Table Of ContentGraphite from the viewpoint of Landau level spectroscopy:
An effective graphene bilayer and monolayer
M. Orlita,1,2,3,∗ C. Faugeras,1 J. M. Schneider,1 G. Martinez,1 D. K. Maude,1 and M. Potemski1
1Grenoble High Magnetic Field Laboratory, CNRS, BP 166, F-38042 Grenoble Cedex 09, France
2Institute of Physics, Charles University, Ke Karlovu 5, CZ-121 16 Praha 2, Czech Republic
3Institute of Physics, v.v.i., ASCR, Cukrovarnick´a 10, CZ-162 53 Praha 6, Czech Republic
(Dated: January 27, 2009)
9 Wedescribeaninfraredtransmissionstudyofathinlayerofbulkgraphiteinmagneticfieldsupto
0 B=34T.Twoseriesofabsorptionlineswhoseenergyscalesas√BandBarepresentinthespectra
0 andidentifiedascontributionsofmasslessholesattheH pointandmassiveelectronsinthevicinity
2
oftheK point,respectively. WefindthattheopticalresponseoftheK pointelectronscorresponds,
n over a wide range of energy and magnetic field, to a graphene bilayer with an effective inter-layer
a coupling2γ1, twice thevaluefor a real graphenebilayer, which reflects thecrystalordering of bulk
J graphitealongthecaxis. TheK pointelectronsthusbehaveasmassiveDiracfermionswithamass
7 enhanced twice in comparison toa true graphene bilayer.
2
PACSnumbers: 71.70.Di,76.40.+b,78.30.-j,81.05.Uw
]
l
l
a Recent interest in graphene [1, 2], a truly two- graphene monolayer and bilayer. This theoretical pic-
h
dimensional system with its simple, but nevertheless, tureisderivedusingadrasticallysimplifiedSWMmodel,
-
s for solids, unconventional electronic states, has focused whichincludesonlytwoparametersγ0 andγ1,describing
e
attention on the properties of Dirac-like fermions in the intra- and inter-layer tunneling respectively. In this
m
condensed matter physics in general. Whereas, two- simplified picture, the dominant contribution to the op-
.
t dimensional massless Dirac fermions [3, 4, 5], character- ticalresponseisprovidedbytheH point,whereelectron
a
isticofgraphenehavebeenwidelyinvestigated,farfewer states closely resemble graphene but with an additional
m
experimentshavebeendevotedtomassiveDiracfermions double degeneracy, and by the K point, where the en-
-
d whicharespecificto agraphenebilayer[6,7], whichrep- ergyspectrumresemblesagraphenebilayer,butwithan
n resents a further example of a two-dimensional system effective coupling of 2γ1, twice enhanced compared to a
o with a highly unusual band structure [8]. Perhaps sur- real bilayer system.
c
[ prisingly, Dirac dispersion relations can also be found in Remarkably, using this simple graphene monolayer
graphite, a three dimensional, bulk material which con- plus bilayer view of graphite, we are able to correctly
1 sists of Bernal-stackedweakly coupled graphene layers. reproduce the magnetic field evolution of all observed
v
The standard Slonczewski-Weiss-McClure (SWM) inter-LL transitions using only the SWM parameters γ
5 0
1 model of electronic states in graphite [9, 10] predicts a and γ1, with values which perfectly match those derived
2 complex form for the in-plane dispersion relation which from studies of realgraphene monolayerand bilayer sys-
4 changes considerably depending upon the value of the tems. Interestingly, the electronic states at K point of
.
1 momentum kz in the direction perpendicular to the lay- graphite are found to mimic those of the graphene bi-
0 ers. Intriguingly, the SWM model predicts that in layer, but with a doubled value of the effective mass, so
9
the vicinity of the H point (k = 0.5) the in-plane thatthey mightbe usefulto further explorethe interest-
z
0
dispersion is linear and thus resembles a Dirac cone. ing physics of massive Dirac fermions.
