Inter-Band Effects of Magnetic Field on Hall Conductivity in Multi Layered Massless Dirac Fermion System $α$-(BEDT-TTF)$_2$I$_3$

0.99 MB·English

by  N. Tajima

APS/123-QED
Inter-Band Effects of Magnetic Field on Hall Conductivity in Multi Layered Massless
Dirac Fermion System α-(BEDT-TTF) I
2 3
Naoya Tajima,∗ Reizo Kato1, Shigeharu Sugawara2, Yutaka Nishio, and Koji Kajita
Department of Physics, Toho University, Miyama 2-2-1, Funabashi-shi, Chiba JP-274-8510, Japan
1RIKEN, Hirosawa 2-1, Wako-shi, Saitama JP-351-0198, Japan
2Department of Physics, Faculty of Science and Technology,
Tokyo University of Science, Noda, Chiba JP-278-8510, Japan
(Dated: January 16, 2012)
2
1 Wehavediscoveredtwo-dimensionalzero-gapmaterialwithalayeredstructureintheorganiccon-
0 ductor α-(BEDT-TTF)2I3 under high hydrostatic pressure. In contrast to graphene, the electron-
2 holesymmetryisnotgoodexceptatthevicinityoftheDiracpoints. Thus,temperaturedependence
of the chemical potential, µ, plays an important role in the transport in this system. The experi-
n
mentalformulaofµisrevealed. Wesucceededindetectingtheinter-bandeffectsofamagneticfield
a
on the Hall conductivitywhen µ passes the Dirac point.
J

3
PACSnumbers: 71.10.Pm,72.15.Gd
1

] Since Novoselov et al. [1] and Zhang et al. [2] ex- thetemperaturedependenceofthechemicalpotential,µ.
l
al perimentally demonstrated that graphene is a zero-gap Bismuth and graphite are the most well-known materi-
h system with massless Dirac particles, such systems have als that serve as testing ground for the interband effects
- fascinatedphysicistsasasourceofexoticsystemsand/or of magnetic fields.[15, 16] To our knowledge, however,
s
e new physics. The Dirac fermion system, on the other α-(BEDT-TTF)2I3 is the first organic material in which
m hand, was also realized in the quasi-two dimensional the inter-bandeffectsofthe magneticfieldhavebeende-
. (2D)organicconductorα-(BEDT-TTF)2I3 (insetofFig. tected.
t
a 1(b)) under highpressures.[3–7] Incontrastto graphene, A crystal of α-(BEDT-TTF)2I3 consists of conductive
m this is the first bulk material with a zero-gap energy layers of BEDT-TTF molecules and insulating layers of
- band. One of the characteristic features of the Dirac I−3 anions as shown in the inset of Fig. 1(b).[17] As the
d fermion system is seen in the magnetic field normal to conductive layers are separated by the insulating layers,
n the conductive layer. In the magnetic field, the energy carriersinthissystemhaveastrongtwo-dimensionalna-
o
of Landau levels (LLs) in zero-gap systems is expressed ture. Two dimensional electron systems on conducting
c
[ as EnLL = p2e~vF2 n B , where vF is the Fermi ve- layers couple with each other by sufficiently weak inter-
locity, n is ±the Landa|u||in|dex, and B is the magnetic layer tunneling to form a layered quasi-2D conductor.
