Pseudospin Magnetism in Graphene

0.21 MB·

by  Hongki Min

Pseudospin magnetism in graphene
Hongki Min,1,∗ Giovanni Borghi,2 Marco Polini,2 and A.H. MacDonald1
1Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA
2NEST-CNR-INFM and Scuola Normale Superiore, I-56126 Pisa, Italy
(Dated: February 1, 2008)
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 band
structure of this system is characterized by a momentum-space vortex which is responsible for
unusualcompetition betweenbandandkineticenergiesleadingtosymmetrybreakinginthevortex
8 core. Wediscussthepossibility ofrealizingapseudospinversionofferromagnetic metalspintronics
0
in graphene bilayers based on hysteresis associated with this broken symmetry.
0
2

n Introduction—The ground state of an interacting elec- in real space transfers charge uniformly between layers.
a tronsystemflowsfromsubtlecompromisesbetweenband Our paper starts by describing a technical calculation
J
and interaction energy minimization. Because of the that supports and elaborateson our prediction and then

8
Pauli blocking effects which underlie Fermi liquid the- discusses anticipated properties of this state.
1
ory however, the consequences of interactions are nor- Chiral two-dimensional electron system Hartree-Fock
] mally only quantitative [1] unless symmetries are bro- Theory—Itisinstructivetoconsideraclassofchiraltwo-
el ken. Recent progress [2] in the isolation of single and
- doublesheetsofcarbonatoms(graphenesheets)haspre-
r
t sented researchers with a new type of interacting elec-
s
. tron system whose properties are now being actively ex-
t
a plored [3, 4, 5], both theoretically and experimentally.
m InthisLetterwearguethatbandenergyminimizationis
- exceptionally frustrating to interactions in graphene bi-
d
layers,andpredictthatbrokensymmetrystatesinwhich
n
charge shifts spontaneously from one layer to the other
o
c occur as a consequence.
[ Graphene bilayers with Bernal stacking have one

low-energy site per unit cell in each layer. When
3
v the layer degree of freedom is described as a pseu-
0 dospin, the continuum limit of the π-orbital band
3 Hamiltonian corresponds [6, 7] to a pseudospin field
15 Bband = [~2k2/(2m⋆)](cos(2φk),sin(2φk),0), where
7. φtukn=nealirncgtaanm(kpyli/tkuxd)e,,man⋆d=vγ1/is(2tvhF2e),eγle1ctisrotnheveinlotceirtlyayaetr FIG. 1: Pseudospin orientation in a graphene bilayer broken
0 F symmetry state. In this figure the arrows represent both the
the Fermi energy in an isolated neutral graphene sheet.
7 magnitudeandthedirectionofthexˆ-yˆprojectionofthepseu-
0 When interactions are neglected the ground state of a dospinorientationnˆasobtainedfromamean-field-theorycal-
: neutral bilayer has a full valence band of pseudospinors culationforaneutral,unbiasedbilayerwithcouplingconstant
v aligned at eachk with this pseudospin field, forming the α=1. The arrows are shorter in the core of the momentum
i
X momentum-space vortex. The vortex exacts a large in- spacevortexbecausethepseudospinsinthecorehaverotated
spontaneously toward thezˆor −zˆdirection.
r teractionenergypenaltybecauseofitsrapidpseudospin-
a
orientationvariation. Weproposethat,likeitsreal-space
counterpart [8], the momentum-space vortex sidesteps
this energy cost by forming a vortex core in which the dimensional electron system (C2DES) models which in-
pseudospin orientation is out of plane in either the zˆ or cludes the continuum limits of single-layer and bilayer
zˆ direction, as illustrated in Fig. 1. The momentum- graphene sheets as special cases. These C2DES models
−
spacevortexstateisnonuniforminmomentumspace,but have band Hamiltonians,
J
k V
Hˆband =−kX,σ′,σ cˆ†k,σ′(ε0(kc)(cid:18)kc(cid:19) hcos(Jφk)τσx′,σ+sin(Jφk)τσy′,σi+ 2g τσz′,σ)cˆk,σ, (1)
2
where σ,σ′ are pseudospin labels, J is the chirality in- sionlesscoupling constantα 1. We use ε (k ) andk−1
∼ 0 c c
dex, τa is a Pauli matrix, k is the model’s ultraviolet as energy and length units in the rest of this paper.
