11/11/2010 Global Focus on Knowledge

TheThe WorldWorld ofof DiverseDiverse MatterMatter —— TheThe DistantDistant JourneyJourney fromfrom SpaceSpace toto EarthEarth The Diversity of Materials Borne from the Collective Actions of Atoms, Electrons, and Molecules 66th LectureLecture Quantum World Nanoscience, Superconductivity, Superfluidity

YasuhiroYasuhiro IyeIye The Institute for Solid State Physics, The University of Tokyo

‡ Material bearing this mark is the property of third parties, so any use or re- production etc of this material requires the direct permission of the copyright holder. TodayToday’’ss TalkTalk

„ AboutAbout QuantumQuantum MechanicsMechanics •• QuantumQuantum Interference,Interference, TunnelTunnel EffectEffect

„ NanoNano--sciencescience •• MesoscopicMesoscopic PhysicsPhysics •• ScanningScanning ProbeProbe MicroscopeMicroscope

„ MacroscopicMacroscopic QuantumQuantum PhenomenaPhenomena •• SuperfluiditySuperfluidity •• BoseBose CondensationCondensation •• SuperconductivitySuperconductivity

„ SummarySummary BriefBrief IntroductionIntroduction toto QuantumQuantum MechanicsMechanics QuantumQuantum MechanicsMechanics Theoretical framework to describe the workings of the micro world

Structure of atoms/molecules Behavior of electrons in solids Light and Substances

Light: both wave-like and particle-like Electron: both particle-like and wave-like

The particle/wave dichotomy only reflects the fact that we have no suitable expressions when we try to describe quantum mechanical behaviors in terms of everyday (classical mechanical) words.

Particle nature: discrete, countable Wave nature: superposition, interference QuantumQuantum MechanicalMechanical ParticlesParticles alsoalso ExhibitsExhibits WaveWave--likelike BehaviorBehavior

h de Broglie Wavelength λ = p Momentum

The de Broglie wavelength of an electron that is accelerated at 100V p 2 E =⇒= 2mEp 2m The de Broglie h h λ == wavelength of a p 2mE classical particle ×1062.6 −34 (e.g. a tennis ball) = −30 −19 is extreeeemely ××××× 106.11001091.02 short = × −10 m1023.1 λ = nm12.0 WaveWave FunctionFunction The state of a quantum mechanical particle is

ψ ,,( zyx described by a wave function ψ tzyx ),,,(

The probability of finding the particle at a position 2 (x,y,z) at time t is given by ψ tzyx ),,,(

The time evolution of a wave function obeys Schrödinger’s equation d ⎛ ⎛ ∂22 ∂2 ∂2 ⎞ ⎞ i ψ tzyx ),,,( ⎜−= h ⎜ + + ⎟+ ⎟ψ tzyxrV ),,,()( h ⎜ ⎜ 2 2 2 ⎟ ⎟ dt ⎝ 2 ⎝ ∂ ∂ ∂zyxm ⎠ ⎠

This is a linear equation => Superposition Principle

ψ , ψ 21 ⇒ ψ +ψ 21 =>Quantum Interference TheThe RoleRole ofof MeasurementsMeasurements inin QuantumQuantum MechanicsMechanics Quantum mechanics provides a probabilistic distribution of the outcome of the measurements of a physical quality when they are repeated in the “same situation”. However, it does not generally give a definite value for an individual measurement.

Immediately after a measurement, the state takes one of the eigenstates of the physical observable. => “Collapse of state”

The unitary time evolution is described by Schrödinger’s equation, whereas measurements give rise to collapse of the state. The standard interpretation of quantum mechanics (Copenhagen Interpretation) Interpretation problem, observation problem of quantum mechanics Many-Worlds Interpretation CharacteristicCharacteristic QuantumQuantum MechanicalMechanical PhenomenaPhenomena

Tunnel Effect • Quantum mechanical particles can go through a potential wall they cannot ? penetrate according to classical dynamics

