The 4th QMMRC-IPCMS Winter School 2 8 Feb 2011, ECC, Seoul, Korea Outline
Lecture 1. Electronic structures of graphene and bilayer graphene
Lecture 2. Electrons in graphitic systems with magnetic fields Graphene and Quantum Hall (2+1)D Physics
Young-Woo Son
Korea Institute for Advanced Study, Seoul, Korea
3 QUANTUM MECHANICS TWO OUT OF MANY FOUNDERS
Lecture 1 Schrödinger Equation Electronic Structures of Graphene and Bilayer Graphene Dirac Equation QUANTUM MECHANICS QUANTUM MECHANICS FOR FREE PARTICLES CONSEQUENCE OF DIRAC’S EQ.
Schrödinger Equation Dirac Equation
(Particle)
(Anti-Particle)
Dirac Equation
(Particle) IF m = 0
(Anti-Particle)
7 8 CarbonWhat is allotropesgraphene? Why Carbon? - Carbon allotropes
Graphene
21C? Intel CPU
1960~
The first 1947~ transistor
This slides is inspired by T. Ohta at LBNL and Fritz-Haber-Institut. = 9 mc Brief history of graphene Electronic structure of graphene 10 - Early works - Isolation of graphene ?
• Intercalated graphite as a route to graphene !
L. M. Vicilis et al, Science 299, 1361 (2003).
• Graphene nano-pencil ? (P. Kim@Columbia)
= mc 12 Electronic structure of graphene 11 Exfoliated Graphene - Isolation of graphene ? - Breakthrough
• Micromechanical cleavage of bulk graphite up to 100 micrometer in size via adhesive tapes !
Novoselov et al , Science 306, 666 (2004)
윤두희, 정현식(서강대)
A. K. Geim Group @ Menchester P. Kim Group @ Columbia KSNK. S. Novose lov etlt al, NtNature 438, 197 (2005) YZhY. Zhang etlt al, NtNature 438, 201 (2005)
020.2 μm BRIEF HISTORY OF GRAPHENE FEW FACTS ON GRAPHENE NOBEL PRIZE Density: 0.77 mg/m2 BkitBreaking streng th42N/th: 42 N/m (Hypothetical steel of graphene thickness~0.4 N/m) Theoreti cal RT mobilit y: 200, 000 cm2V-1s -1 f10for n=1012/cm-2
•Strongggest materials ever measured : Young’s modulus 1.0 Tpa • Thinnest flexible membrane ever created • Impermeable to gases (even atomic hydrogen) • Record value for RT thermal conductivity of ~5000 W/mK • Ballistic transport over micrometers at RT • Current density two order of magnitude higher than that of Cu • Room temperature Quantum Hall Effects • Unique material showing something exotics at RT
15 16 Electronic structure of graphene Electronic structure of graphene - Nature of bonds in graphene - Real space: tight-binding Hamiltonian
A sp2 sp3
a2 π-orbital s1 B s σ-bond 3 s2
a 1 Two sublattices - Bipartite system
Nearest-neighbor tight-binding Hamiltonian for π-orbitals Hexagonal network of Carbon TEM imagg,e, Zettl group at UC Berkeley – sp2 bonding C. Girit et al. Science 323, 170 (2009) 17 Electronic structure of graphene Electronic structure of graphene - Energy spectrum - Linear energy bands
K+
K− • Two inequivalent • Hexagonal BZ with two special Dirac cones at K FiitFermi points, and K’
18
19 20 Electronic structure of graphene Electronic structure of graphene - Real space - ‘Neutrino’ in your pencil?
A B
p ‐t x p y Dirac equa tion w ith zero mass charged ‘neutrino’ in your pencil?
• Relativistic particle : (()Pseudo) (()Pseudo) Two sublatticesSpin Up - BipartiteSpin Down system
c=vF: effective speed of ‘light’ m=0 21 22 Gap in graphene Gap in graphene - Linear bands? - Linear bands?