:
v Such a dispersion has indeed been found in angle re- Thin samples for the transmissionmeasurements were
i
X solved photoemission spectroscopy [11, 12], tunneling prepared by exfoliation of a natural graphite crystal as
r spectroscopy [13, 14], as well as in Landau level (LL)- described in Ref. [15]. Some data is also presented for
a spectroscopy [15]. The latter experiments, mainly fo- highly oriented pyrolytic graphite, which shows prac-
cused on transitions between LLs whose energy scales as tically identical, although slightly less pronounced fea-
√B, are generally believed to exhibit far richer spectra tures, in the magneto-transmission spectra [22]. All
incomparisontotruegraphene[16,17,18],reflectingthe experiments were carried out on macroscopic, roughly
inherent complexity of the SWM model which includes circular-shapedsamples, of severalmillimeters in diame-
no fewer than seven parameters [19, 20]. ter. MeasurementswereperformedintheFaradayconfig-
We show in this Letter that infrared magneto- urationwiththemagneticfieldappliedalongthec-axisof
absorption spectra of graphite, measured over a wide the graphite. All spectra were taken with non-polarized
rangeoftheenergyandmagneticfield,canbeinterpreted light. Tomeasurethemagneto-transmittanceofthesam-
in a very simple, transparent and elegant manner. Our ple in the spectral range of 10-700meV, the radiationof
results confirm, in agreement with theoretical consider- globar was delivered via light-pipe optics to the sample.
ations [21], that graphite can be viewed as an effective Theradiation,detectedbyaSibolometer,placeddirectly
2
totheFermivelocity,c˜=√3a γ /(2~),wheretheatomic
0 0
B = 34 T
distanceisa =0.246nm[23],andinagraphenebilayer,
L −> L 0
-2(-3) 3(2) theinter-layercouplingγ givesanestimateforthemass
32 T L −> L 1
-3(-4) 4(3) of the charge carriers, m=γ /(2c˜2).
L −> L 1
-4(-5) 5(4) Within oursimplifiedapproach,we calculatethe band
30 T L −> L
-5(-6) 6(5) structurealongtheH K H lineoftheBrillouinzone,
− −
i.e. for 0.5 < k < 0.5, which is essential for electri-
z
28 T −
Natural cal and optical properties of bulk graphite. For the in-
graphite plane dispersion of charge carries, we find [9, 10, 20, 24]
on 26 T that it has, for a given momentum kz, the form of a
si T = 2 K graphene bilayer with an effective coupling λγ , where
s 1
mi 24 T λ=2cos(πk ) [20, 24].
z
s
n Inamagneticfield,weobtaintheLLspectrumforeach
a
e tr 22 T effective bilayer, i.e. for each momentum kz:
v
elati 20 T ε±n,µ =±√12h(λγ1)2+(2n+1)E12+
R B
1/2
18 T
µ (λγ )4+2(2n+1)E2(λγ )2+E4 , (1)
q 1 1 1 1(cid:21)
16 T
where sign labels the electron(+) and hole(-) levels.
±
14 T LLsrelatedtothetouchingelectronicbandsareobtained
for µ = 1 and those related to bands split-off in en-
−
12 T ergy by an amount λγ are represented by µ = 1.
1
α C βγ D δ E ε The touching bands c±an be in the parabolic approxima-
tion characterized by the mass m = λγ /(2c˜2). The in-
1
40 60 80 100 120 140 160 plane coupling γ enters Eq. (1) via the Fermi velocity c˜
0
Energy/B1/2(meV.T-1/2) and directly influences the characteristic spacing of lev-
els E = c˜√2e~B. Note, that our approach is a special
1
FIG.1: (coloronline)Transmissionspectraofathingraphite case of the model used by Koshino and Ando [21], who
layer as a function of the magnetic field in the interval B = in an analogous way calculated the spectrum of multi-
12 34T.TheplottedenergyisscaledasE/√Btoemphasize layer Bernal-stackedgraphene with an arbitrarynumber
−
theDiracfermion-likefeaturesinspectra(indicatedbydashed
of layers in an external magnetic field.