Quantum Interference Effect • Superposition of states Yasuhiro Iye 2006 “Alice’s quantum mechanics”(In Japanese) Maruzen‡ that pass through different paths => Quantum Interference QuantumQuantum InterferenceInterference InterferenceInterference ofof LightLight WavesWaves

Young’s double-slit experiment Diffracted Wave (1805) Incident Wave d θ θ

⎧ nλ 強め合いenhance d sinθ = ⎨ 1 打ち消し合いcancel ⎩()n + 2 λ cancel

In-Phase ‡ http://en.wikipedia.org/wiki/File:Ebohr1.svg

Interference of waves that passed through a double slit Out-of-Phase DiffractionDiffraction ofof LightLight Diffraction

θ

In-Phase

Optical Path Difference Out-of-Phase sinθ = nd λ Inter-atomic Spacing ~0.3nm~Wavelength of X-Rays

Diffraction experiment can be also done with electron beams and neutron beams. CrystalCrystal StructureStructure AnalysisAnalysis

θ 2θ

Incident Wave Reflected Wave d θ

sin2 θ = nd λ Bragg Condition Four-Axes X-ray Diffractometer TheThe CaseCase ofof ClassicalClassical ParticlesParticles 1 slit

2 slits

Yasuhiro Iye 2006 “Alice’s quantum mechanics”(In Japanese) Maruzen‡ Probability of a bullet hitting a particular spot ＝Probability of reaching the spot via the right slit ＋Probability of reaching the spot via the left slit

= + yPyPyP )()()( total R L Summation of Probability TheThe CaseCase ofof WavesWaves

y

Summation of Py() Wave Amplitudes

Interference Pattern TheThe CaseCase ofof QuantumQuantum MechanicalMechanical ParticlesParticles

Ψtotal Ψ= + ΨLR Total Wave Function = Wave Function passing through right-hand slit + Wave Function passing through left-hand slit

Probability = |Wave Function| 2

2 2 total Ψ+Ψ=Ψ LR

2 2 * * R L R ΨΨ+ΨΨ+Ψ+Ψ= LRL

Quantum Interference Terms Electron Source InterferenceInterference ofof ElectronsElectrons

Electrons

Electron Beam Biprism

Detector

Dig. 3 Representation of Electron Two-Slit Dr Akira Tonomura Experiment In this experiment an electron beam bi-prism (Hitachi Advanced is used in place of a slit. This type of experiment is technically difficult and at one Research Laboratory) ‡ time was thought of as a thought experiment that could only be done in your head. Double-Slit Experiment with Electrons (Dr Akira Tonomura) InterferenceInterference ofof ElectronsElectrons

electron source Electrons reach the screen one- electron by-one m m

bipris Interference pattern bipris electron beam electron beam emerges

detector

Dr Akira Tonomura Unambiguous proof of (Hitachi Advanced the wave nature of Research Laboratory) electron.

‡ HowHow LargeLarge anan ObjectObject CanCan StillStill Interfere?Interfere? A.Zeilinger Vienna University of Technology ‡

‡

C６０ Fullerene molecule

C60 AharonovAharonov--BohmBohm (AB)(AB) EffectEffect

The phase of electron wave h h changes with magnetic field θ )( d )( ⋅−⋅=Δ drrArrA e L e ∫∫ R (more precisely, with vector potential) e = h )( d h )( ⋅=⋅ dSrBrrA i ∫ )( ⋅drrA e loop e ∫∫ ψ ⇒ψ e h φ h = 2π φ0 = Electron Source φ0 e ‡ Electron Beam Coil Dr Akira Tonomura Biprism (Hitachi Advanced Biprism ‡ Research Laboratory) Phase Difference MesoscopicMesoscopic SystemSystem

