Hofstadter’s Butterfly in the strongly interacting regime Cory R. Dean Columbia University.
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Transcript of Hofstadter’s Butterfly in the strongly interacting regime Cory R. Dean Columbia University.
Graphene on BN
linear dispersion:
Novoselov, Geim, Science (2005), Nature (2005); Zhang, Kim, Nature (2005)
Graphene on BN
linear dispersion:
Dean, et al Nature Nano. (2010)
-40 -20 0 20 400
1000
2000
3000
4000
Vg (Volts)
(
)
T = 1.5K
0
100
200
300
400
500
(e
2/h
)
Novoselov, Geim, Science (2005), Nature (2005); Zhang, Kim, Nature (2005)
Graphene on BN
linear dispersion:
Dean, et al Nature Nano. (2010)
-40 -20 0 20 400
1000
2000
3000
4000
Vg (Volts)
(
)
T = 1.5K
0
100
200
300
400
500
(e
2/h
)
Novoselov, Geim, Science (2005), Nature (2005); Zhang, Kim, Nature (2005)
Mobility
Moire Pattern
0 20 40 60 80 100 120 140 1600.1
1
10
100
1000
10000
Bo
(Tes
la)
unit cell (nm)
a = 15 nm
BL graphene on h-BN
1 mm
Xue et al, Nature Mater (2011);Decker et al Nano Lett (2011);Yankowitz et al, Nat. Phys. (2012)
STM measurement confirms Moiré pattern
BL graphene on h-BN
Xue et al, Nature Mater (2011);Decker et al Nano Lett (2011);Yankowitz et al, Nat. Phys. (2012)
STM measurement confirms Moiré pattern
Bloch Waves:
Length scale: lattice paramater, a
Periodic Potential
BL graphene on h-BN
Park et al, PRL (2008); Nat. Phys. (2008)
Bloch Waves:
Length scale: lattice paramater, a
Periodic Potential
Elecrons subjected to classical forces
Felix Bloch
Bloch Waves:
Length scale: lattice paramater, a
Periodic Potential
Lev Landau
Magnetic Field
, N=0,1,2,3...*m
eBc
Landau levels:
Length scale: magnetic length, lB
Proc. Phys. Soc. Lond. A 68 879 (1955)
Quantum Hall effect in a periodic potential
Harper’s equation for 2D square lattice
“superlattice” DouglasHofstadter
-Hofstadter, PRB (1976)
Harper’s equation for 2D square lattice
When b=p/q, where p, q are coprimes, the LL splits into p sub-bands that are q-fold degenerate
Quantum Hall effect in a periodic potential
“Hofstadter Butterfly”
“superlattice” DouglasHofstadter
Landau energy bands develop fractal structure when magnetic length is of order the periodic unit cell
Harper’s equation for 2D square lattice
When b=p/q, where p, q are coprimes, the LL splits into p sub-bands that are q-fold degenerate
Quantum Hall effect in a periodic potential
“Hofstadter Butterfly”
“superlattice” DouglasHofstadter
-Hofstadter, PRB (1976)
0 1
1
e /W
Hofstadter’s Energy Spectrum
Hofstadter’s Butterfly to Wannier Diagram
magnetic flux per unit cell
0 1
1
e /W
1/21/31/41/5
Hofstadter’s Energy Spectrum
Hofstadter’s Butterfly to Wannier Diagram
magnetic flux per unit cell
0 1
1
e /W
1/21/31/4
1/21/31/41/5
Hofstadter’s Energy Spectrum
0 1
1
Wannier Phys. Stat. Sol. (1978)
Hofstadter’s Butterfly to Wannier Diagram
Electron density per unit cell
magnetic flux per unit cell
t : slopes : intercept
Diophantine equation:
Quantum Hall effect in a periodic potential
•Spectral gaps satisfy a Diophantine equation
(also Streda J. Phys. C. (1982)
Band filling factor Quantum Hall conductance
Quantum Hall effect in a periodic potential
•Spectral gaps satisfy a Diophantine equation
(also Streda J. Phys. C. (1982)
Band filling factor Quantum Hall conductance
Physical observables:
• non-monotonic variation in Rxy but with quantization to integer units of e2/h• corresponding minima in Rxx, but not necessarily at integer filling fractions
0 1
1e
/W
0 20 40 60 80 100 120 140 1600.1
1
10
100
1000
10000
Bo
(Tes
la)
unit cell (nm)
Technical challenge:
Experimental challenges
a = 1nm
0 1
1e
/W
0 20 40 60 80 100 120 140 1600.1
1
10
100
1000
10000
Bo
(Tes
la)
unit cell (nm)
Technical challenge:
Experimental challenges
a = 1nm
Looking for Hofstadter
-Schlosser et al, Semicond. Sci. Technol. (1996)
Albrecht et al, PRL. (2001); Geisler et al, Physica E (2004)
0 20 40 60 80 100 120 140 1600.1
1
10
100
1000
10000
Bo
(Tes
la)
unit cell (nm)
a = 100nm
• Unit cell limited to ~100 nm• limited field and density range accessible• Do not observe ‘fully quantized’ mingaps in fractal spectrum• require low disorder!
