Table Of ContentDirac and Normal Fermions in Graphite and Graphene:
Implications to the Quantum Hall Effect
Igor A. Luk’yanchuk1,2 and Yakov Kopelevich3
1University of Picardie Jules Verne, Laboratory of Condensed Matter Physics, Amiens, 80039, France
7 2L. D. Landau Institute for Theoretical Physics, Moscow, Russia
0 3Instituto de F´ısica ”Gleb Wataghin”, Universidade Estadual de Campinas,
0 Unicamp 13083-970, Campinas, Sao Paulo, Brazil
2 (Dated: February 4, 2008)
n SpectralanalysisoftheShubnikovdeHaas(SdH)magnetoresistanceoscillationsandoftheQuan-
a tumHallEffect(QHE)measuredinquasi-2Dhighlyorientedpyrolyticgraphite(HOPG)[Phys. Rev.
J
Lett. 90,156402(2003)]revealstwotypesofcarriers: normal(massive)electronswithBerryphase0
2 andDirac-like(massless)holeswithBerryphaseπ. Wedemonstratethatrecentlyreportedinteger-
1 and semi-integer QHE for bi-layer and single-layer graphenes take place simultaneously in HOPG
samples.
]
l
l PACSnumbers: 81.05.Uw,71.20.-b
a
h
-
s Observations of the quantum Hall effect (QHE) and bilayer graphene systems. In particular, we demonstrate
e
magnetic-field-drivenmetal-insulatortransitioninquasi- that normal and Dirac-like fermions seen in the bulk
m
2Dhighlyorientedpyrolyticgraphite(HOPG)[1,2]indi- graphite are responsible respectively for the integer and
.
t catethatmuchinthephysicsofgraphitehadbeenmissed semi-integer precursors of the QHE. To check the inde-
a
in the past and had triggered a considerable interest in pendence ofthese conclusionson the windowingand fre-
m
graphite once again. Analyzing the quantum de Haas quency filtration cut-off we used an alternative, insen-
-
d vanAlphen (dHvA) andShubnikovde Haas(SdH) oscil- sitive to filtration procedure method of two-dimensional
n lationsinbulk HOPGsamplesweexperimentallyproved phase-frequency analysis, described in detail in [3].
o [3] that, besides the conventional electronic charge car- We analyze the field dependencies of the basal-plane
c riers with the massive spectrum E = p2/2m, massless resistance R (B) and the Hall resistance R (B) re-
[ xx xy
(2+1)-dimensional fermions (holes) with Dirac-like lin- ported in Ref. [1] for the sample labeled as HOPG-3
2 ear spectrum E = ±v|p| and non-trivial Berry phase π (noting that the HOPG-UC sample, similar to HOPG-3,
v
do exist in graphite. It is likely that these Dirac-type has been studied in Ref. [3]).
7
carriers are responsible for the strongly correlated phe-
3 As seen from Fig. 1, Rxy(B) demonstrates several
0 nomena predicted by theory [4, 5, 6]. The results of [3] QHE-like plateaus at high fields, also observed in vari-
9 were confirmed by the recent angle resolved photoemis-
ous HOPG samples by other groups [13, 14]. However,
0 sion spectroscopy (ARPES) experiments performed on
unlike the conventional QHE, R (B) does not drop to
6 xx
HOPG[7,8],thatprovideanunambiguousexperimental
0 zero at the positions of these plateaus. Only weak SdH
evidence for coexistence of Dirac-like holes and normal
at/ (massive)electronslocatedatHandKpoins ofthe Bril- oscillations ∆Rxx(B) [3] are seen superimposed on al-
m louin zone, respectively.
- Veryrecently,remarkableprogressinthetechnologyof 200
d
ultrathin films consisting of one [9, 10], two [11] or sev- 40 T = 2 K
n
eral [12] graphite monolayers (graphenes) was achieved.
o
c Analysis of unconventional half-integer QHE and Berry
rXiv: cobplfihulDaasysiioeernarπctg-hloriaafkpteShcmdehHnaaesrosgldseecesicmslalaforetrnriiomsetrnrissoaintinnessg[g9rar,aap1np0heh]ew.ennAeteythlptaehedveeotsftoahtmethheeneatiictnmoutnreee--, Ω − Rxy ( m )20 100Ω Rxx ( m )
a
gerQHEinwhichthelast(zero-level)plateauismissing,
and therefore the charge carriers are described by the
chiral fermions having the Berry phase 2π and the en-
ergy spectrum is represented by two touching parabolas 000 11 22 33 44 55 66 77 880
B ( T )
E(p)=±p2/2m [11].
In the present work we apply our filtering and phase FIG. 1: Field dependence of basal-plane magnetoresistance
analysis procedures [3] to analyze the SdH oscillations Rxx(B)andHallresistanceRxy(B)measuredfortheHOPG-
and the QHE effect in HOPG aiming to clarify how 3samplewithmagneticfieldBappliedalongthesamplec-axis
[1].
these effects correlate with those studied in single and
2
most linear R (B) dependence. This can be explained by stop-band filtering of G (B−1), eliminating the fre-
xx xy
either by imperfection of the QHE itself or by the exis- quencywindowsrelatedtotheelectronspectralpeakand
tence of a background resistance of unknown origin. It
is interesting to note that QHE with similar behavior of
Rxx(B) has been reported for Bechgaard salts [15, 16] 5 8
and for (Bi1−xSbx)2Te3 layered semiconductors [17, 18].