NanotechnologyNanotechnology MesoscopicMesoscopic PhysicsPhysics Micro-scale Macro-scale (Microscopic) (Macroscopic) Mesoscopic System Intermediate range between micro and macro (The prefix “meso” stands for intermidiate) Physics characteristic to the systems whose scale is comparable to or smaller than the “characteristic length scale of the relevant physical phenomenon” => mesoscopic physics Examples of the “characteristic” length scale： Wavelength of an electron (Fermi wavelength) Mean free path of an electron (average distance an electron travels before it is scattered) Phase relaxation length of an electron MiniaturisationMiniaturisation ofof SemiconductorSemiconductor DevicesDevices

http://en.wikipedia.org/wiki/File:Replica-of- first-transistor.jpg

50 years

Silicon Wafer First Electrical Transistor Bell Telephone Laboratories (1946) 12 inches (30 cm)

Progress in Semiconductor High-Tech Industry SiliconSilicon SingleSingle CrystalCrystal =>=> WaferWafer =>=> ULSIULSI ULSI (Ultra Large-Scale Integrated Circuit)

Slicing and Polishing Wafer Microfabrication MooreMoore’’ss LawLaw

Gordon Moore (1965) Co-founder of Intel

Typical example of the so- called self-fulfilling prediction

Gordon E.Moore “NO EXPONENTIAL IS FOREVER…” ‡Intel Copyright Permission Department

The degree of integration of LSI (Large Scale Integrated Circuit), namely the number of transistors that can be packed into a given area of semiconductor chip doubles every one-and-a-half or two years. NanoNano--sciencescience // NanotechnologyNanotechnology

„ Miniaturization technology ⇒ More than Moore It is not a simple scaling down • Problem associated with heat generation • Problem associated with quantum fluctuation

„ Nano-science • Observing atoms, Manipulating atoms • Pursuit of new quantum effects in the nano- world

““ThereThere isis plentyplenty ofof roomroom atat thethe bottom.bottom.”” （Richard Feynman) MesoscopicMesoscopic PhysicsPhysics AharonovAharonov--BohmBohm (AB)(AB) EffectEffect

The phase of an electron wave h h changes according to a magnetic θ )( d )( ⋅−⋅=Δ drrArrA e L e ∫∫ R field (more precisely, with a vector potential). e = h )( d h )( ⋅=⋅ dSrBrrA i ∫ )( ⋅drrA e loop e ∫∫ ψ ⇒ψ e h φ h = 2π φ0 = Electron Source φ0 e ‡ Electron Beam Coil Dr Akira Tonomura Biprism (Hitachi Advanced Biprism ‡ Research Laboratory) Phase Difference AharonovAharonov--BohmBohm OscillationsOscillations

2 2 Ψ+Ψ=Ψ −+

2 2 ** + − ΨΨ+ΨΨ+Ψ+Ψ= −+−+

quantum 量子干渉項 interference term Mesoscopic Ring

Interference between two electron waves that travel along the paths on both sides of a ring. Electrical resistance changes periodically with the magnetic flux Φ piercing the ring.

h φ ×== −15 Wb1014.4 0 e Flux Quantum SingleSingle ElectronElectron TunnellingTunnelling Nano-Tunnel Junction

1μm

Coulomb Island (Quantum Dot) The electrostatic potential of the island increases as one electron tunnels into the island => next electron cannot enter the islandCoulomb Blockade QuantumQuantum DotDot (Artificial(Artificial AtomAtom))

“Periodic table” of artificial atoms

‡ Prof. Tarucha, Faculty of Science ObservingObserving AtomsAtoms ManipulatingManipulating AtomsAtoms TheThe TaleTale ofof SmallSmall WorldWorld

１０-12 ｍ １０-9 ｍ １０-6 ｍ １０-3 ｍ １ｍ

１ｐｍ １ｎｍ １μｍ １ｍｍ Picometer Nanometer Micrometer (Micron) Millimeter

Bacteria Virus Size of a Size of nucleus an Atom O-157 Wavelength of visible light Optical Microscope

Electron Microscope

Scanning Probe Microscope SeeingSeeing SmallSmall ObjectsObjects :: ElectronElectron MicroscopeMicroscope

Observe the regular array of atoms with a high-resolution electron microscope

fly’s eye

‡ ObservingObserving thethe ArrangementArrangement ofof AtomsAtoms onon aa SolidSolid SurfaceSurface Were it an array of macroscopic objects you can see (or feel) the array by touching it Would it be possible to achieve the same thing to an array of atoms? => common sense tells us that it is “impossible” Scanning Tunnel Microscope 1984 Binnig and Rohrer

The first image of atomic arrangement on the surface of a silicon crystal.