1
field strength. One important difference between zero- Theratioofthein-planeconductivitytotheout-of-plane
v
gap conductors and conventional conductors is the ap- conductivityismorethan1000. Underambientpressure,
8
2 pearance of a (n=0) LL at zero energy.[8] This special it behaves as a metal down to 135 K at which point it
7 LL is called the zero-mode LL. The characteristic fea- undergoes a phase transition to a charge-orderedinsula-
2 tures of zero-mode Landau carriers,including spin split- tor. In the low-temperature phase, a horizontal-charge-
. ting,areclearlyseenintransport[9–12]andspecificheat stripe pattern for +1 e and 0 is formed.[18–21] When a
1
0 phenomena.[13] crystal is exposed to a high hydrostatic pressure above
2 1.5 GPa, the metal-insulator transition is suppressed,
A weak magnetic field, on the other hand, also gives
1 andtheDirac-fermionsystemisrealizedstably.[3–5]The
us characteristic phenomena. According to the theory
: suppression of the metal-insulator transition by a pres-
v of Fukuyama, the vector potential plays an important
i role in inter-band excitation in electronic systems with sure above 1.5 GPa is accompanied by the disappear-
X
ance of the charge-ordering state as shown by Raman
a vanishing or narrow energy gap.[14] The orbital move-
r experiment.[21]Above1.5GPa,theresistivityisindepen-
a ment of virtual electron-hole pairs gives rise to anoma-
dent of pressure.[5] Dirac fermion system remains stable
lous orbital diamagnetism and the Hall effect in a weak
at least up to 2.0 GPa.
magnetic field.[14] These are called the interband effects
of the magnetic field. This discovery inspired us to ex- Realistic theory predicts that the interband effects of
amine the interband effects of the magnetic field in α- the magnetic field are detected by measuring the Hall
(BEDT-TTF)2I3. In this paper, we demonstrate that conductivity σxy or the magnetic susceptibility at the
these effects give rise to anomalous Hall conductivity in vicinity of the Dirac points (crossing points of the zero-
α-(BEDT-TTF)2I3. Moreover, to detect these effects, gapstructure).[16,22]Inordertodetecttheseeffects,we
we succeeded in developing an experimental formula for should control µ. In this material, however, the multi-
layered structure makes control of µ by the field-effect-
transistor method much more difficult than in the case
of graphene. Hence, we present the following idea.
∗Electronicaddress: [email protected] We find two types of samples in which electrons or
2
−
holesareslightlydopedbyunstableI anions. The dop-
3
ing gives rise to strong sample dependences of the resis- 100 15
tivity or the Hall coefficient at low temperatures (Figs. #1 3
1 and 2). In particular, the sample dependence of the 10
c
Hall coefficient RH is intense. In the hole-doped sample 2h/e) #2
apsossihtoivwenovinerththeeinwsheotloefttehmepleorwaetruprearrtanogfeF(igF.ig2.(a2)(,a)R).HIins 2e)10 R (S5 ###345 2(h/e)2 #6 0 b
t2h(be)e,lehcotwroenv-edr,opthede spaomlaprilteyaissschhoawnngeidn athteloinwsettemofpFeriag-. R (h/S 0 ##670.5 1T (K) 5 10 R S ∝∝∝∝ ln(T)
tures (Fig. 2(b)). The change in polarity of RH is un- 1 1
derstoodas follows. Incontrastto graphene,the present
electron-holesymmetryisnotgoodexceptatthevicinity
of the Dirac points.[23] Thus, µ must be dependent on (a) 1.8GPa (b)
temperature. Ofsignificanceisthefactthataccordingto 0.1 0
0 20 40 60 80 100 0.1 0.51 510
the theory by Kobayashi et al., when µ passes the Dirac
T (K) T(K)
point (µ = 0), RH = 0.[22] Thus, RH at the vicinity of
RH = 0 for electron-doped samples must be determined
to detect the inter-band effects of the magnetic field. FIG.1: (Coloronline)(a)TemperaturedependenceofRS for
seven samples under a pressure of 1.8 GPa. The inset RS
Here we adduce other examples for the effect of un-
at temperaturesbelow 10 K.(b) Temperaturedependenceof
−
stable I3 anions. It was also seen in the superconduct- RS for sample6. It was examineddown to80 mK.Theinset
ing transition of the organic superconductors β-(BEDT- showsthecrystalstructureofα-(BEDT-TTF)2I3viewedfrom
TTF)2I3 (Ref. 24) and θ-(BEDT-TTF)2I3.[25] thea axis.