Becausequantumoscillationsareperiodic functions of a
4
the inverse field B−1, the inverse-field spectral analysis d 4
Rigttsharx−noetyh1uc(esepBpsRae−opcx1ftpx)rcr(,aaoBlrp(r−iriinni1eat)rtebseanoinnsttidhvotooytllchvaoteesofdeHsdSinaidpslHcolorslcicymooinnlsliocdanimtuallitcaoiaettnalisbon.bnecatFsewcikgGoeg.fexnr2yto(hdupBeinrffe−dres1ere)hsenina≡tssts- − G / G xy0xy23 e c xb 0.5 0 ∆Ω − R ( m ) xx
been subtracted) together with analyzed in [3] spectral −4
1
intensity of dHvA oscillations of magnetic susceptibility
Normal QHE
χ(B−1), (χ=dM/dB).
0 −8
Fig. 2 showsthattwopeaks inthe susceptibility spec- 0 1 2 3 4 5
Filling Factor ν
trumobservedat4.68Tandat6.41Tarealsoresolvedin
the spectrum of Gxy(B−1). The higher-frequency mag- FIG. 3: Hall Conductance Gxy (solid line a) and ∆σxx ∼
netoresistancepeak manifests itself as the right-shoulder −∆Rxx (solidlineb)forHOPG-3sampleasafunctionofthe
structure of Rxx(B−1) spectrum. From comparative normalized”fillingfactor”ν =B0/B (wereB0 =4.68T isthe
analysis of dHvA and SdH oscillations in [3] we iden- SdH oscillation frequency). Normal electrons give the dom-
inant contribution in the quantum oscillations. The profile
tified these peaks as originating from normal (electrons)
and Dirac (holes) carriers. Here, we re-affirm this con- of Gxy(ν) is fitted by the integer QHE staircase; theminima
of ∆σxx correspond to the QHE plateaus. Hall conductance
clusion in a different way, namely by means of the com- is normalized on G0xy = 28Ω−1 equal to the step between
parative analysis of longitudinal and Hall conductance QHE plateaus. The results are compared with SdH oscil-
oscillations, which is based on the separate study of the lations in thick film of graphite (dashed line c) [13](having
contributions from both types of carriers. approximately the same B0 = 4.65T) and with the QHE in
bilayergraphene[11]obtainedasafunctionofbothmagnetic
Fig. 3 (curves a and b) shows the raw data of
Gxy(B−1)and∆Rxx(B−1)(afterbackgroundsignalsub- tfiieolnd((ddoatstheeddlliinneeed))((νν ==Bn0//nB0,wBit0h=n207=.8T1).7a·n1d01c2ocnmce−n2traat-
traction)inwhichthenormalelectroncontributiondom-
B=20T)
inates in the magnetotransportquantum oscillations [3].
Extraction of the Dirac-like carriers signal from these
data was the most challenging task that we performed 5 8
a c
4
4
y ( arbitrary units ) e1 h1 Rxx χ − G / G xy0xy23 b −04∆Ω − R ( m ) xx
nsit 1 d
nte G Dirac QHE
al I xy
ctr 00 1 2 3 4 5−8
pe Filling Factor ν
S
FIG. 4: Solid lines a and b : the same as in Fig. 3 (with
2 4 6 8 10 12 14 16 18 20 22 B0 =6.41T, G0xy =19Ω−1) but after filtering elimination of
Frequency B ( T )
0 the normal electrons contribution. The Gxy(ν) profile and
positions of the minima in ∆σxx ∼ −∆Rxx are coherent
FIG.2: SpectralintensityofSdHoscillationsofmagnetoresis- with the Dirac QHE staircase. The results are compared
tance|Rxx(B0)|andofHallconductance|Gxy(B0)|inHOPG- with the Dirac QHE in graphene monolayer obtained both
3sampleincomparisonwithdHvAoscillationsofsusceptibil- as a function of magnetic field (ν =B0/B with B0 =6.67T)
ity |χ(B0)| in HOPG-UC sample [3]. Peaks e1 and h1 corre- (dashedlinec)[10]andcarrier concentration(ν =n/n0 with
spond to the normal electrons and Dirac holes, respectively n0 =1.5·1012cm−2 at B =14T)(dotted line d) [9]
3
its harmonics(seeFig. 2)andusingthe pass-bandfilter- whereµ=∓1forelectrons/holes,andg isthe(iso)-spin
s
ingof∆R (B−1)withthecorrespondingholefrequency degeneracy factor.
