Pseudospin Magnetism in Graphene

Transcription

1 Title Phys. Rev. B 77, (R) (008) Pseudospin Magnetism in Graphene Hongi Min 1, Giovanni Borghi, Marco Polini, A.H. MacDonald 1 1 Department of Physics, The University of Texas at Austin, Austin Texas 7871 NEST-CNR-INFM and Scuola Normale Superiore, I-5616 Pisa, Italy 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 broen symmetry state has a momentum-space vortex, which is responsible for unusual competition between interaction and inetic energies leading to symmetry breaing in the vortex core. We discuss the possibility of realizing a pseudospin version of ferromagnetic metal spintronics in graphene bilayers based on hysteresis associated with this broen symmetry.

2 1. Introduction (1) 1) Graphene Graphene is a two-dimensional honeycomb lattice of carbon atoms. a.46a K M K σ, σ* Energy bands at low energies are described by a D Diraclie equation with linear dispersion near K/K'. Energy (ev) π, π* Min et al., PRB 74, (006) K M K

3 1. Introduction () ) 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. a.46a top Min et al., PRB 75, (007) d 3.35 A U=U top -U bottom bottom

4 . The Model (1) 1) Band Hamiltonian Low energy band Hamiltonian matrix mono H B v F 0 0 H bi B 0 ( ) Chiral two-dimensional electron system 1 m vf px ip m v ( 0 ) in-plane velocity 1 F y H B J 0 ( c ) cos( J ) x sin( J ) c y J τ c ) chirality Pauli matrices cutoff tan 1 ( y / x J=1 : monolayer J= : bilayer 0 ( ) v c F 0( c) c m c

5 . The Model () ) External gate potential 3) Interactions,, 4) Coupling constant ',' q=0 Direct J=1 : monolayer J= : bilayer ',' e H e g e ( c εr0 ε r v F c V ) g ε r v c,', z ',' q= -' Exchange ', J τ c r chirality Pauli matrices cutoff dielectric constant

6 3. Mean-field Theory (1) 1) Hartree-Foc mean-field theory H MF B 0 ( ) B( ) τ τ Pauli matrices representing pseudospin degrees of freedom B() : effective magnetic field ) Self-consistent solution In-plane effective magnetic field is parallel to the band Hamiltonian effective field. B( ) ( B ( )cos( J ), B ( )sin( J ), B ( )) Out-of-plane pseudospin orientation corresponds to the charge transfer between layers. z

7 3. Mean-field Theory () 3) In-plane pseudospin orientation Chiral, J= Non-chiral, J= y c y c x c x c Due to the frustration of in-plane exchange by the chiral character, pseudospin spontaneously rotates to out-of-plane in the chiral model.

8 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 ( up, dn) Out-of-plane pseudospin orientation Ferro Ferri Antiferro Pseudospin polarization Ferro Ferri Antiferro c V g ( 0 c )

9 3. Phase Diagram (1) 1) Stability of the normal state Stability test for a small ( ) 0 at n z 0 V g n n z ( ) ( ) B B z ( ) ( ) n orientation of B n z ( ) 1 d 0 M (, ) n z ( ) in units of c If the largest eigenvalue of the linear integral operator M is larger than 1, the normal state becomes unstable. for constructed from the normal state solution

10 3. Phase Diagram () ) Phase diagram from the stability test J : chirality : coupling constant f : doping N : normal state F : ferro state AF: antiferro state Magnetic order is stable for larger coupling constant, for larger J, for smaller doping.

11 4. Discussion (1) 1) Chirality sum rule Arbitrarily staced graphene multilayers are described by a set of chiral systems. H eff N H J 1 H J H J N D Min et al. PRB 77, (008) C A B B A A J=3 J=+1 N D i 1 J i ) Pseudospin magnets N ABC staced N-layer graphene is described by N-chiral system. Candidate for pseudospin magnets N D Number of doublets Normal Ferro

12 4. Discussion () 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 wor Influence of electronic correlation