Single crystals of α-(BEDT-TTF)2I3 were synthesized
bytheelectrolysismethod.[17]Thetypicalsizeofacrys-
tal was 0.8 0.4 0.04 mm3. The resistivities and the Carrierdensity, n , written as n T2, is a character-
c c
× × ∝
Hall coefficients of seven samples at p = 1.8 GPa were istic feature of 2D zero-gap conductors.[4, 5] Thus, the
measured in magnetic fields of up to 0.01 T at temper- Hall coefficient is proportional to T−2 as shown in Fig.
atures below 20 K. Experiments were conducted as fol- 2(b). Carrier mobility, on the other hand, is determined
lows. Asampletowhichsixelectricalleadswereattached as follows. Accordingto Mott’s argument[26], the mean
wasputinaTefloncapsulefilledwithapressuremedium freepathl ofa carriersubjectedtoelastic scatteringcan
(IdemitsuDN-oil7373),andthenthecapsulewassetina never be shorter than the wavelengthλ of the carrier,so
clamp-type pressurecell made of the hard alloyMP35N. l λ. For the cases in which scattering centers exist at
≥
The pressurelossatlow temperatures waslessthan 0.15 high densities, l λ. As the temperature is decreased, l
∼
GPa. Resistance was measured by the conventional DC becomes long because λ becomes long with the decreas-
method with six probes as shown in the inset of Fig. ing energy of the carriers. As a result, the mobility of
2(a). An electrical current of 10 µA was applied in the carriers increases in proportion to T−2 in the 2D zero-
2Dplanealongtheaaxis. Magneticfieldswereappliedin gap system. Consequently, sheet resistance is given by
the direction normal to the 2D plane. The doping levels RS =h/e2, which is independent of temperature.
were estimated to be 1-10 ppm for seven samples from Below 7 K, the sample dependence due to the effects
the Hall coefficient at lowest temperature. This result ofunstableI−3 anionsappearsstrongly. AriseofRS may
supports the theory of Kobayashi et al. Note that the be a symptom of localization because it is proportional
difference in the sample was notable to be distinguished to logT as shown in the inset of Fig. 1(a). The unsta-
by appearance. ble I− givesrise to a partially incommensurate structure
3
Figure 1 shows the temperature dependence of the re- in the BEDT-TTF layers. As for sample 6, the logT
sistivity ρ per layer (sheet resistance, RS = ρ/C, where law of RS was examined down to 80 mK. The logT law
C = 1.75 nm is the lattice constant along the direction of resistivity, on the other hand, is also characteristic
normal to the 2D plane). Note that the concept of “re- of the transport of Kondo-effect systems. In graphene,
sistivity per layer”applies to this system in which each recently, Chen et al. have demonstrated that the inter-
conductive layer contributes to the transport almost in- action between the vacancies and the electrons give rise
dependently. In this figure, we see that the sheet resis- to Kondo-effect systems.[27] The origin of the magnetic
tancedependsonthetemperatureveryweaklyexceptfor moment was the vacancy. In α-(BEDT-TTF)2I3, on the
theregionbelow7K,atwhichanincreaseinresistanceis otherhand,Kanodaet al. detectedanomalousNMRsig-
observed. It varies from a value of approximately seven nalswhichcouldnotbe understoodbasedonthe picture
times the quantum resistance h/e2=25.8 kΩ at 100 K to ofDiracfermionsystemsatlowtemperatures.[28]Wedo
2
approximatelytwiceh/e at7K.Then,thereproducibil- not yet know which in the localization, the Kondo-effect
ity of the data is checked in six samples. Above 7 K, we or other mechanisms, are the answer for the origin of
findonlyweaksampledependence. Thissheetresistance the logT law of RS. This answer, however, will reply to
is understood as follows. the question of why the sample with the higher carrier
3
FIG. 2: (Color online) Temperature dependence of the Hall FIG.3: (a)SheetelectrondensitynS forfivesamplesplotted
coefficient for (a) hole-doped-type and (b) electron-doped- against temperature at RH = 0. (b) EF was estimated from
type samples. Note that in this figure, the absolute value the relationship nS = EF2/(4π~2vF2) with vF ∼ 3×104 m/s.
