Download Even-denominator fractional quantum Hall effect in bilayer graphene

Even-denominator fractional quantum Hall effect in bilayer graphene
With DongKeun Ki, Dima Abanin, and Volodya Falko
Strain induced universality of transport
through graphene-on-substrate
With Nuno Couto, Davide Costanzo, DongKeun Ki, Takashi Watanabe,
Kenji Taniguchi and Paco Guinea
Alberto Morpurgo
What is the dominant scattering mechanism for graphene
on substrates?
A long standing debate
-Charged impurities on substrates
Long range Coulomb potentials due
To charges at the substrae surface
-Resonant scattering
Very short-range, very strong potentials
Due to adsorbates
das Sarma 2007
-Others
E.g., ripples
Much discussed and studied experimentally
E.g. Fuhrer 2008
Expt.: Maryland, Manchester, Orsay,…
Theory: Ando, das Sarma, MacDonald,
Katsenlson, Guinea, Falko, Peres,….
E.g. Wehling 2010
Quality characterization
Fully developed IQHE for B~ 1T
m ~ 30.000 cm2/Vs commonly
Maximum m ~ 80.000 cm2/Vs
In devices with aligned edges:
Signatures of satellite
Dirac points
Fully broken spin/valley symmetry below 15 T
Quantification of disorder effects
Most relevant phenomena originating from disorder:
elastic scattering and charge inhomogeneity
Width of Dirac peak measures
Mobility m measures
magnitude of charge fluctuations n*
rate of momentum relaxation
Do momentum relaxation and charge inhomogeneity
originate from the same or from different microscopic processes?
If the same microscopic mechanisms cause elastic scattering and charge
inhomogeneity
m and n* measured in different devices should show a correlation
Very clear correlation between m and n*
Kim’s group
Van Wees’ group
SrTiO3 substrate
SiO2 substrate
Correlation satisfied by devices realized in different laboratories and on
different substrates with different surface chemistry and dielectric properties
Evidence for universality of the effects of disorder in graphene
Back to the basics: Scattering times from weak loc.
Here: data from device with m ~ 60.000 cm2/Vs @ 250 mK
(literature: conclusions valid at least for m between 1.000 and 60.000 cm2/Vs)
VG (V)
Ensemble averaging to suppress UCF
McCann et al., PRL 97, 146805 (2006)
Analysis of microscopic times
tf > tiv
That is why we see weak localization
tiv >> t
in the entire density range investigated
Inter-valley scattering is not determining
momentum relaxation.
Intra-valley scattering is the
dominant scattering process
Disorder potential is long-ranged
Other important observation:
t* ~ t for all density of carriers
Proposed sources of long-range disorder
Charged impurities at the substrate surface
Strain and mechanical deformations
Recent interesting paper
Scalar potential
Ax
Vector potential
Ay
What breaks «effective» valley time reversal symmetry
Complex conjugation
Dirac continuum Hamiltonian in a single valley
has an antiunitary symmetry A= isy K
Effective valley time reversal symmetry
Much less robust than true TRS: broken by many perturbations
Gauge potential
due to strain
Trigonal warping:
effect depends on density
+ O(k2) + dt(x,y)
Explains t* ~ t
if strain is the main
scattering mechanism
Potential difference between A and B atoms in graphene unit cell.
could come from random potential due to surface charge
But cannot explain t* ~ t
( long-range potential gives t* >> t
incompatible with observations)
Graphene on SrTiO3 substrates
SrTiO3 dielectric constant
e ~ 200-300
e > 5000
@ room T
@ 4.2 K
Does the substrate dielectric constant
affect transport through graphene?
Graphene on STO: substrate screening due to high e
Very small optical contrast
3L
2L
1L
Optical identification
confirmed by QHE data
e ~ 200-300
@ Room T
e > 5000
@ 4.2 K
SrTiO3 dielectric constant
from capacitance measurements
Dirac peak narrows
with lowering tempeature
Density extracted from Hall effect
Hall resistance
Carrier density as a function of Vg
density –vs-T at Vg = 1 V
Longitudinal conductivity
does not depend on T
between 50 K and 250 mK
e increases ~ 10 x
Rxx @ N=0 : decreases more than 3x with lowering T
Rxx @ N=0 : decreases more than
3x with lowering T
Quantum Hall effect at 50 K
Peak height
N=0
N=+/-1
On SiO2 opposite behavior is seen
Nano-ribbons on STO
Etched nano-ribbons @ charge neutrality
Series of random quantum dots
L= 1 mm
Ti/Au
contacts
W=70 nm
Ti/Au
contacts
Source-drain « gap »
Quantum dot Coulomb energy
Expected much smaller gap on SrTiO3 due to screening
SrTiO3
SiO
SrTiO
2
3
For same dimension nano ribbon (L=1 mm, W=70 nm)
On SiO2:
On SrTiO3:
gap ~ 15 meV
gap ~ 2 meV (also includes effect of single particle level spacing)
Substrate screening
Back to Strain
Can strain explain the correlation qualitatively/quantiatively?