c
momentum cutoff, ε0(kc) is the energy scale of the band The C2DES Hartree-FockHamiltonian canbe written
Hamiltonian,andasumovervalleyandspincomponents (indimensionlessunits)inthefollowingphysicallytrans-
is implicit. In Eq. (1), Vg is an external potential term parent form:
which couples to the pseudospin magnet order parame-
ter and corresponds in the case of bilayer graphene to
an external potential difference between the layers. For HˆHF =− cˆ†k,i,σ′Bσ(i′),σ(k)cˆk,i,σ, (2)
single-layergrapheneJ =1andε (k )=~v k ,whilefor k,Xi,σ′,σ
0 c F c
bilayer graphene J = 2 and ε (k ) = ~2k2/(2m⋆). The
0 c c
dimensionless coupling constantof these C2DESs,which where Bσ(i′),σ =B0(i)(k)δσ′,σ+B(i)(k)·τσ′,σ,
measuresthe interactionstrength,canbe definedasα=
(e2kc/ǫ)/ε0(kc)whereǫistheeffectivedielectricfunction d2k′ 1 f(i) (k′)
duetoscreeningexternaltotheπelectronsystem. Inthe B(i)(k)=α sum , (3)
case of a single graphene layer αmono =e2/(ǫvF~), while 0 Z|k′|<1 2π |k−k′| 2
inthebilayercase,α =2e2/(ǫv ~),wherev =~k /m⋆.
bi c c c
If we choose [9] ~k = √2m⋆γ for the bilayers, we have andthepseudospinfieldB(i)(k)hasbandandinteraction
c 1
α = α . Typically ǫ 2.5 which implies a dimen- contributions,
bi mono
∼
d2k′e−|k−k′|d¯f(i)(k′)
Bx(i)(k)=kJcos(Jφk)+αZ|k′|<1 2π |k−k′| diff2 nx(i)(k′), (4)
d2k′e−|k−k′|d¯f(i)(k′)
By(i)(k)=kJsin(Jφk)+αZ|k′|<1 2π |k−k′| diff2 ny(i)(k′), (5)
V¯ d2k′ 1 f(j)(k′)
B(i)(k)= g +α δ d¯ diff n(j)(k′). (6)
z 2 j Z|k′|<1 2π (cid:18)|k−k′| i,j − (cid:19) 2 z
X
Here i,j label the four valley and spin components of k = 0 tilt away from their band Hamiltonian xˆ-
graphene’s J = 1 and J = 2 C2DESs, n(i)(k) is the di- yˆ plane orientations toward the zˆ direction, i.e.,
±
rection of B(i)(k), fs(ui)m(k′) [fd(iiff)(k′)] is the sum of (dif- n(i)(k) = (n⊥(i)(k)cos(Jφk),n⊥(i)(k)sin(Jφk),n(zi)(k))
ference between) low- and high-energy occupation num- with [n(i)(k)]2 + [n(i)(k)]2 = 1. Pseudospin polariza-
bers, V¯g = Vg/ε0(kc) is the gate potential in units of tion in⊥the zˆ-directzion corresponds to charge transfer
ε0(kc), d¯= kcd is the distance between layers in units between layers. This ansatz yields effective magnetic
of kc−1 in the bilayer case, and d¯= 0 in the monolayer fields whose xˆ-yˆ plane projections are parallel to the
case. The term proportional to d¯on the right-hand side bandHamiltonianeffectivefield. WefindthatB(i)(k)=
otrfaEnqsf.e(r6b)eitswteheenHlaayretrrseeinptohteenbtiialalywerhiccahseo.pposes charge (B⊥(i)(k)cos(Jφk),B⊥(i)(k)sin(Jφk),Bz(i)(k)) with
Local minima of the Hartree-Fock energy func-
1
tional solve Eqs. (4)-(6) self-consistently. Our fo- B(i)(k)=kJ +α dk′F (k,k′)f(i)(k′)n(i)(k′), (7)
⊥ ⊥ diff ⊥
cus here is on the broken symmetry momentum- Z0
space vortex solutions in which pseudospins near
V¯ 1 1
B(i)(k)= g +α dk′ F (k,k′)δ k′d¯ f(j)(k′)n(j)(k′), (8)
z 2 z i,j − 2 diff z
j Z0 (cid:18) (cid:19)
X
where the exchange kernels in Eqs. (7) and (8) are given by
π dφ e−qd¯ with q = q(k,k′,φ) k2+k′2 2kk′cos(φ). The
F⊥(k,k′) = k′ cos(Jφ), ≡ −
2π q
Z0 p
π dφ 1
F (k,k′) = k′ , (9)
z
2π q
Z0
3
pseudospin-chiralityinducedfrustrationisrepresentedby 0.1
thefactorcos(Jφ)inthefirstlineofEq.(9)whichmakes
F much smaller than F .