vertical lines). Arrows denote transitions arising at the K
Thejointdensityofstates(initialandfinalstates),es-
pointwhichevolve(nearlylinearly)withB. DataaboveB =
sentialinourmagneto-opticalexperiments,hasinthefull
23 T were taken on second sample with a higher density of
graphite flakes. aswellasinourreducedSWMmodelsingularitiesattwo
distinct points of the Brillouin zone, at the K (k = 0)
z
andH (k =0.5)points,whereelectronsandholesarelo-
z
behind the sample and cooled down to a temperature cated,respectively. Hence,the magneto-opticalresponse
of 2 K, was analyzed by a Fourier transform spectrome- of bulk graphite should be governed by transitions be-
ter[15,16]. Thetransmissionspectrawerenormalizedby tween LLs defined by Eq. (1) for λ 2 (K point) and
→
the transmission of the tape and by the zero-field trans- λ 0 (H point). Notably, there is no singularity for
→
mission, thus correcting for any magnetic field induced λ = 1, which corresponds to a real graphene bilayer.
variationsin the responseof the bolometer. The missing Consequently, bulk graphite should, in magneto-optical
partsofthetransmissionspectra,indicatedbygreyareas experiments, behave as a combination of a graphene bi-
inFigs.1and2,correspondtothe spectralrangeswhere layer with the effective coupling 2γ and of a graphene
1
± ±
the tape is completely opaque. monolayer,butwithatwofolddegeneracyε =ε ,
n,−1 n+1,1
Prior to presenting our experimental results, we out- in addition to the usual twofold spin and valley degen-
line a simple model of bulk graphite based on SWM eracies. The expected magneto-optical response of bulk
model [9, 10]. Whereas the standard SWM model has graphite should therefore contain hole-related features
seven tight-binding parameters γ ,...,γ , we limit our- whoseenergyevolveslinearlywith√B originatinginthe
0 5
selveshere to only the most importanthopping integrals vicinity of the H point together with absorption lines
γ and γ . In other words,we consider only the parame- whose energy evolves roughly linear with B correspond-
0 1
ters which are relevant for the nature of the band struc- ing to electrons at the K point.
ture in a graphene monolayer and bilayer. In graphene, The magneto-transmissionspectra taken for magnetic
the intra-layer coupling parameter γ is directly related fields B = 12 34 T on a thin layer of bulk graphite
0
−
3
0.6 tHra pnosiintito (nksz=0.5) ζG FεEδ D 4365 Ktra pnosiintito (nksz=0)
βγ 2 γ = 0.375 eV
C 1 1
n = 5 FIG. 2: (color online) (a): Positions of the
m = 0
0.5 absorptionlinesrelatedtotheH pointasa
--12 n = 4 functionof√B. Thesolidanddashedlines
--43 represent expected positions of absorption
V)0.4 --56 n = 3 lines for c˜=1.02×106 m/s (γ0 =3.2 eV).
e (b): K point related absorption lines as
(
y a function of B. The solid lines show
g α
er0.3 L-n(-n-1)−>Ln+1(n) expected dipole allowed transitions in a
En n = 2 graphene bilayer with an effective coupling
B 2γ1calculatedusingEq.(1)forγ0 =3.2eV
0.2 n = 1 and γ1 = 0.375 eV. Grey data points were
takenonhighlyorientedpyrolyticgraphite,
whichexhibitabehaviornearlyidenticalto
natural graphite [22]. The inset schemati-
0.1 c*=1.02×106m.s-1
cally shows theobserved inter-bandtransi-
(γ = 3.2 eV) L−>L tions in the effectivebilayer.