APS ‡ ScanningScanning TunnelTunnel MicroscopeMicroscope (STM)(STM)

(Nano)Tunnel Current

When atoms at the tip of the needle and atoms on the surface are brought Problem of Stability to within ~1 nanometer, tunnelling current flows. By moving the tip Mechanical Vibration laterally while keeping the tunneling Electrical Noise current constant, one can detect the microscopic undulation due to the array of surface atoms. AtomicAtomic ForceForce MicroscopeMicroscope (AFM)(AFM) A cantilever (a flexible beam fixed at one end) probe is use to detect very weak force acting between the atom at the tip of the probe and atoms on the surface.

Detect the change in the atomic force as the tip is brought close to and scanned over the surface.

laser mirror

optical diode cantilever diode specimen

Deflection of the cantilever is detected by the change in the reflected laser beam. ManipulatingManipulating AtomsAtoms

IBM Almaden Lab ‡ ‡ Eigler Group

Artificial arrangement iron atoms on a copper surface. The ripple-like feature reflects the wave interference of the surface electrons MacroscopicMacroscopic QuantumQuantum PhenomenaPhenomena ------BoseBose CondensationCondensation andand SuperconductivitySuperconductivity QuantumQuantum MechanicalMechanical ParticlesParticles Quantum mechanical particles of the same type are intrinsically indistinguishable. Even when two particles are interchanged, the state remains the same as before. The wave function, however, is multiplied by a numerical factor.

Ψ (b,a) = CΨ (a,b)

Ψ (a,b) = CΨ (b,a)= C2Ψ (a,b) ⇒ C2 = 1

⇒ C = 1 or −1

Boson Fermion QuantumQuantum StatisticsStatistics Bose Particle (Boson) Fermi Particle (Fermion) Spin: 0,1,… Spin: ½, 3/2, …

Ψ (b,a) = Ψ (a,b) Ψ (b,a) = −Ψ (a,b) If a=b then Ψ (a,a) = −Ψ (a,a) ⇒ Ψ (a,a) =0

Any number of particles Only one particle is allowed in can occupy the same state each state (Pauli Exclusion Principle) BoseBose--EinsteinEinstein DistributionDistribution andand FermiFermi--DiracDirac DistributionDistribution

E E Bose Particle Fermi Particle

f (E) f (E) 1 1 BE Ef )( = =Ef−μ)( TkE FD Ef )( = −μ)( TkE e −Eμ+ B −1 e B +1 Either distribution is reduced to the classical Maxwell-Boltzmann distribution at high )( =eEf −− μ)( BTkE enough temperatures HeliumHelium IsotopesIsotopes

4He 3He 2 Protons 2 Protons 2 Neutrons 1 Neutrons 2 Electrons 2 Electrons

Total Spin = 0 Total Spin = 1/2

Bose Particle Fermi Particle MakingMaking UltraUltra--LowLow TemperaturesTemperatures Ultra-low temperatures are required to observe phenomena associated with quantum statistics

Liquid Nitrogen 77K Liquid Helium (4He) 4.2K forced evaporation by pumping ~1.2K Liquid Helium（3He) 3.2K forced evaporation by pumping ~0.3K 3He-4He Dilution Refrigerator ~ mK Adiabatic Nuclear Demagnetization Cut-away view of a ~ μK liquid helium vessel PhasePhase DiagramDiagram ofof HeliumHelium

エネenergy ル ギ ー E Helium (at ambient pressure) does not solidify even at absolute zero r => Quantum Liquid distan原子ce b 間etween 距 離 atoms r 0 σ pressure圧力 (bar) 4 −ε Vr() Van der 固相solid He Waals Force 30