Looking for Hofstadter
-Schlosser et al, Semicond. Sci. Technol. (1996)
Albrecht et al, PRL. (2001); Geisler et al, Physica E (2004)
0 20 40 60 80 100 120 140 1600.1
1
10
100
1000
10000
Bo
(Tes
la)
unit cell (nm)
a = 100nm
• Unit cell limited to ~100 nm• limited field and density range accessible• Do not observe ‘fully quantized’ mingaps in fractal spectrum• require low disorder!
Young, et al Nature Phys. (2012)
+10-1 543 987
T=1.5K
12
8
0
4
+6 +10+2-2-6
-10
Conventional QHE in graphene
Agreement with the Diophantine equation
Dean et al, Nature (2012)
0
1
1
-10
slope offset
Wannier diagram:
Diophantine equation:
Symmetry Breaking
Dean et al, Nature (2012)
fully degenerate: S=0,4,8…Sublattice lifted: S=0,2,4,6….
Symmetry Breaking
Dean et al, Nature (2012)
• strong electron hole asymmetry•Valley symmetry breaking•Selected minigaps emerge•Qualitative agreement with theory for BL graphene on BN
fully degenerate: S=0,4,8…Sublattice lifted: S=0,2,4,6….
• strong electron hole asymmetry•Valley symmetry breaking•Selected minigaps emerge•Qualitative agreement with theory for BL graphene on BN
Symmetry Breaking
Dean et al, Nature (2012)
Role of interactions??
What about nesting prediction?
1/20.5
0.6
0.4
f/f o
1/3
0.25
0.35
f/f o
n/no
n/no
-8-6 -4 -2 2 4
6
0
-10 -7-4
2
8
What about nesting prediction?
1/20.5
0.6
0.4
f/f o
1/3
0.25
0.35
f/f o
n/no
n/no
-8-6 -4 -2 2 4
6
0
-10 -7-4
2
8
What about nesting prediction?
1/20.5
0.6
0.4
f/f o
1/3
0.25
0.35
f/f o
n/no
n/no
-8-6 -4 -2 2 4
6
0
-10 -7-4
2
8
1/2
2/3
1/3
Monolayer graphene on BN
Lei et al, Science (2013)
-1 0 10
200
400
600
800
1000
1200
1400
B=1TSiGate = 0V
Rxx absRxy
Vg (Volts)
Rxx
(
)
0
5
10
15
Rxy
(k
)
-1 0 10
200
400
600
800
1000
1200
1400
Rxx Rxy
Vg (Volts)
Rxx
(
)
0
5
10
15
B=1TSiGate = +40V
Rxy
(k
)
Emergence of a gap at CNP
Hunt, et al Science (2014); Gorbachev, et al, Science (2014);Woods, et al, Nature Phys (2014); Chen, et al, Nature Comm (2014); Amet, et al, Nature Comm (2015)
Monolayer graphene on BN
FQHE can exist in a patterned 2DEG with Hofstadter mini-gaps
When FQHE and mini-gap states intersect, state with the larger gap emerges
Kol, A. & Read, PRB (1993).
Pfannkuche, D. & Macdonald (1997)
Ghazaryan, Chakraborty, & Pietilainen, arXiv (2014).
Detailed look at high field features
gaps and mini gaps allowed by Diophantine equationconventional fractional quantum Hall effectanomalous
Detailed look at high field features
0.80 0.82 0.84 0.86 0.88 0.90
1.15
1.20
n/n
o
o
Equation y = A + B*xAdj. R-Square 0.99876
Value Standard Errorconsider the center A 0.27276 0.00705
consider the center B 1.04165 0.00821
0
1
2
3
Sig
ma
xy(e
2 /h)
0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.20
3000
6000
9000 40T
Rxx
()
n/no
0
1
2
3
4
Sig
ma
xy(e
2 /h)
Detailed look at high field features
0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.20
3000
6000
9000 40T
Rxx
()
n/no
0
1
2
3
4
Sig
ma
xy(e
2 /h)
sXY plateau = 1 e2/hSlope (“t”) = 1.009 +/- 0.007n/no intercept (“s”) = 0.3 +/- 0.01
1/3 1/2-1/3
Observation of fractional Bloch index
Detailed look at high field features
• Generalized Laughlin-like state?• Spontaneous quantum Hall ferromagnetism?• Charge density wave?
MacDonald and Murray, PRL (1985)Kol, A. & Read, PRB (1993).Pfannkuche, D. & Macdonald (1997)Ghazaryan, Chakraborty, & Pietilainen, arXiv (2014).Chen et al PRB (2014)
Observations
• Direct evidence of mini-gap structure following Wannier description in density-flux space• Observation of two quantum numbers associated with each gap in the QHE spectrum• Evidence of recursion within the diagram• Evidence of fractional Bloch index
Open questions• role of electron-electron interactions/origin of symmetry breaking• relationship between observed gap hierarchy and superlattice symmetry• nature of the FBQHE ground state??
Acknowledgements
Theoretical discussions
A. MacDonald, T. Chakraborty, M. Koshino, I. Aleiner, V. Fal’ko, A. Young
Collaboration
James Hone, Ken Shepard, Philip Kim,Masa Ishigami, M. Koshino, P. Moon
L. Wang, Y. Gao, P. Maher, F. Ghahari, C. Forsythe, Y. Gao, J. KatochZ. Han, B. Wen
Takashi Taniguchi, Kenji Watanabe
NIMS Japan (h-BN crystals)