0 0 1
0.0
0 1 2 3 4 5 6 0 5 10 15 20 25 30 35
(a) B1/2(T1/2) (b) B(T)
are presented in Fig. 1. The transmission is plotted as a tained for a real graphene bilayer [28, 29] as well as on
functionofenergydividedby√B tofacilitatetheidenti- bulk graphite[30, 31]. Our data follow well the theoreti-
fication of spectral features originating from around the cal predictions up to the highest energies, in contrast to
H and K points. Plotted in this way, the transitions thethedeviationreportedforarealgraphenebilayer[7].
denoted by Roman and Greek letters, do not shift for Hence, the electrons in the vicinity of the K point can
spectra recorded at different magnetic fields. Thus they be describedwithareasonableaccuracyusingthe model
scalelinearlywith√B,seeFig.2a,andarerelatedtothe of a graphene bilayer, but with coupling strength twice
H point. The second set of lines marked by vertical ar- enhancedas comparedto a true bilayer. The strengthof
rowsshiftwithmagneticfieldandactuallyfollowanearly the coupling 2γ directly reflects the long-range Bernal
1
linear dependence with B, as can be seen in Fig. 2b. stackingofgraphitealongthecaxis. Thetwiceenhanced
The transitions following a √B dependence, corre- coupling in the effective bilayer can be understood using
sponding to massless holes around the H point, have the example of semiconductor superlattices, which are
been thoroughly analyzed in our previous work [15, 22]. three-dimensional but nevertheless strongly anisotropic
Whereas, the absorption lines denoted by Roman let- systems resembling in some aspects the band structure
ters have their direct counterpart in spectra of true of bulk graphite. The energy difference, ∆SAS, between
graphene [16, 17, 18, 25, 26, 27], the Greek lines are in the bonding and anti-bonding state in a symmetric dou-
principle dipole forbidden in a pure 2D system of Dirac blequantumwellissimply halfoftheminibandwidthof
fermions. Nevertheless, these transitions can be consis- the superlattice created from the same wells [32].
tently explained with the same selection rule ∆n = 1, Our results also show that the parabolic approxima-
when the twofold degeneracy of LLs, ε± = ε± ±, at tion which is widely used for the touching electronic
n,−1 n+1,1
the H pointofbulk graphiteisproperlyconsidered. The bandsin abilayerandwhichdirectly leadsto LLs whose
±
Fermi velocity is extracted to be c˜ = (1.02 ± 0.02) × energy evolves linearly with magnetic field: εn,− ≈
106 m/s, giving a rather precise measure of the in-plane ~ω n(n+1) [8, 13] is a good approximation only
c
±
hopping integral γ0 =(3.20 0.06) eV in bulk graphite, in thepvicinity of the charge neutrality point. As can
±
which is the only parameter needed to describe all ab- be seen in Fig. 2(b), the small deviation from a linear
sorption lines originating at the H point. dependence, predicted at higher energies by the simpli-
Here we focus on transitions denoted by arrows in fied SWM model for an effective bilayer, is reproduced
Fig.1,whoseenergyevolvesnearlylinearlywithB. Tak- in our data. The bilayer character of K point elec-
ing into account the selection rule ∆n = 1 and using trons also explains the non-linear evolution with B of
±
thein-planecouplingestimatedabovetoγ =3.2eV,we the magneto-reflection spectra published recently [19].
0
caninterpretalltheabsorptionlinesinFig.2basdipole- Nevertheless,withintheparabolicapproximation,weob-
allowedtransitionsinagraphenebilayerwithaneffective tainaneffective mass ofK point electronsin graphiteof
coupling 2γ with γ = (375 10) meV. The deduced m= γ /c˜2 0.063m in good agreement with other cy-
1 1 1 0
± ≈
value for γ is in a very good agreement with results ob- clotronresonanceexperiments[33]. Thismassisafactor
1
4
of two higher when compared to a true bilayer [7]. PartofthisworkhasbeensupportedbyEuroMagNET
While, only two tight-binding parameters γ and γ II under the EU contract, by the French-Czech Project
0 1
are required to obtain a reasonable description of the BarrandeNo.19535NF,bycontractANR-06-NANO-019
magneto-opticalresponseofthe K pointelectronsovera and by projects MSM0021620834and KAN40010065.
wide range of energy and magnetic field, the influence of
the remaining hopping integrals γ ,...,γ , merits some
2 5
consideration. Ingeneral,additionalcouplingparameters
should lift the electron-holesymmetry of the bilayer and
∗
Electronic address: [email protected]ff.cuni.cz
thetrigonalwarping(γ )shouldleadtoamixingofLLs,
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