λ線λ line Helium atoms are light. 20

液 相 Ⅱ ordinary 液相Ⅰ triple superfluid()超 流動相 ()常流動fluid 相 三重点 Helium atoms interact only 10 point weakly with each other. ga Kinetic energy > Interaction energy 気相ga 0124 3 5 温度 (K) １気boiling 圧point での沸 at 1 点 temperature (K) atm SuperfluiditySuperfluidity ofof LiquidLiquid HeliumHelium

The University of Tokyo Cryogenic Research Center ‡ SuperfluiditySuperfluidity ofof LiquidLiquid HeliumHelium

GlassGlass DDewarewar figure 1

photo 1 Vacuum Two-Fluid Model Superfluid Two-Fluid Model Component Thermomechanical Effect (Internal Convection)

Normal fluid Heat Component Source

Fountain Effect the University of Tokyo Cryogenic Research Center

Superfluid component flows without friction through a capillary (superleak) which is too narrow for the normal fluid to pass. QuantumQuantum VortexVortex

iθ Macroscopic Wave Function Ψ=Ψ 0e Quantization of Circulation Rotating Bucket Experiment h h h κ s dsv ds 2πθ==⋅∇=⋅≡ nn C m ∫∫ C m m

Permanent Current Quantum Vortex

v ()r エネルギー s

vs ()r

r

-2 -1 0 12 κ ()h/m 0 BoseBose--EinsteinEinstein CondensationCondensation

p p p y y y

p p p x x x

T = 0 T =TBE T ＞TBE Bose condensation occurs when the Thermal de Broglie Wavelength thermal de Broglie wavelength becomes comparable to the interparticle distance 2 2/1 ⎛ 2π h ⎞ 2 λT = ⎜ ⎟ 1 2 ⎜ ⎟ 3 BTmk − 2π ⎛ n ⎞ ⎝ ⎠ 3 T h λT ≈ n BE = ⎜ ⎟ mkB ⎝ 612.2 ⎠ LaserLaser CoolingCooling ofof AtomicAtomic GasesGases Collect and cool an atomic (e.g. Rb) gas (vapor) in a trap Doppler Cooling

hν

・Use a laser light with frequency slightly below the resonance frequency of the atoms ・For an atom travelling in the opposite direction to the light, the Doppler shifted light frequency becomes closer to the resonance frequency, so the absorption probability is higher. The momentum of the atom is reduced by absorbing the incident light (photon). ・When the atom reemits the light, it does so isotropically, so that on average the atom is slowed down. ・By a set of six laser beams along the x-, y-, z- axes, Doppler cooling is achieved for all directions. ・With this technique, temperature on the order of ~100μK is attained. => Further reduction of temperature by 3 to 4 orders of magnitude is needed. BoseBose--EinsteinEinstein CondensationCondensation ofof AtomicAtomic GasesGases Reducing the temperature and achieving the conditions of Bose-Einstein Condensation by Evaporative Cooling of atomic gas cooled in a magnetooptical trap λ ≈ n− 31 T ~ 10-7K T Upon turning off the trap potential, the cloud of cold atoms drops with gravity, and at the same time swells according to the velocity distribution. ‡ http://cua.mit.edu/ketterle_group /research.htm

p y

p x

T =TBE SuperfluiditySuperfluidity ofof LiquidLiquid 33HeHe Phase diagram of 3He Being Fermi particles, 3He atoms do not undergo Bose condensation per se. However, they can become superfluid by forming pairs. (This is the same mechanism as superconductivity)

3He becomes a superfluid at an ultra-low temperature of ~2mK BasicBasic PropertiesProperties ofof SuperSuper--ConductionConduction

Perfect Conductor (Zero Resistance) Quantization of Magnetic Flux Electrical h −15 Resistance Φ Φ φ0 ×== Wb1007.2 = n 0 2e

Same as quantization of Superconductor circulation in superfluid Transition Temp.

