Bloch, Landau, and Dirac : Hofstadter’s Butterfly in Graphene Philip Kim, Department of Physics, Columbia University

Fractional & Integer Quantum Hall effect in graphene: electron correlation

Dean et al, Nature Phys (2011)

Heteroepitaxy of Layered Materials: Andreev reflection between NbSe2 and graphene

B (

T)

0

2

Vsd (mV) 0 1 -1

dI/dV(μS)

Efetov et al (2013)

graphene

NbSe2

Graphene Materials and Applications

Large-Scale CVD Graphene

+ Graphene

Nanoplatelet Composites

Composites

Cars, Aerospace Appliations

Semi-conductors

Ultrafast Transistors, RFIC,

Photo/Bio/Gas Sensors

Transparent Electrodes Flexible/Transparent

Electrodes/Touch Panels

Printable Inks Conductive Ink,

EMI shields

Energy Electrodes

Super Cap./Solar Cells Secondary Batteries

Fuel Cells

Heat Dissipation

LED Lights, BLU ECU, PC …

Gas Barriers

Gas barriers fo Displays, Solar Cells

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

Will graphene appear in market soon?

hBN samples: T. Taniguchi & K. Watanabe (NIMS)

Theory: P. Moon & M. Koshino (Tohoku)

Prof. Jim Hone

Prof. Ken Shepard

Dr. Cory Dean Lei Wang Patrick Maher Fereshte Ghahari Carlos Forsythe

Bloch, Landau, and Dirac : Hofstadter’s Butterfly in Graphene

k

e

k

e

n(E)

Bloch Waves: Periodic Structure & Band Filling

Felix Bloch

Periodic Lattice

a

A0 : unit cell volume eF

Zeitschrift für Physik, 52, 555 (1929)

Block Waves: Band Filling factor

MacDonald (1983)

Landau Levels: Quantization of Cyclotron Orbits

Lev Landau

Zeitschrift für Physik, 64, 629 (1930)

Free electron under magnetic field

Each Landau orbit contains magnetic flux quanta

Energy and orbit are quantized:

Massively degenerated energy level

2-D

Density o

f Sta

tes

e

2-dimensional electron systems

Landau level filling fraction:

eF

Harper’s Equation: Competition of Two Length Scales

Proc. Phys. Soc. Lond. A 68 879 (1955)

a

Tight binding on 2D Square lattice with magnetic field

Harper’s Equation

Two competing length scales:

a : lattice periodicity

lB : magnetic periodicity

Commensuration / Incommensuration of Two Length Scales

Spirograph

lB a

a / lB = p/q

Hofstadter’s Butterfly

When b=p/q, where p, q are coprimes, each LL splits into q sub-bands that are p-fold degenerate

Energy bands develop fractal structure when magnetic length is of order the periodic unit cell

Harper’s Equation

energy (in unit of band width)

f (

in u

nit

of f

0)

0 1

1

Energy Gaps in the Butterfly: Wannier Diagram

0 1

1

e /W

n0: # of state per unit cell

f : magnetic flux in unit cell

n : electron density

1/2 1/3 1/4

1/2 1/3 1/4 1/5

Hofstadter’s Energy Spectrum Tracing Gaps in f and n

0 1

1

Wannier, Phys. Status Solidi. 88, 757 (1978)

Diophantine equation for gaps

t : slope, s : offset

Streda Formula and TKNN Integers

What is the physical meaning of the integers s and t ? Band Filling factor

Quantum Hall Conductance

0 1

1

e /W

Experimental Challenges

0 20 40 60 80 100 120 140 1600.1

1

10

100

1000

10000

Bo (

Tesla

)

unit cell (nm)

Obvious technical challenge:

a = 1nm

Disorder, temperature

Hofstadter (1976)

Experimental Search For Butterfly

-Schlosser et al, Semicond. Sci. Technol. (1996) Albrecht et al, PRL. (2001);

Geisler et al, PRL (2004)

• Unit cell limited to ~100 nm • limited field and density range accessible, weak perturbation • Do not observe ‘fully quantized’ mingaps in fractal spectrum

Electrons in Graphene: Effective Dirac Fermions

K K’

kx

ky

E

DiVincenzo and Mele, PRB (1984); Semenov, PRL (1984)

