Philip Kim

Physics Department, Columbia University

Bloch, Landau, and Dirac:

Hofstadter’s Butterfly in Graphene

Acknowledgment

Funding:

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

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

Prof. Jim Hone

Prof. Ken Shepard

Prof. Cory Dean

(now at CUNY)Lei Wang Patrick Maher Fereshte Ghahari Carlos Forsythe

Bloch Waves: Periodic Structure & Band Filling

Felix Bloch

Zeitschrift für Physik, 52, 555 (1929)

Periodic Lattice

a

A0 : unit cell volume

k

e

Block Waves:

Bloch Waves: Periodic Structure & Band Filling

Felix Bloch

Periodic Lattice

a

A0 : unit cell volume

k

e

k

e

eF

n(E)

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

lBa

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

off

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

1/21/31/4

1/21/31/41/5

Hofstadter‟s Energy SpectrumTracing 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)

-Schlosser et al, Semicond. Sci. Technol. (1996)

Experimental Search For Butterfly

Albrecht et al, PRL. (2001);

Geisler et al, PRL (2004)

• Unit cell limited to ~40-100 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

Novoselov et al. (2004)

Graphene: Under Magnetic Fields

N = 1

N = 2

N = 3

N = -3

N = 0

N = -1

N = -2

Energ

y

DOSkx' 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

Hexa Boron Nitride: Polymorphic Graphene

Boron Nitridegraphene

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 nma0 = 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-manipulatedDeposition

Remove polymerAnealing

5o 10

o

15o20

o30

o

Graphen/hBN Moire Pattern

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

momentum

Transport Measurement Graphene with Moire Superlattice

~ 30 V ~ 30 V

lT = 14.6 nm

1 mm

Moiré lAFM = 15.5 nm

UHV AFM (Ishigami group)

Dirac point

Second Dirac point E

k Zone folding and mini-gap formation

Abnormal Landau Fan Diagram in Bilayer on hBN

Special Samples with Large Moire Unit Cell

VG (Volts)

B(T

esla

)

Rxx (kW)Vxx

I

|Rxy| (kW)

B(T

esla

)

VG (Volts)

I

Vxy

T=300 mK T=300 mK1.7 K1.7 K

Low Magnetic field regime

How to “Read” Normal Landau Fan Diagram?

Landau Fan Diagram for “typical” graphene

0

5

10

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

B(T

)

0

5

Rxx (kW)Rxx

0

1Rxx (kW)

0

5

10

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

B(T

)

0

15

|Rxy| (kW)| Rxy |

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

n =-2n =-6

n =-10

n =2n =4

-3-4-5

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(T

esla

)

Rxx (kW)Vxx

I

T=300 mK

Rxx

Landau level filling factor

Quantum Hall conductance

Quantum Hall Effect in Graphene Moire

Quantum Hall Effect with Two Integer Numbers

(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)

n/n0: density per unit cell; f : flux per unit cell

(-3, 1)

0 15Rxx (kW)Quantization of xx and xy

Quantum Hall Effect with Two Integer Numbers

RXX RXY

RXX (kW) RXY (kW)

B(T

)

B(T

)

t = -4t = -8

t = -12

t = -16

t = -4t = -8

t = -12

t = -16

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

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

t =-8

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

t =-8

RXX RXY

RXX (kW) RXY (kW)

B(T

)

B(T

)

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

t =-6

t =-8

t =-6

t =-8

Quantum Hall Effect with Two Integer Numbers

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

Quantum Hall Effect with Two Integer Numbers

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 and Outlook

Energy spectroscopy is needed for the next step

• 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

Fractal Gap Size

83 K

Large odd integer gap indicates (fractal) quantum Hall ferromagentism!!

SU(4) Quantum Hall Ferromagnet in Graphene

q

Valley spinSpin

X

SU(4)<

yy

yy

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

Partial list of references

Nature of Quantum Hall Ferromagnetism in Graphene

Electron Interaction Fractal Spectrum

Graphene Franctional Quantum Hall Effect

5 mm

Dean et al. Nature Physics (2011)

Fractional Quantum Hall Gaps

Vg (V)

B(T

)

xy (mS)0 1-1

Hall Conductivity

Fractional Quantum Hall Effect in Moire Superlattice

Single layer graphene on hBN @ 20 mK up to 35 T

Fractional Quantum Hall Effect

2/3

4/35/3

7/3

Rxx

Rxy

xx (e2/h)

(n)

4/3 5/3 7/38/3

13/310/3

Acknowledgment

Funding:

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

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

Prof. Jim Hone

Prof. Ken Shepard

Prof. Cory Dean

(now at CUNY)Lei Wang Patrick Maher Fereshte Ghahari Carlos Forsythe