⊥ z 0.08
Pseudospin magnet phase diagram— We test the sta-
J = 2
bility of the “normal” state [n(i)(k) 0 at V = 0]
z g
solution of the Hartree-Fock equation≡s by linearizing 0.06 J = 2 N/F
the self-consistency condition; n(i) = B(i)n(i)/B(i) f N/AF
Bz(i)/B⊥(i)|nz(i)≡0. This gives a k-spzace intezgral⊥equa⊥tion→ 0.04 J = 1
1
n(i)(k)= dk′ M (k,k′)n(j)(k′), (10) 0.02
z i,j z
j Z0
X
0
where 0 0.5 1 1.5 2 2.5 3
α
α[F (k,k′)δ k′d¯/2]f(j)(k′)
M (k,k′)= z i,j − diff . (11)
i,j 1
kJ +α dk′′F⊥(k,k′′)fd(iiff)(k′′) aFnIGd.J2:=(1C.oFloorrotnhleinJe)=P2habsieladyieargrcaamseowfeCh2aDvEeSta’skewnitdh¯=J 0=.22.
Z0 Pseudospinmagnetism occursatstrongcouplingαandweak
The normal state is stable when the largest eigenvalue dopingf. (1+f =n↑+n↓wherethepseudospindensitynσ =
of the linear integral operator M in the right-hand side Pk,ihcˆ†k,i,σcˆk,i,σi/N and N = Pk,i1.) In the J = 2 bilayer
of Eq. (10) is smaller than 1. Eigenvalues larger than 1 case,theHartreepotentialfavorssmallertotalpolarizationso
are possible only because F is smallerthan F , i.e., be- that the initial normal (N) state instability (blue separatrix)
⊥ z
is to antiferromagnetic (AF) states in which the pseudospin
cause of pseudospin chirality. Phase diagrams for J = 2
polarizations of different valley and spin components cancel.
and J = 1 are plotted in Fig. 2. The pseudospin mag-
Atlargerα,thenormalstateisunstable(greenseparatrix)to
net is more stable for larger coupling constant because
ferromagnetic (F) pseudospinstates. IntheJ =1monolayer
it is driven by interactions, for larger J because the case d¯=0 so thephase boundaries (red separatrix) of Fand
typical value of the band energy term proportional to AFbroken-symmetrystates coincide.
