Title

Hongki Min1, Giovanni Borghi2, Marco Polini2, A.H. MacDonald1

1Department of Physics, The University of Texas at Austin, Austin Texas 787122NEST-CNR-INFM and Scuola Normale Superiore, I-56126 Pisa, Italy

Pseudospin Magnetism in Graphene

We predict that neutral graphene bilayers are pseudospin magnets in which the charge density contribution from each valley and spin spontaneously shifts to one of the two layers. The broken symmetry state has a momentum-space vortex, which is responsible for unusual competition between interaction and kinetic energies leading to symmetry breaking in the vortex core. We discuss the possibility of realizing a pseudospin version of ferromagnetic metal spintronics in graphene bilayersbased on hysteresis associated with this broken symmetry.

Phys. Rev. B 77, 041407 (R) (2008)

1. Introduction (1)

1) Graphene

• Graphene is a two-dimensional

honeycomb lattice of carbon

atoms.

• Energy bands at low energies are described by a 2D Dirac-like equation with linear dispersion near K/K'.

K

K M

A46.2a

Ene

rgy

(eV

)

σ , σ*

π, π*

Min et al., PRB 74, 165310 (2006)K M K

1. Introduction (2)

2) Graphene bilayer

• Graphene bilayer is composed of a pair of coupled graphene monolayers.

• A band gap opens if on-site energy difference U between two layers is non-zero.

• U can be controlled by doping or an external electric field.

A35.3d

bottom

topA46.2a

U=Utop-Ubottom

Min et al., PRB 75, 155115 (2007)

2. The Model (1)

1) Band Hamiltonian

• Low energy band Hamiltonian matrix

• Chiral two-dimensional electron system

0

0

FmonoB vH

yx

J

ccB JJ

k

kkH )sin()cos()(0 kk

0)(

)(0

2

12

2

mH bi

B

in-plane velocity yx ipp m 2

1 2 Fv

Fv

J=1 : monolayer

J=2 : bilayer

cFc kvk )(0

mkk cc 2)( 220

chirality

Pauli matrices

cutoff

τ ck

J

k )/(tan 1xy kk

2. The Model (2)

2) External gate potential

3) Interactions

4) Coupling constant

zg

g

VH

2

J=1 : monolayer

J=2 : bilayer

chirality

Pauli matrices

cutoff

dielectric constant

τ ck

J Fr vεe 2

cr vεe 22

)(02

crc kεke

r

q=0

k,

k, k','

k','Direct

k, k',

k,'

q= k-k'

k','

Exchange

3. Mean-field Theory (1)

1) Hartree-Fock mean-field theory

2) Self-consistent solution

• In-plane effective magnetic field is parallel to the band Hamiltonian effective field.

• Out-of-plane pseudospin orientation correspondsto the charge transfer between layers.

τkBk )()(0BHMF

))(),sin()(),cos()(()( kBJkBJkB zkkkB

: effective magnetic field)(kB

τ Pauli matrices representing pseudospin degrees of freedom

3. Mean-field Theory (2)

3) In-plane pseudospin orientation

• Due to the frustration of in-plane exchange by the chiral character, pseudospin spontaneously rotates to out-of-plane in the chiral model.

Non-chiral, J=2

cx kk

cy kk

Chiral, J=2

cx kk

cy kk

3. Mean-field Theory (3)

4) Out-of-plane pseudospin orientation

• Spin (↑/↓) and valley (K/K') degeneracy

• Ferro (4 up), ferri (3 up, 1 down), antiferro (2 up, 2 dn)

Out

-of-

plan

e ps

eudo

spin

orie

ntat

ion

ckk

FerroFerriAntiferro

Pse

udos

pin

pola

riza

tion

)(0 cg kV

FerroFerriAntiferro

3. Phase Diagram (1)

1) Stability of the normal state

• Stability test for a small at

• If the largest eigenvalue of the linear integral operatorM is larger than 1, the normal state becomes unstable.

in units of forck k

0)( knz 0gV

orientation of Bn

)(),()(1

0

knkkMkdkn zz

)(

)(

)(

)(

kB

kB

kn

kn zz

constructed from

the normal state solution

3. Phase Diagram (2)

2) Phase diagram from the stability test

• Magnetic order is stable for larger coupling constant, for larger J, for smaller doping.

: chirality

: coupling constant

: doping

N : normal state

F : ferro state

AF: antiferro state

f

J

4. Discussion (1)

1) Chirality sum rule

• Arbitrarily stacked graphenemultilayers are described by a set of chiral systems.

2) Pseudospin magnets

• ABC stacked N-layer grapheneis described by N-chiral system.Candidate for pseudospin magnets

DNJJJeffN HHHH

21

NJDN

ii

1

B

A

A

B

C

A

J=3 J=2+1

Ferro

Normal

Number of doubletsDN

Min et al. PRB 77, 155416 (2008)

4. Discussion (2)

3) Pseudospintronics

• Similar to behavior for an easy-axis ferromagnet in an external magnetic field along the hard axis.

• The pseudospin ferromagnet can be switched between metastable states with gate voltages much smaller than thermal energies, similar to standard CMOS but uses much less power .Can exhibit a pseudospin version of giant magnetoresistance and spin-transfer torque.

4) Future work

• Influence of electronic correlation