Temp. Perfect Diamagnetism (Meissner Effect) Persistent Electrical Current

T>Tc T

Type I Superconductor Type II Superconductor Critical magnetic field －M －M Lower critical Upper critical field field

0 0 Hc H Hc1 Hc Hc2 H

Mixed B B state Normal state Meissner state

0 0 Hc H Hc1 Hc2 H Normal state Meissner state Superconducting materials of practical use are type II superconductors. QuantumQuantum MagneticMagnetic FluxFlux inin TypeType IIII SuperconductorsSuperconductors (Vortex(Vortex LineLine))

Quantum magnetic flux Mixed state of type II (Vortex line) superconductors

h φ ×== −15 Wb1007.2 0 2e

Repulsive force acts between vortex lines ‡ ‡ Lorentz Microscope Triangular lattice Bitter Method （Tonomura Akira, 1992) (Essmann & Traueble,1968) MechanismsMechanisms ofof SuperconductivitySuperconductivity

Cooper Pair Origin of Attraction? Configuration Electron Particle Interaction + + + +

+ + + + − + + + + If the inter-electron attractive force mediated by electron- + + + + lattice interaction surpasses the repulsive Coulomb force, the net attractive inter- electron interaction results in If attractive force acts Cooper pair formation. between two electrons on the Fermi surface, they Superconducting Transition Temperature form a bound state (Cooper Pair). ⎛ 1 ⎞ Tc D exp14.1 ⎜−Θ= ⎟ ⎝ )0( VN ⎠ VicissitudeVicissitude ofof thethe HighestHighest SuperconductingSuperconducting TransitionTransition TemperatureTemperature Is room-temp superconductivity possible?

(Under high pressure)

(Under high pressure)

Room-temp liquid Resistance Mercur LaFeAsO y

MgB2 Liquid Neon Liquid Temp(K) Hydrogen PredictionPrediction ofof thethe FutureFuture TechnologicalTechnological AdvancesAdvances vs.vs. ScientificScientific DevelopmentsDevelopments

(Under high pressure)

(Under high pressure

Liquid Hydrogen LaFeAsO

MgB2

critical Liquid Neon temperature

Roadmap ⇒ It is impossible to foretell the Self-fulfilling Prediction development of science (discovery, in particular). SummarySummary

„ CharacteristicCharacteristic ofof QuantumQuantum MechanicsMechanics • Quantum interference, Tunnel effect „ MesoscopicMesoscopic PhysicsPhysics • Quantum conductance, e2/h-physics • Quantum Interference AB Effect • Single electron tunneling Effect „ NanotechnologyNanotechnology andand nanosciencenanoscience • Observing atoms, Manipulating atoms • Scanning probe microscope „ MacroscopicMacroscopic QuantumQuantum PhenomenaPhenomena • Superfluidity • Bose Condensation • Superconductivity TheThe RoleRole ofof MaterialsMaterials ScienceScience andand CondensedCondensed MatterMatter PhysicsPhysics

„ Understanding the diverse properties of various substances based on the basic principles of physics (quantum mechanics and statistical mechanics)

„ Phase transition and emergent phenomena: seeking universality and unity in diversity and complexity

„ Performing experiments that approach the essence of quantum mechanics

„ Toward the construction of “materials perspective”: Some of the concepts formed share commonality with other subfields of physics, such as elementary particle physics or astrophysics.

„ Forms the basis of engineering TheThe RoleRole ofof MaterialsMaterials ScienceScience sandand! ing th g CondensedCondensed MatterMatter PhysicsPhysicstin es er „ Understanding the diverse propertiesnt of various substances y i based on the basic principlesa nof physics (quantum based on the basic principlesm of physics (quantum mechanics and statistical so mechanics) Bil re at a „ Phaseer transition andre emergent phenomena: al e Pu Th seeking universalityrsu and unity in diversity and complexity i t o „ Performing experimentsf H that approach the essence of quantum mechanicsum quantum mechanics an iti „ Toward the construction ofe s“materials perspective”: Some of an the concepts formed share commonalityd with other subfields Sc of physics, such as elementary particleien physics or of physics, such as elementary particlec physicse or astrophysics. s!

„ Forms the basis of engineering