of RH is plotted. Thus, the dips in (b) indicate a change in From this curve, thetemperature dependenceof µ is written
the polarity. The inset in the upper part of (a) shows the approximatelyasµ/kB =EF/kB−AT withA∼0.24whenwe
configuration of the six electrical contacts. The schematic il- assumeAisindependentofEF. (c)Temperaturedependence
lustrationoftheFermilevelsforthehole-dopedtypeandthe of µ for EF =0. Our experimental formula is quantitatively
electron-doped type are shown in the inset of the lower part consistent with the theoretical curve of Kobayashi et al. [22]
of (a) and theinset of (b),respectively. at temperaturebelow 100 K.
density (lower RH saturation value in Fig. 2) exhibits with A 0.24. This experimental formula reproduces
the higher RS at low temperature. Further investigation wellther∼ealistictheoreticalcurvebyKobayashietal.,[22]
should lead us to interesting phenomena. as shown in Fig. 3(c). Our simple calculations, on the
other hand, also reproduce well this curve when we as-
Another impressive phenomenon is seen in RH, as sume vh/ve 1.2, where vh and ve are the holes and
shown in Fig. 2(b). The polarity of RH at low tem- F F ∼ F F
electrons of the Fermi velocity, respectively. This is the
perature indicates which was doped: electrons or holes.
electron-hole symmetry in our system.
The saturation value at the lowest temperature, on the
other hand, depends on the doping density, nd, as nd = ThesecondstepistocalculatetheHallconductivityas
nS/C = 1/e/RH, where nS is the sheet density. In this σxy =ρyx/(ρxxρyy+ρ2yx). Inthiscalculation,weassume
work, to detect the inter-band effects of the magnetic ρxx = ρyy for the following reasons. According to band
field, we focused on the behavior of RH in which the po- calculation,theenergycontouroftheDiracconeishighly
larity is changed. anisotropic.[7, 23] In the galvano-magnetic phenomena,
however,theanisotropyisaveragedandthesystemlooks
The first step is to examine the temperature depen-
very much isotropic. A simple calculation indicates that
denceofµ. Asmentionedbefore,webelieveµpassesthe
thevariationinthemobilitywiththecurrentdirectionis
DiracpointatthetemperatureT0,shownasRH =0.[22] withinafactorof2. Experimentally,Iimorietal. showed
ThesheetdensitynS,ontheotherhand,isapproximately
proportionaltoT2asshowninFig. 3(a). Thisresultsug- that the anisotropy in the in-plane conductivity is less
0
than 2.[29]
gests that µ is to be writtenas µ/kB =EF/kB AT =0
at T0 approximately because EF √nS, where−A is the Based on this assumption, we show the temperature
∝ dependence of σ for samples 1 and 7 in Fig. 4(a) as
fittingparameterdependingontheelectron-holesymme- xy
try (vFh/vFe). Thus, we obtain the EF versus T0 curve an example. We see the peak structure in σxy at the
vicinityofσ =0. Thispeakstructureistheanomalous
in Fig. 3(b). EF is estimated from the relationship xy
nS =EF2/4πv¯F2, where the averagedFermi velocity v¯F, is Hall conductivity originating from the interband effects
estimatedtobeapproximately3.3 104m/sfromthein- of the magnetic field. The realistic theory indicates that
× theHallconductivitywithouttheinterbandeffectsofthe
terlayer magnetoresistance.[10] Note that the weak sam-
magnetic field has no peak structure [22].