Linear band? Is that true? Yes from QHE
( If including NNI t’, )
MOST direct answer : ARPES measurement of suspended graphene does NOT published until now (now - I wrote in Dec 2008)
ELECTRONS IN GRAPHENE Quasi-particle spectrum of graphene 24 DIRECT OBSERVATION - IR measurements Angle Resolved Photoemission Spectroscopy
Graphene Typical Semiconductor Z. Q. Li et al, Nature Phys 4, 532 (08) ((pEpitaxial ggpraphene on SiC(000-1) (Indium Nitride surface states)
• Strong renormalization of group velocities near Dirac points
Sprinkle et al, Colakerol et al, Phys. Rev. Lett. 103, 226803 (2009) Phys. Rev. Lett. 97, 237601 (2006) 25 Electronic structure of graphene Electronic structure of graphene - Energy spectrum - Total Hamiltonian
K+
s=+ K- K+ K+ s= -
K−
• Hexagonal BZ with two special FiitFermi points, K− with
σ acts on sublattice A (B) and τ on K+ (K-) 26
28 Electronic structure of graphene Electronic structure of graphene - Gap generation in graphene - Consequences of massless Dirac fermions
Linear Density of States
E E
0 0 k N2D(E)
Onsite energy difference Mixing between K+ and K- - Mixing pseudo-spins - Mixing chiralities
|2α| |2β|
27 Zhang et al, Novoselov et al (05) Transport properties of graphene 29 Electronic structure of graphene - Mobility of graphene - Low energy dispersions
Mobilit y o f suspen de d graphene ~ 200, 000 cm2/Vs Graphene Usual Semiconductor
Observation of nearlyypg ballistic transport regime/ FQHE
X. Du et al, Nature Nanotech. 3, 491 (2008). X. Du et al, Nature 462, 192 (2009). K. I. Bolotin et al. SSC 146, 351 (2008) K. I. Bolotin et al. Nature 462, 196 (2009)
31 Electronic structure of graphene Electronic structure of graphene - Pseudospin and chiral states - Berry’ s phase
Eigenfunctions : Spinor representation
K py K py + + for a path of C at K+ and K- px θp px θp
Pseudo-spin up (down): A (B) sublattice 33 Electronic structure of graphene Electronic structure of graphene - Chiral states: charged “neutrino” in your pencil - Consequence of chirality
HliitHelicity operator : • Eigenstates :
• Conduction band : • For a long-range disorder where K+ a elastic scattering matrix element is
: Complete absence of backscattering σ = -1/2 σ =+1/2= +1/2 x -e x (di(pseudospin conserva ti)tion) p < 0 p > 0 x x • Klein paradox: M. I. Katsnelson et al,,y Nature Phys. 2,,() 620 (2006)
• Veselago lens: V. V. Cheianov et al, Science 315, 1252 (2007) 34
= mc 36 Electronic structure of graphene Electronic structure of graphene - Scattering - Tunneling
K+
K- K+
K−
• Intra-valley scattering: small momentum transfer, lattice distortion, etc. • ItInter-valley scatt eri ng: large momen tum t ransf er, sh ort range at omi c impurities, etc A. K. Geim & P. Kim Scientific American, Apr. 2008 Electronic structure of graphene 37 Electronic structure of graphene 38 - Klein paradox - Klein tunneling
Graphene
particle
~ 2 mc 2
antiparticle
• Klein paradox: Unimpeded penetration of relativistic particles through very high potential barriers. Katsnelson et al, Nature Phys. 2, 620 (2006) 2 = 8 • Potential drop ~ 2 mc over mc : ~10 V/Å V. V. Cheianov et al, Science 315, 1252 (2007) … → Event horizon of Black hole C. Park, Y.-W. Son et al, Nature Phys. (2008), → Supercritical massive atoms Phys. Rev. Lett. (2008), O. Klein, Z. Phys. 53, 157 (1929) Nano Lett (2008)
Electronic structure of graphene 39 Transport properties of graphene 40 - Klein tunneling - Klein tunneling
Pabry-Ferot interference: observation ofBf Berry’ s ph ase
A. F. Young & P. Kim, Nature Phys. 5, 222 (2009). N. Stander, B. Huard, D. Goldhaber-Gordon, Phys. Rev. Lett. 102, 026807 (2009)
Katsnelson et al, Nature Phys. 2, 620 (2006) V. V. Cheianov et al, Science 315, 1252 (2007) Electronic structure of graphene 41 Electronic structure of graphene 42 - Opacity - Opacity
graphene
SiScience 320, 1308 (2008) ω TittTransmittance ~ 97.7%
44 What is bilayer graphene? Electronic structure of graphene - Pseudospin and chiral states
Nearest neighbor hopping (t) between A and B sublattices
Normal material Neutrino Graphene Bilayer Graphene
• Only single layer graphene has a linear dispersion. • All others are massive, i.e., almost normal metals But, Pseudo-spin up (down): A (B) sublattice 45 46 Bilayer graphene Bilayer graphene : Massive chiral particles : Massive chiral particles
H. Min et al, PRB 75, 155115 (07)
Min ima l mo de l: Two coupled single layer graphene Low energy effective Hamiltonian projected on a spinor space for the with dimer couplings A1-B2 (Bernal two layers of bilayer graphene (σ acts on layers): stacking) spin up (down) upper (lower) layer
47 Single and bilayer graphene Bilayer graphene : Massless and massive chiral particles : Massive chiral particles
Single layer Graphene
Pauli matrices on ∆ A and B sublattices Elect ri c fi eld
MASSLESS CHIRAL -∆ A B Sublattice Sublattice
Bilayer Graphene Low energy effective Hamiltonian by integrating out high energy dimer part
Pauli matrices on upper and lower layers Transverse electric field can generate energy gaps in spectrum !!
