Topological Crystals and thequantum effects:as a newParadigm
Satoshi Tanda

Dept. of Appl. Phys.,
Center of topological Science&tech.
SapporoHokkaido Univ.,,Japanhttp://exp-ap.eng.hokudai.ac.jp/index.html
Cambridge, Newton, 6 Dec. 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,SuperStrips

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
Berry Phase

Strategy
Is it Possible ?

Soap-film Möbius strip
Goldstein,Moffatt,Pesci,
Ricca PNAS (2010)For synthesize of topological crystals
Requirements★ One-dimensional wire

Organic Protein
polymer
DNAToo flexible(CH)xNetwork★ Rigidity

Inorganicpolymer

MX3,MX2C

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

chain

Semi-flexible !

Covalent : Intra-chain…

…

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 Yamaya
Hatakenaka

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!!

TaS3
1999

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.

NbQuartz 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
Hertman-Nirenberg
Theorem (2002)

2002 Nature

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

Möbius Crystals (π twist)

Copyright ©
NatureNbSe3Copyright © 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

Bending

Twisting
ComplianceΘ

Twist Angle： ΘMonoclinic , triclinic
Twist
Bent

Cross SectionComplianceＺ

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

ＹＸ

Ｓ１１

Extensionandcompression

Ｓ１２
Ｓ１６
Ｓ４６

Ｓab:Compliance

Cubic・ : 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 oftopology 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 real
space (lattice) to k-space (Bragg spots)

Reciprocalspacespots

Quasicrystal

Crystals and quasicrystal
show sharp Bragg spots

Topological crystal?
Copyright © 2002 Nature

X-ray Diffraction: Ring
Imaging PlateRing

X-ray

X-ray

Bragg RingLine is homogeneous

Ring
Single Crystals !!

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

ω= 1/2π
ω*= 0πTopologicalCrystals

ω= 2π
ω*= 0πNew 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, andS. 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 symmetry

1D,2D CrystalsCrystal
Aperiodic
5-fold symmetry,

SolidsPolymer
LiquidCrystals

LiquidIncommensurate

inorganic

Topological Crys.

organic

Rotational symmetry
Topology, Knots, Links

Plastic
Crystals

Amorphous

Embedding manifolds

Incomme. Crys.
Quasicrystals
Ring Crystals
Lk=0

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

Hopf-link Crys.

Topologicalmatter !!

EmbeddingSpherical（C60, Onion)

Kroto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
Berry Phase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
loop

time
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 peaks in CDW loops
Idc+Iacsin(2pfext)

T = 120 K

1W

Outer radius：120 mmInner radius:10 mmdVdc / dIdc [W]Vac = 0 mV

NbSe3 ring

1/1-I CDWp f ex
2eNqp/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

02/1-

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 p 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
I2eN
q100

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 line002

p/q-p/q–3

6.256x10
p/q –35.904x10+5.348x104Shapiro Index p/q

Discovery of Beat peak only in the loop CDW !!

Idc–3

Circulating current
I CDW  II CDW  II CDW

I CDWδII 

Temporallychangingpf exq

I CDW  I p
 f ex
2eNq

I CDWp
 f ex
2eNq
dVdc/dIdcp/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 byCircumference： 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 supercurrent
T. 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 crystalth

Dt  0.1mm
h  1mmD  27 mm

Result
T = 5.1 K
300 mV

2102I (pA)

1200 mV

100–1002he

100 mV

100–100h2e1Power Spectrum (a.u.)

100–1000 20 40 60–11/B (Gauss )100
200
B (mGauss)

300

Periodic oscillations were observed.

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

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

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

Quantum Phase Slip in CDW rings
Applied fields

Periodicity

h2eApplication of topological crystals
We obtain NbS3 topological crystals

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

Room-temperature SQUID

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

Cycloid crystals (cut-ring)

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

Insulatotor
Pinning of CDW at the edgeFrölich superconductor
AB effect of CDW

Superconducting vortices Abrikosov lattice

Cylindrical vortex in ringcrystals
Knot vortex in Mobius and 8 ?

Topology defines the properties of condensed matter

Application of topology

Thank you for your attention !
色即是空

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

色=空