Condensed Matter Physics At Low Dimensions

Philip Kim

Department of PhysicsColumbia University

Condensed Matter Physics

~1023 electrons~1023 ions

SP2 Carbon: 0-Dimension to 3-Dimension

Atomic orbital sp2s

p

Fullerenes (C60) Carbon Nanotubes GraphiteGraphene

0D 1D 2D 3D

Benzenemolecule

x

E(k2D)

kx

ky

Electronic Band Structure of Graphene

kvE F

Band structure of graphene (Wallace 1947)

Zero effective mass particles moving with a constant speed vF = c/300

2D Brillouin Zone

kx

ky

K

K’

ccBAtHmn

mn .ˆ),(

kx'

ky'

E

empty

filled

A B

Single Layer Graphene: Pseudo Spinor

‘A’ sublattice: pz orbitals

‘B’ sublattice : pz orbitals

A

B

q

Spinor Representation

Pseudo spin

Graphene Lattice Structures

BeA iqSuperposition:

Two inequivalent lattice sites!

[ ] Cs

Spin

X

Dirac Fermions in Graphene : “Helicity”

KK’

momentumpseudo spin

E

kx

ky

E

kx

ky

E

ky

kx

G. Semenoff PRL (1984)

Effective Dirac Equations

kvvH FFeff

s

0

0

yx

yx

ikkikk

qk = tan-1(ky / kx)

21

k = eik r.

eiqk

1

ccBAtHmn

mn .ˆ),(

Vxy

Hall Effect

I

Vxx

+

+

+

+

-

-

--

B

Quantized Cyclotron Orbit

Rxy

Rxx

s

s

Rxx = Vxx / I

Rxy = Vxy / I = B/enQuantum Hall Effect:

Klitzing (1980)

Hall (1879)

Quantum Hall Effect in Graphene (2005)

Quantization:4 (n + )Rxy

=-1 ___ eh

2

21

spin (2) X pseudo-spin (2)

pseudo-spin rotation

kx' ky'

E

Klein Tunneling (1928)

Step Potential problemV>m: transmission via negative energy states

x

V

0

Klein result: barrier sharpness ~Compton wavelength

eT Em

2p

x

V

0x

V

0E

filed

1T

Klein Tunneling and Pseudo spin

Chiral tunneling in graphene pn junctions

Katsnelson et al. (2006)

n

np n

p

p

VBG > 0VTG < 0

VBG < 0VTG > 0

graphene

electrode

1 mm

20 nmYoung et al. (2009)

|n2| (1012 cm-2)

1 2 3 4

0

2

B (T

)

Gosc (e2/h)-1 0 +1

Magnetic field modulation of FP

Spin ½ and Electron Interaction

“Triplet”

“Singlet”

Pseudo Spin

“Quantum Hall bilayer”

“Valley spin”

…

21 cc

ji ssjjiiji

Xedd ,

**

,

2

)()()()(||2

1 rrrrrr

rr

Exchange Interaction:

SU(4) Quantum Hall Ferromagnet in Graphene

q

Valley spinSpin

X

SU(4)<

K

K’

K’

K

Yang, Das Sarma and MacDonal, PRB (2006);

K’ K’

kx

ky

E

Magnetic Wave Function

Under magnetic fields:pseudospin = valley spin

Degree of freedom:

Spin (1/2), Valleys

Charge Density WaveKekule DistortionAnti FerroMagneticFerroMagnetic

Possible SU(4) Quantum Hall Ferromagnetism at the Neutrality

Spin & pseudo spins: many body physics in graphene

5 mm

Dean et al. Nature Physics (2011)

• SU(4) hierarchical Fractional Quantum Hall Effect

• Spin and Pseudospin Ferromagnetic Quantum Hall Effect

• Spin Skyrmion and Valley Skyrmions

Mobility > 300,000 cm2/Vsec

Phase Transitions Among Fractional Quantum Hall States

Maher*, Wang* et al. submitted

Bilayer GrapheneEncapsulated with top & bottom gate

mobility > 106 cm/Vsec

sxx (S)

0.5 1.0 1.5 2.00

2000

4000

6000

8000

10000

12000B=18TSiGate = +40V Rxx

Rxy

Vg (Volts)

Rxx

()

0

10

20

30

40

50

60

70

432

Rxy

(k

)

1

2/34/3

5/37/3

8/3 10/311/3

20 mK

Bilayer graphene: Fractional Quantum Hall effect Rxx (k)2/31 2 3 4 5 6

5/3 8/3

E-field tunable FQHEPhase Transitions in Lowest Landau Levels

X

X

M

C

BN

graphene

hexa-BN

Metal-Chalcogenide

M = Ta, Nb, Mo, W, Eu …

X = S, Se, Te, …

Bi2Sr2CaCu2O8-x

Charge Transfer Bechgaard Salt

(TMTSF)2PF6

Lead Halide Layered Organic

Semiconducting materials: WSe2, NbS2, MoS2, … Complex-metallic compounds : TaSe2, TaS2, … Magnetic materials: EuS2, EuSe2 ,…Superconducting: NbSe2, Bi2Sr2CaCu2O8-x, ZrNCl,… A

