2π
ω*= 2πLk = 1

Global Wedge +Global Twisting
Knots Crystals !!
1*fω,ω
dV = Lk
crystal2π

Discover Hopf-Link Crystals

Topologically linked crystals
T.Matsuura, M.Yamanaka, N. Hatakenaka,T. Matsuyama, and
S. Tanda, Journal of Crystal Growth 297, 157 (2006).

Hopf-link crystals cannot be categorized
by the linking number of Knots crystals andusual point groups → New Classification

Classification By Embedding manifolds
Lk = 0

Lk = 1/2

Lk = 1

Hopf-Link Crystals
Matsuura,Matsuyama,Tanda：JCG (2006)

RingMobius

8？
SphereKlein-Bottle

Torus

Embeddingmanifolds
Double-Torus

We propose New Classification with Embedding manifolds

Category of Crystals
Bragg reflection

International Union of Crystallography(1991)

Translational

Superlattice

Point group symmetry1D,2D Crystals

CrystalAperiodic
5-fold symmetry,

SolidsIncommensurate

inorganicPolymer
LiquidCrystals

Liquidorganic

Topological Crys.
Topology, Knots, Links
Embedding manifolds

Incomme. Crys.
Quasicrystals
Ring Crystals
Lk=0

Mobius Crys.
Lk=1/2Fig-8 Crys.
Lk=1

Plastic
Crystals

Amorphous

Hopf-link Crys.

Topologicalmatter !!

EmbeddingSpherical（C60, Onion)

Our Goal

1. Topological Crystallography (Ladder)
2. Quantum topological effects in the
topological crystals (Ants)
Macroscopic quantum coherence
in nontrivial topological space

Topological gauge fields

NbSe3 : Charge Density Waves
CDW: Charge-Density-Waves

Chainaxis

Covalent wireSe

Tc=145K

Nb

Tc=52K

Fiberstructure

Transition
A periodic charge modulation

Macroscopic wave function

Sliding of Macroscopic wave function
Fröhlich superconductors : phason

CDW sliding
sliding

CDW
looptime
Eternal current

Real space topology
I. Circulating Sliding Current of CDW
By Shapiro Steps in the Loop
II. AB-Effect of CDW in the Loop

Shapiro steps （Sliding state)
potential

E>ET interference of phase
E=d 2dne20sin( E dc Eac sin t )0
E<ET dt 2
dtm*E>ET  at  b sin(t   )CDW:Shapiro steps : Cut-ring and needle NbSe3
Vac= 0.006 V

T=120K

dV/dI [W] (relative)

Josephson Junction
Shapiro steps

FieldsIac:–20

10MHz
7MHz
5MHz
2MHz

15

Shapiro, PRL, (1963)–0.002Zettel et.al, SSC, (1983)

0Idc[A]

1MHz0.002

★★ Shapiro peaks in CDW loops
Idc+Iacsin(2fext)

T = 120 K

1W

Outer radius：120 mm
Inner radius:10 mm

dVdc / dIdc [W]Vac = 0 mVNbSe3 ring

1/1-I CDWp f ex2eNq

p/q=1/1
1/2

1/1+

1/2 1/1

2/1

Vac = 65 mV

fex = 200 MHz

2/1 3/1

4/1 5/1
80 MHz

0

2/1-

2/1+

2dc bias current Idc [mA]

Subpeaks at both sides ofShapiro peaks are observed!

4

Analysis of peak positions
fex = 40, 80, 100, 200, 300 MHz
2Idc + Iac sin 2  fext

1+
2/–
1
2/ 2/1

Ip / fex [mA/MHz]

3–

3/2

2

1/1 +

1/1 1/1 –p/q+
p/q
p/q0.02

Coefficient of p/q

2/3
1/2 +

1

1/2 1/2 –1/300

I CDWp f ex
I2eNq100

200

Frequency fex [MHz]300

dVdc/dIdc

Peak Position Ip = Idc – Ith [mA]

Vac = 65 mV

5/4/1 4/1+4/1–
3/1 3/ 3/1
1 +

0.04

Fall into
3 line002p/q-p/q–36.256x10p/q –3
5.904x10+5.348x104Shapiro Index p/q

Discovery of Beat peak only in the loop CDW !!

Idc

–3

Circulating current
I CDW  I

I CDW  II CDW

I CDWδII 

Temporallychangingpf exq

I CDW  I p
 f ex
2eNq

I CDWp
 f ex
2eNq
dVdc/dIdc

p/qp/q-p/q+

I

IIdc

Additional current
must exist in theloop CDW

Damping time of Circulating current
If CDW is circulating, thedamping time can be estimated by
Circumference： 10-4 m
δＩ

Phason velocity： 104 m/s
Circumference / Phason velocity = 10-8 s＞＞ NbSe3 Phason damping time: 10-11 s
Richard and Chen, Solid State Commun. 86 485 (1993)

Elongation of damping time !!

