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Unconventional Sequence of Fractional Quantum Hall States in Suspended Graphene Sudeep Kumar Ghosh CCMT, Department of Physics, Indian Institute of Science. January 31, 2013 Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 1 / 36 Outline of the talk 1 2 Review of Quantum Hall Effect QHE in 2DEG QHE in Graphene The single electron transistor scanning electrometer(SETSE) QH measurement in 2DEG using SETSE 3 Unconventional FQHE seen in suspended graphene using SETSE 4 Summary of results 5 Future directions 6 References and acknowledgements Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 2 / 36 Review of quantum Hall effect Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 3 / 36 Landau levels Electrons in a magnetic field: H= 1 ~ ~ 2 (P + e A) 2m (1) In the Landau gauge the eigenstates are: (y −y0 )2 e ikx x e − 2l 2 Hn (y − y0 ) ψn,kx (x, y ) = (22n π(n!)2 )1/4 (2) with energies 1 En = (n + )~ωC ∝ B, 2 ~ eB 2 where ωC = m , l = eB and y0 = l 2 kx . Finite sample of area A implies degeneracy of each LL = where φ0 = h/e. Filling factor ν = ρ/( φB0 ). (3) A 2πl 2 = φ φ0 Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 4 / 36 Edge states and broadening of Landau levels Figure: Edge states Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 5 / 36 Edge states and broadening of Landau levels Figure: Edge states Figure: Broadening of Landau levels Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 5 / 36 The quantum Hall effect of 2DES Figure: Overview of Hall(RH ) and longitudinal(R) resistances respectively. (No plateau is associated with f = 1/2). Source: H. L. Stormer, Rev.Mod.Phys. 71875889(1999). Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 6 / 36 Understanding the plateaus IQHE The presence of the localized bands explains the plateaus.(Can be seen by changing ρ at fixed B) Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 7 / 36 Understanding the plateaus IQHE The presence of the localized bands explains the plateaus.(Can be seen by changing ρ at fixed B) In the plateaus n extended edge branches carry the current. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 7 / 36 Understanding the plateaus IQHE The presence of the localized bands explains the plateaus.(Can be seen by changing ρ at fixed B) In the plateaus n extended edge branches carry the current. FQHE Laughlin’s brilliant guess for the ground state wave function for ν = 1/3. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 7 / 36 Understanding the plateaus IQHE The presence of the localized bands explains the plateaus.(Can be seen by changing ρ at fixed B) In the plateaus n extended edge branches carry the current. FQHE Laughlin’s brilliant guess for the ground state wave function for ν = 1/3. Explains the Laughlin fractions ν = 1/n (n is an odd integer) and their particle hole symmetric counter parts ν = 1 − 1/n. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 7 / 36 Understanding the plateaus IQHE The presence of the localized bands explains the plateaus.(Can be seen by changing ρ at fixed B) In the plateaus n extended edge branches carry the current. FQHE Laughlin’s brilliant guess for the ground state wave function for ν = 1/3. Explains the Laughlin fractions ν = 1/n (n is an odd integer) and their particle hole symmetric counter parts ν = 1 − 1/n. But can not explain ≈ 30 other fractions seen for ν < 1. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 7 / 36 Understanding the plateaus IQHE The presence of the localized bands explains the plateaus.(Can be seen by changing ρ at fixed B) In the plateaus n extended edge branches carry the current. FQHE Laughlin’s brilliant guess for the ground state wave function for ν = 1/3. Explains the Laughlin fractions ν = 1/n (n is an odd integer) and their particle hole symmetric counter parts ν = 1 − 1/n. But can not explain ≈ 30 other fractions seen for ν < 1. Jain’s idea of composite fermions ⇒ FQHE of electrons is like IQHE of composite fermions. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 7 / 36 Idea of composite fermions A composite fermion is the bound state of an electron and an even number of quantised vortices. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 8 / 36 Idea of composite fermions A composite fermion is the bound state of an electron and an even number of quantised vortices. Figure: Cartoon of composite fermions(taken from Jain’s book). Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 8 / 36 Idea of composite fermions A composite fermion is the bound state of an electron and an even number of quantised vortices. Figure: Cartoon of composite fermions(taken from Jain’s book). Other flavors are 4 CF , 6 CF etc. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 8 / 36 FQHE as IQHE of composite fermions Composite fermions = electrons + 2p flux quanta. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 9 / 36 FQHE as IQHE of composite fermions Composite fermions = electrons + 2p flux quanta. Experience an effective magnetic field: B ∗ = B − 2pρφ0 (4) where, p is an integer. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 9 / 36 FQHE as IQHE of composite fermions Composite fermions = electrons + 2p flux quanta. Experience an effective magnetic field: B ∗ = B − 2pρφ0 (4) where, p is an integer. CFs fill Λ (Landau like) levels in this effective magnetic levels. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 9 / 36 FQHE as IQHE of composite fermions Composite fermions = electrons + 2p flux quanta. Experience an effective magnetic field: B ∗ = B − 2pρφ0 (4) where, p is an integer. CFs fill Λ (Landau like) levels in this effective magnetic levels. Then in the above equation 0 using B = ρφν 0 and B ∗ = ρφ ν∗ we obtain the standard CF sequence, ν= ν∗ . 2pν ∗ ± 1 (5) Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 9 / 36 FQHE as IQHE of composite fermions Composite fermions = electrons + 2p flux quanta. Experience an effective magnetic field: B ∗ = B − 2pρφ0 (4) where, p is an integer. CFs fill Λ (Landau like) levels in this effective magnetic levels. Then in the above equation 0 using B = ρφν 0 and B ∗ = ρφ ν∗ we obtain the standard CF Figure: Left column shows ellctron g.s sequence, of IQHE and the right column shows ∗ C.F g.s of FQHE with ν = n/(2n ± 1) ν ν= . (5) (taken from Jain’s book). 2pν ∗ ± 1 Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall StatesJanuary in Suspended 31, 2013 Graphene 9 / 36 QHE in graphene Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 10 / 36 Graphene band structure and effective Hamiltonian Figure: Band structure of graphene Near the Dirac points : ξ Heff = ξ~vF (qx σx + qy σy ), (6) 0 where, the valley pseudospin ξ = +1(−1) denotes the K (K ) point. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 11 / 36 Landau levels of Dirac electrons Dirac Hamiltonian in a magnetic field : H = ξvF (Πx σ x + Πy σ y ) (7) where, Πi = (Pi + eAi ) and ξ is the valley pseudospin. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 12 / 36 Landau levels of Dirac electrons Dirac Hamiltonian in a magnetic field : H = ξvF (Πx σ x + Πy σ y ) (7) where, Πi = (Pi + eAi ) and ξ is the valley pseudospin. The LL spectrum is λ,n = λ √ ~vF √ 2n ∝ B l where, l 2 = (8) ~ eB . Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 12 / 36 Landau levels of Dirac electrons Dirac Hamiltonian in a magnetic field : H = ξvF (Πx σ x + Πy σ y ) (7) where, Πi = (Pi + eAi ) and ξ is the valley pseudospin. The LL spectrum is λ,n = λ √ ~vF √ 2n ∝ B l where, l 2 = (8) ~ eB . Each LL is four fold degenerate ⇐ 2-fold spin + 2-fold valley degeneracy. Figure: Relativistic Landau levels. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 12 / 36 The relativistic QHE SU(4) symmetry of each LL implies changing the filling by 4 b/w adjacent plateaus. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 13 / 36 The relativistic QHE SU(4) symmetry of each LL implies changing the filling by 4 b/w adjacent plateaus. Particle - hole symmetry leads to half filled zero energy LL at n = 0. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 13 / 36 The relativistic QHE SU(4) symmetry of each LL implies changing the filling by 4 b/w adjacent plateaus. Particle - hole symmetry leads to half filled zero energy LL at n = 0. In the absence of Zeeman effect and electronic interaction no QHE at ν = 0. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 13 / 36 The relativistic QHE SU(4) symmetry of each LL implies changing the filling by 4 b/w adjacent plateaus. Particle - hole symmetry leads to half filled zero energy LL at n = 0. In the absence of Zeeman effect and electronic interaction no QHE at ν = 0. Completely filled n = 1 LL occurs only for ν = ±2, hence the sequence is, 1 ν = ±4(n + ) . 