kJ decreases with J, and for smaller doping because
f(i)(k)isthennonzeroinalargerregionofk-space. The
diff
eigenvectors of specify the instability channel. The
M
component-index structure of implies that the eigen- Pseudospintronics— In Fig. 4 we illustrate typical re-
M
valuesoccuringroupsoffour,threeofwhich[labeledan- sults for the pseudospin (layer) polarization ζ = (n↑
−
tiferromagnetic (AF) in Fig. 2] correspond in bilayers to n↓)/(n↑+n↓) of a graphenebilayer as a function of gate
sTthaetesfewrriothmangonenteitcc(hFa)rgiensttraabnislifteyr,iin.e.w,hicjhna(zjl)l(kco)m≡po0-. v[Zo2ltagSeUV¯(g4.)]T/[hSeUA(2F)grSouUn(d2)s]tbartoekaetnVs¯ygm=m0e,trwyhbicehcahuases
× ×
P ofthefreedomtochooseanytwospinorpseudospincom-
nents are polarized in the same sense is opposed by the
ponentsfor(say)positivepolarization,isgraduallypolar-
Hartree potential and delayed to larger coupling con-
izedbythegatevoltage,buteventuallybecomesunstable
stant. For both AF and F instabilities, n (k) is peaked
z
in favor of polarizing more layers in the sense preferred
atsmallkwherethexˆ yˆpseudospin-planeexchangeen-
− by the gate voltage. At sufficiently strong gate voltages,
ergies are most strongly frustrated by chirality, and the
the F ground state in which all layers are polarized in
kineticenergytermwhichopposespseudospinmagnetism
the same sense becomes the ground state. As the gate
is weakest.
voltageisvariedlocalminimaoftheHartree-Fockenergy
The physics that drives pseudospin magnetism in
functional become saddle points which are in the basin
graphenebilayersis illustratedinFig.3whichpartitions
of attraction of another local minima. In this way, the
the condensation energy into band, Hartree, intralayer
self-consistent solutions exhibit hysteretic behavior.
exchange, and interlayer exchange contributions. Spon-
taneous layer polarization lowers the intralayer interac- If only the fully polarized solutions existed these re-
tion energy at a cost in all other components. The over- sults for pseudospin polarization as a function of gate
all energy change is negative, and the broken symmetry voltage would be very much like the behavior expected
state occurs, because the interlayer exchange energy of for an easy-axis ferromagnet in an external magnetic
the normal state is weakened by the band-Hamiltonian field along the hard axis. In magnetic memories bista-
induced frustration explained earlier. The cost in inter- bility enables information storage. In magnetic metal
layerexchangeenergyofpseudospinrotationistherefore spintronics the dependence of the resistance of a circuit
much smaller than the gain in intralayer exchange en- containing magnetic elements on the magnetization ori-
ergy and the overall energy is reduced. The energy gain entation of those elements gives rise to sudden changes
is considerably larger for AF broken symmetry states. in resistance with field (giant magnetoresistance) which
4
0.2 0.4
Ferromagnetic
0.3
0.1
0.2
0
0.1
δε -0.1 ζ 0
Band -0.1
-0.2
Hartree
IntralayerExchange -0.2 Ferromagnetic
-0.3 InterlayerExchange -0.3 Ferrimagnetic
Total Antiferromagnetic
-0.4 -0.4
0 0.5 1 1.5 2 2.5 3 -0.2 -0.1 0 0.1 0.2
α V¯
g
0.2
Antiferromagnetic FIG.4: (Coloronline)Metastableconfigurationsofthepseu-
0.1 dospin ferromagnet as a function of bias voltage Vg [in units
of ε0(kc)] with α = 1 and f = 0. We find self-consistent so-
lutions of the gap equations (7)-(8) in which the pseudospin
0
polarization has the same sense in all four components (fer-
romagnetic),inthreeofthefourcomponents(ferrimagnetic),
ε -0.1
δ or in half of the four components (antiferromagnetic).
Band
-0.2
Hartree
IntralayerExchange
by experiment. Indeed, it is well known that Hartree-
-0.3 InterlayerExchange
Fock theory (HFT) often overestimates the tendency to-
Total
ward broken symmetry states. For example HFT pre-
-0.4
0 0.5 1 1.5 2 2.5 3 dicts that a non-chiral 2DES is a (real-spin) ferromag-
α
netatmoderatecouplingstrengths,whereasexperiments
and accurate quantum Monte Carlo calculations suggest
FIG. 3: (Color online) Condensation energy per electron δε that ferromagnetism occurs only at a quite large value
[in unitsof ε0(kc)]as afunction of α for an undoped(f =0) of the coupling constant [1]. Nilsson et al. [11] have re-
J =2C2DES.ThisfigureshowsresultsforbothF(top)and
cently claimed that a similar ferromagnetic instability
AF(bottom) states.
occurs in weakly-doped graphene bilayers, presumably
only at a much stronger coupling constant than implied
by HFT. We believe that the momentum-space vortex
can be used to sense very small magnetic fields. Cur- instability identified here, which is unique to the pecu-
rents running through such a circuit can also be used to liar band-structure of bilayer graphene, is qualitatively
changethe magneticstatethroughspin-transfertorques. more robust than the real-spin ferromagnetic instability.