→ Precursor of Fröhlich supercurrentT. Matsuura, K. Inagaki, and S.Tanda, Phys.Rev.B 79, 014304 (2009).

Real space topology
1. Circulating current of CDW
By Shapiro Steps in the Loop
2. AB Effect of CDW in the Loop

Experimental Setup
FIB cut

TaS3 ring/tube crystal

thDt  0.1mm
h  1mmD  27 mm

Result
T = 5.1 K
300 mV

2102I (pA)

1200 mV

100–1002he

100 mV

100–100h2e1

Power Spectrum (a.u.)

100–1000 20 40 60–11/B (Gauss )100200B (mGauss)

300

Periodic oscillations were observed.

Estimation of unit charge h1
B  
S e*  r2

B: The period of the oscillation
S: The area of the ring crystal
Sample Diameter Area m2 PeriodCharge
A

27 mm

5.6 x 10-10 39.7 mGauss 3.0 x 10-19 C

B

17 mm

2.3 x 10-10 95.2 mGauss 3.1 x 10-19 C

Unit charge corresponds to 2e (=3.2 x 10-19 C)

Quantum coherence at temperature of liquid-nitrogen

T=5.1K

T=79K

We also observed AB-effect of CDW sliding at 79K in TaS3ring crystals !
EPL, 97 (2012) 57011

AB Amplitude
300

T = 5.1 K

ICDW (pA)

CDW Current
200

100

Two curves behave similarly

VT0Ipp (pA)

20AB Amplitude

The amplitude of theinterference is relatedwith sliding CDW.

10VT
00100

200
V (mV)

300

Edge dislocation loop : soliton

The dislocation loop of CDW cannot escape from a
CDW chain, and provides 2e charge.
M. Tsubota, K. Inagaki, S. Tanda, Physica B: Condensed MatterVol. 404, pp416-418, (2009), EPL (2012) in Press.

Quantum Phase Slip in CDW ringsApplied fields

Periodicity

h2ePeriodicity

h2e

Sliding regime

AB amplitude
 0.1
CDW current

Ring CDW phase slips

Application of topological crystals
We obtain NbS3 topological crystals

Tc=340K, which have room-temperature Macroscopic-wave
function (CDW) !

Room-temperature SQUID

Application of topology

chaos
Homology

Homotopy

Geometry

Duality, Non Abelian
Geometry

Knottheory

Cosmology Strings
CS Gravity,
Einstein eq.
Black hole

Biology Ribosome Entanglement

Information

FormConformation

RNA,DNA,Topology
SolitonPolymer,Softmatter

MaterialScience

C60, CNT
Organic mat.
Topological Crystals
SDW,CDW
MX3,MX2

Topological Defects

Chern Simon
Jones eq.
QHE
Quantump-d-wave
Anomaly
Spectrum Flow
Superconductor Superconductor
Dirac Monopole
Siberg-Witten
StripesVortex‘thooft Monopole
M-Theory
Helicity
He3,texture TopologicalInsulator

Field Theory

Condensed
MatterTopology change
Local

Global effects

Simply connected space :(Cut-ring)Multiply connected space :Crystals

Cycloid crystals (cut-ring)

Ring, Mobius, 8, Hopf-link
Knots & links crystals
Embedding manifold

CDWs

Pinning of CDW at the edge

Frölich superconductor
AB effect of CDW

Superconducting vortices

Abrikosov lattice

Cylindrical vortex in ring crystals
Knots & Link vortex in 8 crystals

Topology defines the properties of condensed matter

Many Thanks !
色即是空

色=obverse
空=reverse“Zen”: Hakuin (Buddhist)used Mobius strip in 1758

色=空

Collaboration
AP.Tsuneta Graduate Student
AP.Toshima Graduate Student
AP. Matsuura Appl. Phys of H.U.

Hokkaido
University Group

Prof. Inagaki Appl. Phys of H.U.
Prof. Yamaya Appl.Phys of H.U.

Nobel Prize
2010

Dr. Uji National Research Inst. of materials, Japan
Prof. Paalanen Helsinki Univ.of Tech., Finland
Prof. Hakonen Helsinki Univ.of Tech., Finland
Prof. Volovik Helsinki Univ.of Tech., Finland
Prof. Hatakenaka Hiroshima Univ.,JapanProf. Niemi, Uppsala Univ., Sweden
Prof. Osipov Bogoliubov Institute, Russia
Prof. Matsuyama Nara edu Univ., JapanFinland

Theory