Upper Lower MASSIVE CHIRAL Layer Layer StrainedBilayer graphene bilayer graphene StrainedBilayer graphene bilayer graphene - Energy gap under perpendicular electric field - Energy gap under perpendicular electric field
Min, McDonald et al, PRB 75, 155115 (2007) McCann, PRB 74, 161403 (R) (2006)
Oostinga et al, Nature Mat. 7, 151 (2007) Zhang et al, Nature 459, 820 (2009) T. Ohta et al, Science 313, 951 (2006)
Min, McDonald et al, PRB 75, 155115 (2007) McCann, PRB 74, 161403 (R) (2006)
51 52 Bilayer graphene Bilayer graphene : Next nearest neighbor inter-layer hopping : Next nearest neighbor inter-layer hopping
1.1 meV
The next nearest neighbor hopping breaks a global U(1) symmetry into Z3
(magnitude of kDi is about 0.4% of distance from Γ to K point) Bilayer graphene in AB-stacking 53 54 - First principles calculation with vdW corrections
Lecture 2
1.1 meV Electrons in Graphitic Systems with Magnetic Fields
κxx=×−ΓΓ100 (k K) / K
κ yy=×Γ100k / K Contour line interval =0.5 meV
SEMI-CLASSICAL APPROACH SEMI-CLASSICAL APPROACH 2 Dimensional Electron Gas in B-field 2 Dimensional Electron Gas in B-field
Drude Model : RH
I I B-field B-field
Y RL
X Rxx = RL Rxy = RH Longitudinal Hall Resistance Resistance B-field (Classical) Hall Effect (1879)
Drude Model : QUANMTUM MECHANICS LANDAU QUANTIZATION 2 Dimensional Electron Gas in B-field 2 Dimensional Electron Gas in B-field
RH
I I B-field B-field
Y RL
X Rxx = RL Rxy = RH Longitudinal Hall B Resistance Resistance C B-field
Shubnikov-de Hass effect (()1930)
LANDAU LEVELS INTEGER QUANTUM HALL EFFECT 2 Dimensional Electron Gas in B-field BIG BIG BREAKTROUGH !!
RL
B C B-field
At least , one cyclotron orbit before scattering INTEGER Quantum Hall Effect INTEGER Quantum Hall Effect 2 Dimensional Electron Gas in B-field 2 Dimensional Electron Gas in B-field
RH
I I B-field II B-field B-field
Y RL
X Rxx = RL Rxy = RH Longitudinal Hall Resistance Resistance B-field Fermi energy Rxx = RL Rxy = RH Y Longitudinal Hall Resistance Resistance X
Fractional Quantum Hall Effects (1982) 2 Dimensional Electron Gas in B-field
movie 65 Electronic structure of graphene Electronic structure of graphene - Pseudospin and chiral states - Berry’ s phase
Eigenfunctions : Spinor representation
K py K py + + for a path of C at K+ and K- px θp px θp
Pseudo-spin up (down): A (B) sublattice
67 Electronic structure of graphene LANDAU QUANTIZATION Graphene in B-field - Shubnikov-de Hass Oscillation and Berry’ s phase
Landau orbit near Normal 2DEG: Fermi level LANDAU LEVELS Quasi-particle spectrum of graphene 70 Graphene in B-field - Transport measurements : QHE
Normal 2DEG: In the presence of magnetic field,
En = sgn(n) × vF 2e= n B
π Cyclotron mass: m* = n vF
Novoselov et al (05), Zhang et al (05)
Quasi-particle spectrum of graphene 71 Evolution of Landau Levels 2 DEG vs. Graphene in B-field - Magneto-transport
n=3
n=2
n=3 n=1 n=2
n=0 n=1 n=0 0 0 B-field 6 B-field • Magneto-oscillation in tunneling conductance: vF=(1.070±0.006) x 10 m/s • Graphene on SiC(000-1): rotational stacking fault
J. Stroscio group, Science 324, 924 (2009) 73 Electronic structure of graphene Bilayer graphene - Landau levels in perpendicular B-field : Massive chiral particles
Parabolic band (normal metal) NlNormal Singl e l ayer Metal Graphene
Linear band (single layer graphene)
75 Electronic structure of graphene Zero energy states SINGLE LAYER GRAPHENE - Consequences of chiral massless Dirac fermions
Half-integer Quantum Hall Effect (Room T) ! (Manifestation of Berry’s phase of pseudospin)
Zhang et al (05), Novoselov et al (05) Kim & Geim et al (07) Haldane (88), T. Ando (02) Zero energy states Bilayer graphene GENERALIZATIONS : Massive chiral particles
NlNormal Singl e l ayer Bilayer Metal Graphene Graphene
J different states are zero energy states