Pseudospin ferromagnetism in graphene bilayers could This should be especially true in neutral bilayers since
potentially lead to very appealing electrical analogs of the momentum-space vortex instability occurs at a cou-
both of these effects. Because of the collective behav- pling constant (α 0) for which correlation corrections
→
ior of many electrons, the pseudospin ferromagnet can to HFT are weak. This is not a strong-coupling insta-
be switched between metastable states with gate volt- bility like ferromagnetism, but much more akin to the
agesthataremuchsmallerthanthethermalenergykBT, veryrobustattractive-interactionweak-couplinginstabil-
potentially enabling electronics which is very similar to ity which leads to superconductivity. The condensation
standardcomplementarymetal-oxide-semiconductorbut energy per electron associated with the formation of a
usesmuchlesspower. Thispossibilityisanalogoustothe momentum-space vortex core is e2k /ǫ, much larger
c
∼
property that a magnetic element can be switched be- than the e2k /ǫ condensation energy for the spin-
F
∼
tween magnetic states by Zeeman field changes that are polarized state. Because this broken symmetry state is
extremely small compared to the thermal energy kBT. mostrobustforuniformneutralbilayers,thesmoothbut
Pseudospin-transfer torques, which are expected to oc- strong disorder potentials responsible for inhomogene-
cur inelectronicbilayersystems[10],canalsobe used to ity [12] in nearly neutral graphene sheets may need to
switch the pseudomagnetic state. be limited to allow this physics to emerge.
Discussion— The proposals made here are based on ap- WorkatUTAustinwassupportedbytheWelchFoun-
proximatecalculationsandmustultimatelybeconfirmed dation, NRI-SWAN, ARO, and DOE.
5
M. I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, and
A. K. Geim, Nature Phys.2, 177 (2006).
[7] E.McCannandV.I.Fal’ko,Phys.Rev.Lett.96,086805
∗ Electronic address: [email protected] (2006).
[1] G. F. Giuliani and G. Vignale, Quantum Theory of [8] See for example V. S. Pribiag, I. N. Krivorotov, G. D.
the Electron Liquid (Cambridge University Press, Cam- Fuchs, P. M. Braganca, O. Ozatay, J. C. Sankey, D. C.
bridge, 2005). Ralph,andR.A.Buhrman,NaturePhysics3,498(2007)
[2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, and work cited therein.
Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. [9] H.Min, B.Sahu,S.K.Banerjee, andA.H.MacDonald,
Firsov, Science 306, 666 (2004). Phys. Rev.B 75, 155115 (2007).
[3] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, [10] Saeed H. Abedinpour, Marco Polini, A. H. MacDonald,
M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and B.Tanatar,M.P.Tosi,andG.Vignale,Phys.Rev.Lett.
A.A. Firsov, Nature 438, 197 (2005). 99, 206802 (2007).
[4] Y.Zhang,Y.-W.Tan,H.L.Stormer,andP.Kim,Nature [11] J. Nilsson, A.H. Castro Neto, N.M.R. Peres, and F.
438, 201 (2005). Guinea, Phys.Rev. B 73, 214418 (2006).
[5] ForrecenttechnicalandpopularreviewsseeA.K.Geim [12] J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H.
and K. S. Novoselov, Nature Mat. 6, 183 (2007) and A. Smet, K. von Klitzing, and A. Yacoby, Nature Physics
K. Geim and A. H. MacDonald, Phys. Today 60(8), 35 advance online publication, 25 November 2007 (DOI
(2007). 10.1038/nphys781).
[6] K.S.Novoselov,E.McCann,S.V.Morozov,V.I.Fal’ko,