Enter the Theorist
Graphene-on-substrate is an elastic membrane
with (frozen) random height flucuations that cause strain
1) distortion (scalar) potentials; 2) random hopping integrals (gauge potentials)
Calculate scattering time (Fermi golden rune) and mobility
Scalar = screened
Gauge = NOT screened
Calculate charge inhomogeneity (due to scalar potential only)
m and n* are related because
&
Both come directly from
It works quantiatively without free parameters
Only depends on properties of
graphene and fundamental
constants
Constants describe the effect of
deformation of
graphene on electronic energies
Values known from literature
Strain: not only explains all qualitative aspects but also works quantiatively
Even-denominator fractional quantum Hall effect
In bilayer graphene
Landau levels in bilayer graphene
Monolayer
Bilayer
One zero mode (N=0)
0
Two zero modes (N=0, 1)
0
H=
0
0
Mono- and Bi-layer have 4x degeneray (2x spin and 2x valley)
Ballistic transport and FQHE in suspended monolayers
Very narrow Dirac peak -> high mobility
Andrei’s group
Kim’s group
Fractional quantum Hall effect in
two-terminal measurements
Multi-terminal devices cannot cleaned by current annealing
Multi-terminal device configurations have been tried by all groups
• Yield of working devices very,
very small
• Problem: current annealing
not uniform
• Contact probes are too invasive
• Virtually all groups gave up
Multi-terminal: The solution of the problem
Separating spatially the bulk metallic contacts
from the region probed by transport
Annealing: current from (2,3) to (1,4):
all contact configurations show sharp Dirac peak at same VG
BTW: where are
The E=0 edge states?
Separating longitudinal and transverse resistance
Multi-terminal configuration enables measurements
not possible in two-terminal devices
Graphene bilayer
• Very high quality QHE
at low magnetic field
• Broken symmetry states
starting at ~ 0.2 T
Complete vanishing of longitudinal
resistance in the IQHE
Transverse resistance
quantized at correct values
Zooming in on low field
Very high-quality quantum Hall effect at low field (starting at 300 G)
Criterium for visibility
of QHE: m B > 1
m > 330.000 cm2/Vs
Clear plateaus
at least up to
n=36
Negative 4-terminal resistance
Ballistic transport on sample dimensions
n ~ 10 10 cm-2
le ~ 1 mm
Crossover
Diffusive/ballistic
Ballistic
9
cm-2
n ~ 10
1) Charge puddles
2) lF = 2p/kF ~
sample width
le > 2 mm
Diffusive
m ~ 800.000 cm2/Vs
Magnetic field dependence of negative resistance
confirms quasiclassical ballistic transport
Independent measurement of rxx and rxy
At large B fractional quantum Hall effect becomes visible
Even denominator: Clear plateau at ½ e2/h
Fan diagram of FQHE states
Integer QH robust:
electron hole symmetric
Fractional QH weaker:
seen only on hole side (due to disorder)
Plotting as a function of filling factor and B:
helps to identify QH features
5/2
Two fully developed FQH states at 1/2 and 4/3
• 1/2 and 4/3: have plateaus in Rxy at expected values and minima in Rxx
• 1/2 and 4/3: seen fully developed in the two devices measured
• Other fractions: seen rather unambiguously, but not same data quality
1/2
4/3
Energy scales
Activation energies obtained experimentally from minimum in rxx
Not fully developed states
Several other features occur
at a fixed filling factor
Observed fractions:
-4/3; -1/2
fully developed
Others
-2/3; -8/5; -5/2; +2/3
Minima in Rxx
Same fractions seen with comparable quality in two different devices
Predicted:
Intrinsic electron-hole asymmetry
Predicted: -1,-4/3, -5/3, -8/5, -1/2
Predicted:
approximate symmetry n-> n + 2
Predicted:
Basic physics comes from N=0,1 degeneracy
Observed
Observed: all except -5/3
(we see -2/3 not predicted)
Observed:
-5/2; -1/2 = -5/2 + 2
+2/3; +2/3 = -4/3 + 2
Explains:
Difference between mono- and bi-layer
Prediction: states at -1/2 and -1/2 +/- 2 are of Moore-Read type
Conclusions
Correlation between mobility and charge inhomogeneity:
Strain accounts for all our expt observation and implies universality
First observation of FQHE in bilayer graphene
New even-denominator
FQH state at filling factor 1/2