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American Physical Society March Meeting, Mar 16, 2009. High-field ground state of graphene at Dirac Point. J. G. Checkelsky, Lu Li and N.P.O. Princeton University. Ground state of Dirac point in high fields Thermopower at Dirac point. Checkelsky Li, NPO, PRL 08 Checkelsky, Li, NPO, PRB 09

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High field ground state of graphene at dirac point

American Physical Society March Meeting, Mar 16, 2009

High-field ground state of graphene

at Dirac Point

J. G. Checkelsky, Lu Li and N.P.O.

Princeton University

  • Ground state of Dirac point in high fields

  • Thermopower at Dirac point

Checkelsky Li, NPO, PRL 08

Checkelsky, Li, NPO, PRB 09

Checkelsky, NPO, arXiv 08

National High Magnetic Field Lab., Tallahassee

NSF-MRSEC DMR 084173


High field ground state of graphene at dirac point

Zero-energy mode, n=0 Landau Level

For valley K

,

2 valleys K, K’

For n=0 Landau Level

1) K-K’ valley index same as A-B sublattice index

2) Electrons on B sites only


High field ground state of graphene at dirac point

Broken Symmetry ground states

Coulomb interaction energy in high-field state

Nomura, MacDonald, PRL 06

Goerbig, Moessner, Doucot, PRB 06

Kun Yang, Das Sarma, MacDonald, PRB 06

Ezawa, JPSJ 06

Abanin, Lee, Levitov, PRL 06, 07

Fertig, Bray, PRL 06

Shimshoni, Fertig, Pai, 08

Alicea, Fisher, PRB 06

Gusynin, Miransky, Sharapov, Shovkovy, PRB 06

Khveshchenko, PRL 02


High field ground state of graphene at dirac point

Quantum Hall Ferromagnet and CPE modes

Edge potentl

zig-zag

K

Exchange

K

m

4-fold

degen.

K’

K’

Abanin, Lee, Levitov

Fertig, Bray

Bernevig, Zhang

QHF state

Counter-propg edge (CPE) modes

At Dirac point, CPE modes protected

Rxx ~ (h/e2) in intense fields (dissipative to very large H)

Realization of quantized spin-Hall system


High field ground state of graphene at dirac point

Magnetic Catalysis (field induced gap)

E(k)

E(k)

ky

ky

kx

kx

neutrino

electron

Applequist et al. 1986

Miransky et al.

1995, 2003, 2005, 2008

D. Khveshchenko 2002

Field H creates gap

Mass term

Chiral-symmetry breaking

In (2+1) QED3

Field-enhancement of DOS at Dirac Point

Exciton condensation of electron hole pairs

Order parameter for H > Hc

Equivalent to sublattice CDW


High field ground state of graphene at dirac point

The resistance Rxx and Hall conductivity in graphene

Checkelsky, Li, Ong, PRL ‘08

Rxx diverges with H at Dirac Point

s


High field ground state of graphene at dirac point

Divergent R0 and zero-Hall step

Checkelsky, Li, Ong, PRL ‘08

R0 increases as T decreases

At fixed H and T, n=0 peak

dependent on V0 (offset voltage)

sxy = 0 at n=0

At 14 T, n=0 LL split into 2 subbands

Split resolved at 100 K


High field ground state of graphene at dirac point

Offset Gate voltage V0 -- an important parameter

V0

Small offset V0 correlates with low disorder


High field ground state of graphene at dirac point

R0 saturates below 2 K

Conductance saturates

for T < 2 K

Divergence in R0 steepest

At 0.3 K

Sample dependent


High field ground state of graphene at dirac point

Contour map of R0 at Dirac point in H-T plane

R0 is exponentially sensitive to H

but T-independent below 2 K


High field ground state of graphene at dirac point

Spurious features from self-heating

Power dissipatnP > 10 pW 

thermal runaway at Dirac point

A major experimental obstacle


High field ground state of graphene at dirac point

Adopt ultralow dissipation measurement circuit

Checkelsky, Li, NPO, PRB ‘09

Hold Vtot = 40 mV

Phase detect I,

Vxx and Vxy

Power dissipated

10 aW to 3 fW


High field ground state of graphene at dirac point

Checkelsky, Li,

NPO, PRB ‘09

At 0.3 K, R0 diverges to 40 MW

Divergence in R0 seems singular


High field ground state of graphene at dirac point

Fits to Kosterlitz-Thouless correlation length over 3 decades

3-decade fit to

KT correlation x


High field ground state of graphene at dirac point

Kosterlitz Thouless fit in Samples K52, J18 and J24 decades

Checkelsky, Li, NPO, PRB ‘09

(29 T)

(36 T)

Slopes b are similar

Hc sample dependt

(32 T)


High field ground state of graphene at dirac point

Temperature dependences of decadesR0 at fields H < Hc

Checkelsky, Li, NPO, PRB ‘09

J18 Hc~ 32 T

Thermally activated

R0 = A exp(bD)

D = 14 K

At “low” H, R0

Saturates below 2 K


High field ground state of graphene at dirac point

Inference from ultralow dissip. measurements decades

  • At Dirac Point

  • Field drives a transition to insulating state at Hc

  • Divergence in R0 is exponential and singular.

  • CPE modes are not present (or destroyed in high fields)

  • Saturation of R0below 2 K for H < Hc

  • (Not caused by self-heating, we believe).


High field ground state of graphene at dirac point

Phase diagram in decadesT-H plane at Dirac point

H

activated

Hc

saturates

KT like

T

Constant-R0 contours

Ordered insulator


High field ground state of graphene at dirac point

Thermopower in zero field decades

E(k)

m

ky

kx

S is T-linear and large


High field ground state of graphene at dirac point

Thermoelectric response functions decades

E

y

x

Current response

Measured E field

Total charge current is zero (open boundaries)

(thermopower)

(Nernst effect)


High field ground state of graphene at dirac point

Thermopower at 5, 9 and 14 T decades

  • Features of thermopower

  • Charge antisymmetric

  • Peak at n=1 nearly indpt of H

  • Very weak T dependence below 50 K


High field ground state of graphene at dirac point

Nernst Signal (off-diagonal decadesSyx)

  • Nernst signal in graphene

  • Charge symmetric

  • Nernst signal is anomalously large

  • at Dirac Point

  • 3. Peaks at n=0, but dispersive for

  • n=1,2,3…

  • At n=0, sign is positive

  • (similar to vortex-Nernst signal in

  • superconductors)


High field ground state of graphene at dirac point

Quantization of thermoelectric current decades

S

1

2

3

m

2

3

1

axy

m

Girvin Jonson 1982

1) Current response to DT is off-diagonal

Dirac

axy is quantized indpt of n

= 93 nA/K


High field ground state of graphene at dirac point

Convert to thermoelectric conductivities decadesaxx and axy

93 nA/K

  • Peaks well defined

  • axx is antisym. In e

  • axy is symmetric

93 nA/K

axy peak very narrow at n=0

Within 20% of quantized value

(need to estimate gradient

more accurately)


High field ground state of graphene at dirac point

  • 1. Field induces transition to insulating state decades

  • at Hc that correlates with V0.

  • For H>Hc, R0 is thermally activated (D ~ 14 K),

  • H<Hc, R0saturates below 2 K.

  • 3. H<Hc, divergence fits KT form exp[2b/(h-1)1/2].

  • 4. Large Nernst signal at Dirac point.

  • 5. axy consistent with quantization

  • ~ 90 nA/K indpt of T, H and n.


High field ground state of graphene at dirac point

END decades