xx
windowing. The resulting contributions from Dirac-like The phase factor γ is defined for an arbitrary spec-
carriers are presented in Fig. 4 (curves a and b). trumE(p)bythequasiclassicalquantizationconditionof
Note that the sign of ∆Rxx is reversed in order to the Fermisurface cross-sectionS(Ef)=(n+γ)2πeB/~c
presenttheoscillatingpartoftheconductivity∆σxx,be- (n≫1). This factor is uniquely related to the topologi-
cause in our case of ρxx ≫ ρxy and ∆σxx is related to cal Berry phase ΦB =kπ acquired by a fermion, moving
∆Rxx as: around S(Ef) [23]: γ equals either 1/2 for even k (nor-
mal carriers) or 0 for odd k (Dirac-like fermions). For
ρ ∆ρ
xx xx
∆σ =∆ ≃− ∼−∆R . (1) example, the two-parabola spectrum of bilayer graphite
xx ρ2 +ρ2 ρ2 xx
xx xy xx E(p) = ±p2/2m⊥ is quantized in magnetic field as:
ThesituationisjustoppositetotheconventionalQHE En = ±(e~/m⊥c)Bpn(n−1) [24]. This gives an un-
conventional doubly degenerate LLL with E = E = 0
case with giant oscillations of R (B) when ρ ≪ ρ 0 1
xx xx xy
butforhigher nthe systemrecoversthe behaviorofnor-
andσ behavesinthe same wayas R . Note alsothat
xx xx
mal carriers having γ =1/2 (with n→n−1 in (3)).
the amplitude of SdH oscillations associated with Dirac-
likefermions(Fig. 4,curveb)isthreetimessmallerthan We obtain the phase factors γ and δ by plotting the
that for normal carriers (Fig. 3, curve b). inverse field values at conductivity oscillation maxima
To facilitate the comparative analysis of the results (minima) as a function of their number (number - 1/2),
obtained for various graphite (graphene) samples, we as shown in Fig. 5. The linear extrapolation of data
presentallthedataasafunctionofthenormalizedfilling points to B−1 = 0 unambiguously determines the phase
factorν =B /B,whereB correspondstotheSdHoscil- factors: γ = 1/2, δ ≃ −1/8 for data depicted in Fig.
0 0
lation frequency. The conductance G is normalized by 3 (curve b) and γ = 0, δ = 0 for data shown in Fig. 4
xy
G ,(≃28Ω−1 forelectronsand19Ω−1 forholes)corre- (curveb). Thus,weidentifyherenormal(3D)andDirac-
0xy
sponding to the step between subsequent QHE plateaus. like (2D) charge carriers in agreement with our previous
Fig. 3 (curve c) includes SdH oscillations reported for results obtained for HOPG-UC sample [3].
an8µmthick HOPGsampletakenfromthe inset ofFig. Curves (a) in Figs. 3 and 4 provide evidence for the
4in[13]afterthe polynomialbackgroundsignalsubtrac- staircasebehavior(quantization)ofGxy forbothnormal
tion. The coincidence of the SdH oscillation frequency, and Dirac-like fermions. As expected for QHE regime,
phase and the asymmetric form of the signal with those the minima in longitudinal conductivity σxx coincide
measured in our HOPG sample suggests that the same with the plateau positions in Gxy.
Fermi surface electron pocket is responsible for SdH os- In the case of normal carriers , Fig. 3 (curve a), the
cillations in both samples. quantization of Gxy corresponds to the integer QHE, as
Beforeweproceedwiththedataanalysis,wenotethat
Landau Level (LL) quantization spectra for normal and
Dirac-like carriers are different. In the normal carrier
case the equidistant LLs En =(e~/m⊥c)B(n+1/2) [19] 1
are separated by the gap E0 = e~B/2m⊥c from E = 0 N
whereas in the Dirac-like case En = ±vp2e~Bn/c [20], 0.8
aantdEthe=L0o.weTsthiLsalneaddasutLoevtwelo(iLmLpLo)rtiasnltoceaxtpederiemxaencttlayl −1T )0.6
conse0quences. −1B ( D
(i) SdH oscillations of conductivity [3] 0.4
B
∆σ (B)≃−A(B)cos[2π( 0 −γ+δ)] (2) 0.2
xx
B
(where A(B) is the non-oscillating amplitude) acquire −01 −1/|2 0 1 2 3 4 5
Landau level n
the phase factor either γ =1/2 or γ =0 for normal and
Diraccarriers,respectively. Theadditionalphasefactorδ FIG. 5: Values of B−1 for the normal (N) and Dirac-like
governedbythecurvatureofFermisurfaceisquitesmall:
(D)charge carriers at which nth Landau level crosses the
|δ|<1/8 (in 2D δ =0) and can be neglected. Fermi level. These values are found as maxima (o) in the
(ii) QHE plateaus occur either in integer (normal car- SdHoscillations ofmagnetoconductivity ∆σxx(B). Themin-
riers)orinsemi-integer(Diracfermions)wayandcanbe ima positions (x) in ∆σxx(B) are shifted by 1/2 to the left.
expressed via the phase factor γ as [21, 22]: Linear extrapolation of the data to B−1 = 0 (solid lines) al-
lowsforthephasefactor(−γ+δ,Eq. (2))definition: γ =1/2,
e2 1 δ≃−1/8fornormalelectronsandγ =0,δ=0forDirac-like
Gxy =µgs (n+ −γ) (3) holes.
h 2
4
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