Effects of Interaction and Disorder in Quantum Hall region of Dirac Fermions in 2D Graphene

Donna Sheng (CSUN)

In collaboration with:

Hao Wang (CSUN), L. Sheng (UH), Z.Y. Weng (Tsinghua) F. D. M. Haldane (Princeton) and L. Balents (UCSB)

Supported by DOE and NSF

Introduction: Experiment observations of “half-integer” and odd integer QHE in graphene

an atomic layer of carbon atoms forming honeycomb lattice

Carbon nanotubes are a beautiful material. Problem is how to make them in wafers and with reproducible properties. You can't distinguish between semiconducting and metallic nanotubes. You've got uncontrollable material. Graphene is essentially the same material, but unrolled carbon nanotubes.

K. S. Novoselov et al., Nature (2005)Y. Zheng et al., Nature (2005)

One of applications is to make graphene transistors of 10nm size.

Experiment observations of “half-integer” integer QHE in graphene

K. S. Novoselov et al., Nature (2005)

Y. Zheng et al., Nature (2005) Columbia Univ.

Theoretical work using continuous model for Dirac fermions can account/predicted such quantizations: Gusynin et al. Peres et al., Zheng and Ando

Relation with band structure of honeycomb lattice model?

iii

iA

jAiBCCwcheCCH ij .t-

Three regions of IQHE in the energy band IQHE for Dirac fermions in the middle

D.N.Sheng, Phys. Rev. B73, 233406 (2006)

)(0

0/2

a

ailVH

BF

Effect of disorder and phase diagram: PRB 73 (2006)

Phase diagram

Disorder splits one extended levelat the center of n=0 LL to 2 criticalpoints, leaving n=0 region an insulating phase

Experiment discovers IQHE and “=0” insulating phase

Y. Zhang et. al., PRL 2006

Interaction has to be taken into account to explain the =1IQHE---Pseudospin Ferromagnet

delocalization of Dirac fermions at B=0 (a different issue)

Transfer matrix calculation of the “finite size localization length” for quasi-1D system with width M, it indicates “delocalization” at Dirac point E_f=0 at E<1.0t

E_f=0

M

Experiment discovers IQHE and “=0” insulating phase

Y. Zhang et. al., PRL 2006Interaction has to be taken into account to explain the =1 IQHE---Pseudospin Ferromagnet

More experiments by Z. Jiang et. al. on activation gap of IQHE

For , E is proportional to e^2/ l

Interaction and pseudospin FM state:

Theoretical works Nomura & MacDonald Stoner criteria for pseudospin FM

Alicea & Fisher lattice effect is relevant

Yang et al., Gusynin et al. Toke & Jain, Goerbig et al. continuum model, SU(2)*SU(2) symmetry

Haldane’s Pseudo-Potential gives rise to incompressible state, SU(4) invariant

Our motivation: detailed nature of quantum phases, quantitative behavior of systems

Competitions between: Coulomb interaction, lattice, and disorderscattering effect based on exact calculations

Exact diagonalization using lattice model

Only keep states inside the top-Landau level, large lattice size and keep a degeneracy of Ns around 20

iii

iA

jAiBCCwcheCCH ij .t-

Energy spectrum for pure system: PFM ordering

A-sub, B-sub,

Ising PFM

No pseudospin conservationfor higher LL!!

= 3 XY plane PFM

(zero-energy states)

CDW

the excitation energy gap (with double occupation)

the excitation gap scales with 1/Ne, possibly extrapolates to zero

at large Ne limit

W

|S_z|

Directly look at the transport property instead of “gaps”

The destruction of odd IQHE is due to the mixing of various Chern numbers

boundary phase

Just get at all nodes of mesh of 100-1000 points,

overlap of at nearest points x, y)

Chern number “IS” Hall conductance

2

0

D. J. Thouless et al 1982, J. E. Avron et al. 1883

D.N. Sheng et al., PRL 2003;Sheng, Balents, Wang Xin et al. PRB (FQHE)

)( mC

Fm

C

C is for many-body

The importance of mobility gap (activation gap of experiment): from direct Hall conductance and Chern number calculations

States inside mobility gap

the size of theMobility gap

finite size scaling confirms a finite transport gap at large size limit (more data are coming)

e^2l

Fluctuation of Chern numbers determine a mobility edge

typing

Example of comparing mobility gap with experiments for 1/3 FQHE: D. N. Sheng et al (2003) PRL, Xin et al. (2005) PRB

Phase diagram for Odd IQHE states

0 2 4 6 8 10 12-1.55

-1.50

-1.45

-1.40

-1.35

-1.30

-1.25

-1.20

E(e

2 /l)

Momentum J

n=3(LL=2&3) Graphene, 12/24 systerm, a/b=0.74,q*=(0,±0.595)

-10-5

05

10

0.0

1.5

3.0

4.5

-10

-50

510

Symmetry broken states (stripes and bubbles) in n=3 and n=4 Dirac LLs

0.050.060.070.080.090.100.11

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S0(

q*)

K y (

2/b

)

Kx (2/a)

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S0(

q*)

K y ( 2/b

)K

x ( 2/a)

0.020.040.060.080.100.120.14

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0.020.040.060.080.100.120.14

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S0(

q*)

K y ( 2/b

)Kx ( 2/a)

ust=0.2 ust=2.4 ust=3.2

Disorder-Caused Phase Transition

0 1 2 3 4

1

2

3

4S

0(q* )

ust ( e2/l)

8/24 system, LL2+3, a/b=0.65, bubble phase

Structure factor:S0(q) Correlation function:G(r)

8/24 system, LL2+3, a/b=0.86, stripe phase

Structure factor:S0(q) Correlation function:G(r)

Summary:

The IQHE state is an Ising and valley polarized PFMThe IQHE state is a valley mixing, xy plane polarized PFM

Critical Wc, mobility gap, and quantum phase diagram

Symmetry broken states (stripes and bubbles) are predicted to be The ground state in higher (n=3 and n=4) Dirac LLs

Open questions: Is there a Non-Abelian FQHE at n=1 and n=2 Dirac LLs?