Bloch, Landau, and Diraconline.itp.ucsb.edu/online/colloq/kim1/pdf/Kim1_Colloquium_KITP.pdf ·...
Transcript of Bloch, Landau, and Diraconline.itp.ucsb.edu/online/colloq/kim1/pdf/Kim1_Colloquium_KITP.pdf ·...
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