2 (9) Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 13 / 36 The relativistic QHE SU(4) symmetry of each LL implies changing the filling by 4 b/w adjacent plateaus. Particle - hole symmetry leads to half filled zero energy LL at n = 0. In the absence of Zeeman effect and electronic interaction no QHE at ν = 0. Completely filled n = 1 LL occurs only for ν = ±2, hence the sequence is, 1 ν = ±4(n + ) . 2 Figure: RQHE observed in exfoliated graphene.Adapted from Novoselov, (9) Geim, Morozov et al., 2005, Nature 438, 197. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 13 / 36 The single electron transistor scanning electrometer(SETSE) Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 14 / 36 The Single Electron Transistor Figure: SET Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 15 / 36 The Single Electron Transistor Figure: SET Figure: Coulomb oscillations(period e/Cg ) Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 15 / 36 SET scanning electrometer over 2DEG SETSE, uses the SET as a probe to sense the electrically induced charge on its small (100nm) metal island held in proximity to the sample surface. It can detect ≈ 1% of an electron charge. Figure: SETSE over 2DEG Figure: Magnified view of SETSE Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 16 / 36 Typical data Figure: Typical current oscillations ISET (Vb ) of a SET. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 17 / 36 Typical data Figure: Typical current oscillations ISET (Vb ) of a SET. Figure: Complete data set of ISET (Vb ) for 2µm by 2µm raster scan. The top of the data block maps the surface electric field. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 17 / 36 Surface potential map Figure: VS measured along edge. (Inset)VS map in this region. Figure: VS map showing two metal stripes and the intermediate 2DEG region. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 18 / 36 QH measurement in 2DEG using SETSE Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 19 / 36 Density profile near edge Figure: Density profile near edge Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 20 / 36 Incompressible strips Figure: (A) and (G ) are surface potential maps. Others are transparency images measure of the 2DEG compressibility. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 21 / 36 QH measurement in suspended graphene using SETSE Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 22 / 36 SETSE over suspended graphene Suspended graphene Typically have linear dimension 1 − 2µm. Mobility ranges from 20000 to 150000cm2 /Vs at low temperature.(The low temperature mobility of devices on Si/SiO2 substrates are typically 2000 to 15000cm2 /Vs). Figure: Typical device Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 23 / 36 SETSE over suspended graphene Suspended graphene Typically have linear dimension 1 − 2µm. Mobility ranges from 20000 to 150000cm2 /Vs at low temperature.(The low temperature mobility of devices on Si/SiO2 substrates are typically 2000 to 15000cm2 /Vs). Figure: Typical device Figure: Measurement setup Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 23 / 36 Measurement of local electronic compressibility Modulate the charge carrier density using the back gate and monitor the resulting change in the SET current. Measure the local chemical potential and the change in carrier density. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 24 / 36 Measurement of local electronic compressibility Modulate the charge carrier density using the back gate and monitor the resulting change in the SET current. Measure the local chemical potential and the change in carrier density. Obtain the inverse compressibility(κ) using κ−1 = n2 (dµ/dn). Figure: Landau fan Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 24 / 36 Observations B = 0 has incompressible peak due to vanishing d.o.s at the charge neutrality point. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 25 / 36 Observations B = 0 has incompressible peak due to vanishing d.o.s at the charge neutrality point. For B > 0 strong incompressible behavior occurs at ν = 4(n + 1/2) ⇒ monolayer nature of the sample. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 25 / 36 Observations B = 0 has incompressible peak due to vanishing d.o.s at the charge neutrality point. For B > 0 strong incompressible behavior occurs at ν = 4(n + 1/2) ⇒ monolayer nature of the sample. Additional peaks at intermediate integer filling factors ν = 0, 1, 3, 4, 5, 7, 8 and 9 are seen. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 25 / 36 Observations B = 0 has incompressible peak due to vanishing d.o.s at the charge neutrality point. For B > 0 strong incompressible behavior occurs at ν = 4(n + 1/2) ⇒ monolayer nature of the sample. Additional peaks at intermediate integer filling factors ν = 0, 1, 3, 4, 5, 7, 8 and 9 are seen. These broken symmetry states arise from interactions among electrons and are visible well below 1T indicating high quality of the sample. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 25 / 36 Observations B = 0 has incompressible peak due to vanishing d.o.s at the charge neutrality point. For B > 0 strong incompressible behavior occurs at ν = 4(n + 1/2) ⇒ monolayer nature of the sample. Additional peaks at intermediate integer filling factors ν = 0, 1, 3, 4, 5, 7, 8 and 9 are seen. These broken symmetry states arise from interactions among electrons and are visible well below 1T indicating high quality of the sample. Most intriguing is the appearance of the incompressible peaks at fractional filling factors, the strongest of which emerges around B = 1T . Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 25 / 36 Compressibility as a function of filling factor Figure: The vertical features correspond to QH states, whereas localized states curve as the magnetic field is changed. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 26 / 36 Finer compressibility image for ν between 0 and 1 Incompressible peaks occur at ν = 13 , 32 , 52 , 35 , 73 , 74 and 49 reproducing standard composite fermion sequence n ν = 2n±1 observed in GaAs. Figure: Finer measurement of κ for filling factor ν between 0 and 1 Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 27 / 36 Finer compressibility image for ν between 0 and 1 Incompressible peaks occur at ν = 13 , 32 , 52 , 35 , 73 , 74 and 49 reproducing standard composite fermion sequence n ν = 2n±1 observed in GaAs. Strongest incompressible states at ν = 13 and 23 can be resolved down to B ≈ 1T although ν = 23 state weakens considerably below 4T . Figure: Finer measurement of κ for filling factor ν between 0 and 1 Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 27 / 36 Finer compressibility image for ν between 0 and 1 Incompressible peaks occur at ν = 13 , 32 , 52 , 35 , 73 , 74 and 49 reproducing standard composite fermion sequence n ν = 2n±1 observed in GaAs. Strongest incompressible states at ν = 13 and 23 can be resolved down to B ≈ 1T although ν = 23 state weakens considerably below 4T . Figure: Finer measurement of κ for filling factor ν between 0 and 1 With the increase in the filling factor denominator the field at which the state emerges also increases, with ν = 49 being apparent only above B ≈ 9T . Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 27 / 36 Finer compressibility image for ν between 1 and 2 Different pattern of incompressible behavior is observed. Figure: Finer measurement of κ for filling factor ν between 1 and 2 Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 28 / 36 Finer compressibility image for ν between 1 and 2 Different pattern of incompressible behavior is observed. No FQH states with odd numerators occur in this regime. Figure: Finer measurement of κ for filling factor ν between 1 and 2 Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 28 / 36 Finer compressibility image for ν between 1 and 2 Different pattern of incompressible behavior is observed. No FQH states with odd numerators occur in this regime. Instead the FQH states occur 14 only at ν = 34 , 85 , 10 7 and 9 . Figure: Finer measurement of κ for filling factor ν between 1 and 2 Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 28 / 36 Finer compressibility image for ν between 1 and 2 Different pattern of incompressible behavior is observed. No FQH states with odd numerators occur in this regime. Instead the FQH states occur 14 only at ν = 34 , 85 , 10 7 and 9 . The peaks at ν = 43 and 85 are the most robust, persisting down to ≈ 1T to ≈ 1.5T . Figure: Finer measurement of κ for filling factor ν between 1 and 2 Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 28 / 36 Finer compressibility image for ν between 1 and 2 Different pattern of incompressible behavior is observed. No FQH states with odd numerators occur in this regime. Instead the FQH states occur 14 only at ν = 34 , 85 , 10 7 and 9 . The peaks at ν = 43 and 85 are the most robust, persisting down to ≈ 1T to ≈ 1.5T . Figure: Finer measurement of κ for filling factor ν between 1 and 2 The even numerator fractions can be explained by considering the standard CF sequence and ν ∗ = 2 − ν. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 28 / 36 Average compressibility Averaging over magnetic field reduces the influence of localized states and shows clear incompressible peaks centered at ν = 13 , 25 , 37 , 49 , 47 , 35 , 2 4 8 10 14 3 , 3 , 5 , 7 and 9 . Figure: dµ/dn between ν = 0 and 1, averaged over 9 to 11.9 T (blue), and between ν = 1 and 2, averaged over 4.9 to 6.4 T (red). Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 29 / 36 Possible explanation of the unconventional hierarchy The absence of odd-numerator fractions indicates that a robust underlying symmetry enables low-lying excitations, preventing the formation of incompressible states. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 30 / 36 Possible explanation of the unconventional hierarchy The absence of odd-numerator fractions indicates that a robust underlying symmetry enables low-lying excitations, preventing the formation of incompressible states. The differing behavior above and below ν = 1 suggests an intriguing interplay between the inherent symmetries of graphene and electronic correlations in the lowest LL. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 30 / 36 Possible explanation of the unconventional hierarchy The absence of odd-numerator fractions indicates that a robust underlying symmetry enables low-lying excitations, preventing the formation of incompressible states. The differing behavior above and below ν = 1 suggests an intriguing interplay between the inherent symmetries of graphene and electronic correlations in the lowest LL. May be the Zeeman effect lifts spin degeneracy, but valley symmetry remains intact, allowing large valley skyrmions to form with a minimal energy penalty at odd-numerator filling factors. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 30 / 36 Possible explanation of the unconventional hierarchy The absence of odd-numerator fractions indicates that a robust underlying symmetry enables low-lying excitations, preventing the formation of incompressible states. The differing behavior above and below ν = 1 suggests an intriguing interplay between the inherent symmetries of graphene and electronic correlations in the lowest LL. May be the Zeeman effect lifts spin degeneracy, but valley symmetry remains intact, allowing large valley skyrmions to form with a minimal energy penalty at odd-numerator filling factors. The total electron density, and not just the filling fraction, may play an important role in electronic interactions in the lowest LL. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 30 / 36 Steps in chemical potential Integrate the inverse compressibility w.r.t carrier density to extract the step in chemical potential ∆µν associated with each FQH state and thereby determine the corresponding energy gap ∆ν . Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 31 / 36 Steps in chemical potential Integrate the inverse compressibility w.r.t carrier density to extract the step in chemical potential ∆µν associated with each FQH state and thereby determine the corresponding energy gap ∆ν . √ ∆µν are expected to scale as B but the behavior seen is unclear. Also the extracted energy gaps do not match with that of theory. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 31 / 36 Spatial dependence of compressibility Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 32 / 36 Summary of results For filling factor ν < 1 the standard FQH sequence is observed. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 33 / 36 Summary of results For filling factor ν < 1 the standard FQH sequence is observed. Between ν = 0 and 1, the compressibility is approximately symmetric about ν = 1/2, suggesting that the fourfold spin and valley degeneracy is fully lifted. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 33 / 36 Summary of results For filling factor ν < 1 the standard FQH sequence is observed. Between ν = 0 and 1, the compressibility is approximately symmetric about ν = 1/2, suggesting that the fourfold spin and valley degeneracy is fully lifted. In contrast, between ν = 1 and 2 only even numerator fractions are observed. This indicate that one symmetry persists between ν = 1 and 2. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 33 / 36 Summary of results For filling factor ν < 1 the standard FQH sequence is observed. Between ν = 0 and 1, the compressibility is approximately symmetric about ν = 1/2, suggesting that the fourfold spin and valley degeneracy is fully lifted. In contrast, between ν = 1 and 2 only even numerator fractions are observed. This indicate that one symmetry persists between ν = 1 and 2. √ The chemical potential steps do not show expected B dependence. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 33 / 36 Summary of results For filling factor ν < 1 the standard FQH sequence is observed. Between ν = 0 and 1, the compressibility is approximately symmetric about ν = 1/2, suggesting that the fourfold spin and valley degeneracy is fully lifted. In contrast, between ν = 1 and 2 only even numerator fractions are observed. This indicate that one symmetry persists between ν = 1 and 2. √ The chemical potential steps do not show expected B dependence. The theoretically predicted energy gaps do not match with the ones obtained. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 33 / 36 Summary of results For filling factor ν < 1 the standard FQH sequence is observed. Between ν = 0 and 1, the compressibility is approximately symmetric about ν = 1/2, suggesting that the fourfold spin and valley degeneracy is fully lifted. In contrast, between ν = 1 and 2 only even numerator fractions are observed. This indicate that one symmetry persists between ν = 1 and 2. √ The chemical potential steps do not show expected B dependence. The theoretically predicted energy gaps do not match with the ones obtained. Graphene provides a rich platform to investigate correlated electronic states. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 33 / 36 Future directions Further experiments are necessary to elucidate the exact spin and valley ordering of each state; for example, tilted field measurements decouple Zeeman splitting from orbital effects and could provide insight into spin polarization. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 34 / 36 Future directions Further experiments are necessary to elucidate the exact spin and valley ordering of each state; for example, tilted field measurements decouple Zeeman splitting from orbital effects and could provide insight into spin polarization. Rigorous theoretical explanation is necessary for the unconventional sequence of FQH states observed. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 34 / 36 Future directions Further experiments are necessary to elucidate the exact spin and valley ordering of each state; for example, tilted field measurements decouple Zeeman splitting from orbital effects and could provide insight into spin polarization. Rigorous theoretical explanation is necessary for the unconventional sequence of FQH states observed. Theoretical understanding is necessary for the unusual behavior of the chemical potential steps observed. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 34 / 36 References and acknowledgements References 1 Feldman et.al., Science 337, 1196-1199 (2012). 2 Yacoby et.al., Solid state communications 111, 1-13 (1999). 3 Csaba et.al., PRB 74, 235417(2006). 4 M. O. Goerbig, RMP 83, 1193(2011). 5 Castro Neto et. al., RMP 81, 109(2009). 6 Composite Fermions – Jainendra K. Jain. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 35 / 36 References and acknowledgements References 1 Feldman et.al., Science 337, 1196-1199 (2012). 2 Yacoby et.al., Solid state communications 111, 1-13 (1999). 3 Csaba et.al., PRB 74, 235417(2006). 4 M. O. Goerbig, RMP 83, 1193(2011). 5 Castro Neto et. al., RMP 81, 109(2009). 6 Composite Fermions – Jainendra K. Jain. Acknowledgements I acknowledge Prof. Vijay B. Shenoy and Subhomay Ghatak for many fruitful discussions. Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 35 / 36 Thank You! Sudeep Kumar Ghosh (CCMT, Department ofUnconventional Physics, IndianSequence Institute of Fractional Science.) Quantum Hall States January in Suspended 31, 2013Graphene 36 / 36