Effective Dirac Equations

kvvH FFeff

0

0

yx

yx

ikk

ikk

Graphene, ultimate 2-d conducting system

Band structure of graphene

Paul Dirac

Graphene: Under Magnetic Fields

N = 1

N = 2

N = 3

N = -3

N = 0

N = -1

N = -2

Energ

y

DOS kx' ky'

E

Quantization Condition

Energ

y (

eV)

B (T)

0 10 20

0

-0.1

-0.2

0.1

0.2

Novoselov et al (2005)

Zhang et al (2005)

Quantum Hall Effect

Bilayer Graphene

Low energy approximation in 1st BZ Bernal stacked bilayer graphene

Guinea, Neto, & Peres, Phys. Rev. B 73, 245426 (2006)

McCann, Phys. Rev. B 74, 161403 (2006)

Novoselov et al.,Nature Physics (2006)

Energ

y (

eV)

B (T)

0 10 20

0

-0.1

-0.2

0.1

0.2

Hofstadter’s Butterfly in Twisted Graphene

-Moon and Koshino, PRB (2012); See also Bistrizer and MacDonald (2011)

Moiré wavelength is determined by angle between the graphene layers

Moire Pattern in Twisted Graphene Layers

STM observation

of Moire pattern

Twisting angle

~1.16,

a = 7.7 nm

Hexa Boron Nitride: Polymorphic Graphene

Boron Nitride graphene

Band Gap Dielectric Constant Optical Phonon Energy Structure

BN 5.5 eV ~4 >150 meV Layered crystal

SiO2 8.9 eV 3.9 59 meV Amorphous

Comparison of h-BN and SiO2

a0 = 0.250 nm a0 = 0.246 nm

Stacking graphene on hBN

Mobility > 100,000 cm2V-1s-1

Dean et al. Nature Nano (2009)

• Co-lamination techniques • Submicron size precision • Atomically smooth interface

Polymer coating/cleaving/peeling

Micro-manipulated Deposition

Remove polymer Anealing

Moire pattern in Graphene on hBN:

a new route to Hofstadter’s butterfly?

Xue et al, Nature Mater (2011);

Decker et al Nano Lett (2011)

Graphene on BN exhibits clear Moiré pattern

q=5.7o q=2.0o q=0.56o

9.0 nm

13.4 nm

Minigap formation near the Dirac point due to Moire superlattice

0 60 20 40 -20 -40 -60

1

2

3

0

Vg (V)

r (

kW

)

Typical Bilayer Graphene

-40 -30 -20 -10 0 10 20 30 400

2

4

6

8

10

12

r

Vgate (Volts)

r (

kW

)

T = 300 mK

Some Bilayer Graphene on hBN

1 mm

Bilayer graphene on BN substrates shows strong signature of satellite peaks…some times… (~ 30%)

Vg

Abnormal Landau Fan Diagram in Bilayer on hBN

Special Samples with Large Moire Unit Cell

VG (Volts)

B (

Tes

la)

Rxx (kW)

Vxx

I

|Rxy| (kW)

B (

Tes

la)

VG (Volts)

I

Vxy

T=300 mK T=300 mK 1.7 K 1.7 K

Low Magnetic field regime

How to “Read” Normal Landau Fan Diagram?

Landau Fan Diagram for “typical” graphene

0

5

10

0 -1 -2 1 n (cm-2)

B (

T)

0

5

Rxx (kW) Rxx

0

1 Rxx (kW)

0

5

10

0 -1 -2 1 n (cm-2)

B (

T)

0

15

|Rxy| (kW) | Rxy |

-2 n =-6 n =2 -3 -4 -5

n =-2 n =-6

n =-10

n =2 n =4

-3 -4 -5

Abnormal Quantum Hall Effect

Quantum Hall-like Transport

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60.0

0.5

1.0

1.5

2.0

RXX

RXY

filling fraction (n)

RX

X(k

W)

B=18T

-16

-12

-8

-4

0

4

8

12

16

RX

Y (

kW

)

VG (Volts)

B (

Tes

la)

Rxx (kW)

Vxx

I

T=300 mK

Rxx

Landau level filling factor

Quantum Hall conductance

a

Yankowitz et al., (2012)

Dirac point

Miniband I

Miniband II

T = 300 mK B=0

Zero Field Transport

Cg = 118 aF/mm2

-40 -20 0 20 400

5

10

15

r (

kW

)

Vg (V)