BA

CA

Assembly of Various 2D Systems

-40 -20 0 20 400.96

0.98

1.00

1.02

1.04

1.06

1.08

VSD(mV)

(dI/d

V)/(d

I/dV(

10K)

)

Andreev Reflection btw graphene/NbSe2

1.5 K2.5 K3.5 K4.5 K5.5 K6.5 K6.8 K7.0 K7.2 K7.5 K

Andreev Reflections – between NbSe2 & GrapheneEfetov et al. (2014)

NbSe2Tc = 7 KHc2 = 4.5 T

graphene

5 mm

Tomasch Oscillations

Andreev Reflection

Andreev Reflection into QH edge states are more efficient!

Superconductivity and QHE

Vertical & Lateral Channels

- Al contact to MoS2 for electron injection

- Pd contact to WSe2 for hole injection

-60 -40 -20 0 20 40 6010-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

WSe2

MoS2

I ds (A

)

Vg (V)

Vds

= 0.5 V

Lateral Transport in Channels

C. Lee et al, submitted

Atomically Thin vdW p-n junction

Forward

Lateral and vertical electron band alignment

Interlayer recombination by inelastic tunneling process

Gate Tunable Diode Characteristic

Graphene Materials and Applications

Large-ScaleCVD Graphene

+Graphene

NanoplateletComposites

Composites

Cars,AerospaceAppliations

Semi-conductors

Ultrafast Transistors, RFIC,

Photo/Bio/Gas Sensors

Transparent ElectrodesFlexible/Transparent

Electrodes/Touch Panels

PrintableInks Conductive Ink,

EMI shields

EnergyElectrodes

Super Cap./Solar CellsSecondary Batteries

Fuel Cells

HeatDissipation

LED Lights, BLUECU, PC …

GasBarriers

Gas barriers fo Displays,Solar Cells

Images: Royal Swedish Academy Courtesy: B. H. Hong

CERN Electro-Positron Collider

Relativistic QM: High Energy Physics

Conclusions

Kim Lab @ Columbia in City of New York

Quasi Relativistic QM: Low Energy Physics

Dirac Equation:.

Majorana Equation: ??

Acknowledgement

Past MembersMelinda Han (Ph.D. 2010, Frontier of Science Fellow, Columbia University)Meninder S. Purewal (Ph.D. 2008) Josh Small (Ph.D. 2006) Yuanbo Zhang (Ph.D. 2006, Professor, Fundan University)Yuri Zuev (Ph.D. 2011, IBM Fishkill)Kirill Bolotin (Assistant Professor, Department of Physics, Vanderbilt University)Byung Hee Hong (Associate Professor, Department of Chemistry, Seoul National University) Pablo Jarillo-Herrero (Assistant Professor, Department of Physics, MIT)Keunsoo Kim (Assistant Professor, Department of Physics, Sejong University)Namdong Kim (Research Scientist, POSTECH)Barbaros Oezyilmaz (Assistant Professor, Department of Physics, National University of Singapore)

Current MembersYue ZhaoMitsuhide TakekoshiAndrea YoungDmitri EfetovFereshte GhahariPatrick MaherYoung-Jun Yu (jointly with GRL, POSTECH)Vikram Deshpande (jointly with Hone group)Paul Cadden-Zimansky (Columbia Frontier of Science Fellow)Chenguang Lu (jointly with Hone and Herman

Collaborating Students/postodcsCory Dean, Inanc Meric, Lei Wang, Sebastian Sorgenfrei, Kevin Knox, Nayung Jung, Seok Ju Kang, Jun Yan, Yanwen Tan, Kevin Knox

CollaboratorsHorst Stormer, Aron Pinczuk, Tony Heinz, Abhay Pasupathy, Latha VenkataramanLouis Brus, George Flynn, Colin Nuckolls, Jim Hone, Ken Shepard, Louis Campos, Rick OsgoodT. Taniguchi, K, WatanabeAndre Geim, Kostya Novoselov, Sanka Das Sarma

Kim group and friends (2011)

Funding:

Amelia Barreiro Chul-ho Lee (jointly with Nuckolls group)Jean-Damien Pillet Jayakanth Ravichandran Adam Wei Tsen (jointly with Pasupathy group)Dmitri EfetovFereshte Ghahari

Patrick MaherCarlos ForsytheGiselle Elbaz (jointly with Brus group)Austin Cheng Frank Zhao Xiaomeng Liu

Collaborations: Brus, Dean, Heinz, Hone, Nuckolls, Shepard