Size of the Moire Supper Lattice in Graphene

Ishigami group (UCF)

Confirmed by UHV AFM

a=15±1 nm ~ 30 V ~ 30 V

Vgsb Vg

sb

a = 14.3 nm

Normalized Fan Diagrams

RXX |RXY|

n /

nO

f / fO f / fO

|RX

Y| (k

W)

-4

-2

-4 -3

-1

+2

+4

-4

-2

-4

-2

-4 -3

-1

+2

+4

-4

-2 n /

nO

RX

X (

kW

)

Normalized Fan Diagrams

RXX |RXY|

n /

nO

f / fO f / fO

|RX

X| (k

W)

-4

-2

-4 -3

-1

+2

+4

-4

-2

-4

-2

-4 -3

-1

+2

+4

-4

-2 n /

nO

RX

X (

kW

)

Wannier diagram:

Tracing gaps in

Hofstadter’s Butterfly

slope offset

Landau Fan in Low Magnetic Field Regime

RXX RXY

RXX (kW) RXY (kW)

B (

T)

B (

T)

t = -4 t = -8

t = -12

t = -16

t = -4 t = -8

t = -12

t = -16

Vg (V) Vg (V) s =-4 s =-4 s = 0 s = 0 s =-2 s =-2

t =0 t =-2 t =-4 t =-6

t =-8

t =0 t =-2 t =-4 t =-6

t =-8

Landau Fan in Low Magnetic Field Regime

RXX RXY

RXX (kW) RXY (kW)

B (

T)

B (

T)

Vg (V) Vg (V) s =-2 s =-2

t =-6

t =-8

t =-6

t =-8

Landau Fan in Low Magnetic Field Regime

RXY

RXY (kW)

B (

T)

Vg (V) s =-2

t =-6

t =-8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 140

1

2

3

4

5

(1/6)h/e2

RX

Y (

kW

)

B (Tesla)

t =-6, s =-2

0 1 2 3 4 5 6 7 8 9 10 110

1

2

3

4

RX

Y (

kW

)

B (Tesla)

(1/8)h/e2

t =-8, s =-2

Hall Conductance in Fractal Band

RXY

|RXY| (kW)

n /

nO

f / fO

2

1

3

1

4

1

-12 -8 -4 0 4 8 120

5

-15

-10

-5

0

5

10

15

RX

Y (

kW

) R

XX (

kW

)

f/f0 1/2

1/3

1/4

RXY

RXX

n/n0

Strong suppression of Rxy while Rxx is finite!

Recursive QHE near the Fractal Bands

σxy

Higher quality sample with lower disorder

1/2

1/3

Hall conductivity across Fractal Band

0.0 0.2 0.4 0.6 0.80

50

100

f / f0

xx

(e

2/h

)

-20

-10

0

10

20

30

xy (e

2/h )

At the Fractal Bands Sign reversal of xy

Large enhancement of xx

1/2

1/3

1/4

1/5

Recursive QHE!

Summary

• Graphene on hBN with high quality interface created Moire pattern with supper lattice modulation

• Quantum Hall conductance are determined by two TKNN integers.

• Anomalous Hall conductance at the fractal bands

Open Questions: • Elementary excitation of the fractal gaps? • Role of interactions, Hofstadter Butterfly in

FQHE?

Fractal Gaps: Energy Scales

(t, s)

(-4, 0)

(-8, 0)

(-12, 0)

(-16, 0)

(4, 0)

(8, 0)

(12, 0)

(16, 0)

(3, 0)

(-2, -2)

(-4, 1)

(-2, 1)

(-1, 1)

(-3, 2) (-4, 2)

(-2, 2)

(-1, 2)

Fractal Quantum Hall Effect

(-3, 1)

0 15 Rxx (kW)

Temperature Dependence

Fractal Gap Size

83 K

Single Layer Graphene Hofstadter’s Butterfly

Rxx (Ohm) Rxy (Ohm)

B1/2

-6 -4

-2 +2

+4

Similar physics is observed in single layer graphene on hBN

Rxx Rxy

Acknowledgement

Funding:

hBN samples: T. Taniguchi & K. Watanabe (NIMS)

Theory: P. Moon & M. Koshino (Tohoku)

Prof. Jim Hone

Prof. Ken Shepard

Dr. Cory Dean Lei Wang Patrick Maher Fereshte Ghahari Carlos Forsythe