pledependenceofbothRSandRHattemperaturesabove
7 K (Figs. 1 and2) stronglyindicates that the vF values In the last step, we redraw σxy in Fig. 4(b) as a func-
of all samples are almost the same. When we assume tion of µ using the experimental formula, µ = EF AT
−
that A is independent of EF, A is estimated to be ap- with A=0.24. It should be compared with the theoret-
proximately 0.24 from Fig. 3(b). Thus, we examine the ical curve,[22] σxthyeory at T = 0. Our experimental data
temperature dependence of µ as µ/kB = EF/kB −AT are roughly expressed as σxy ∼ gσxthyeory, where g is a
4
damping energy and the tilt of the Dirac cones.[22]
Lastly, we briefly mention the zero-gap structure in
this material. The smooth change in the polarity of σ
xy
isalsoevidencethatthismaterialisanintrinsiczero-gap
conductor. Nakamura demonstrated theoretically that
in a system with a finite energy gap, σ is changed in a
xy
stepwise manner.[30]
In conclusion, α-(BEDT-TTF)2I3 under high hydro-
staticpressureisanintrinsiczero-gapconductor. Incon-
trasttographene,theelectron-holesymmetryisnotgood
exceptatthevicinityoftheDiracpoints. Thus,thetem-
perature dependence of µ plays an important role in the
transport in this system. We revealed that µ depends
on temperature as µ/kB = EF/kB AT with A 0.24.
− ∼
We succeeded in detecting the interband effects of the
FIG. 4: (Color online) (a) Temperature dependence of the magnetic field on the Hall conductivity when µ passes
Hallconductivityforsamples1and7. (b)Chemical-potential the Dirac point. Good agreement between experiment
dependence of the Hall conductivity for samples 1 and 7. and theory was obtained. This system offers a testing
Solid lines and dashed lines are the theoretical curves with groundforanewtypeofparticles,namely,masslessDirac
and without the interband effects of the magnetic field by
fermions with a layered structure and anisotropic Fermi
Kobayashi et al.,respectively.[22]
velocity.
We thank Prof. T. Osada,Prof. A. Kobayashi,Dr. S.
parameter that depends on temperature because the ef- Katayama, Prof. Y. Suzumura, Dr. R. Kondo, Dr. T.
fect of thermal energy on the Hall effect is strong. Note Morinari, Prof. T. Tohyama, and Prof. H. Fukuyama
that σ depends strongly on temperature. The energy for valuable discussions. This work was supported by
xy
between two peaks, is the damping energy that depends Grants-in-Aidfor Scientific Research(No. 22540379and
on the density of scattering centers in a crystal. The in- No.22224006) from the Ministry of Education, Culture,
tensity of the peak, on the other hand, depends on the Sports, Science and Technology, Japan.
[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, [16] H. Fukuyama and R. Kubo, J. Phys. Soc. Jpn. 28, 570
M.I.Katsnelson,I.V.Grigorieva,S.V.DubonosandA. (1970).
A.Firsov, Nature438, 197 (2005). [17] K.Bender,I.Hennig,D.Schweitzer,K.Dietz,H.Endres
[2] Y. Zhang, Y. W. Tan, H. Stormer and P. Kim, Nature andH.J.Keller,Mol.Cryst.Liq.Cryst.108,359(1984).
438, 201 (2005). [18] H. Kino and H. Fukuyama, J. Phys. Soc. Jpn. 64, 1877
[3] N.Tajima, M.Tamura,Y.Nishio,K.KajitaandY.Iye, (1995).
J. Phys. Soc. Jpn. 69, 543 (2000). [19] H.Seo, J. Phys.Soc. Jpn.69, 805 (2000).
[4] N.Tajima, S.Sugawara,M.Tamura,R.Kato,Y.Nishio [20] Y. Takano, K. Hiraki, H. M. Yamamoto, T. Nakamura
and K. Kajita, J. Phys. Soc. Jpn. 75, 051010 (2006). and T. Takahashi, J. Phys.Chem. Solid 62, 393 (2001).
[5] N.Tajima, S.Sugawara,M.Tamura,R.Kato,Y.Nishio [21] R. Wojciechowski, K. Yamamoto, K. Yakushi, M.
and K. Kajita, EPL 80, 47002 (2007). Inokuchi and A. Kawamoto, Phys. Rev. B 67, 224105
[6] A. Kobayashi, S. Katayama, K. Noguchi and Y. Suzu- (2003).
mura, J. Phys.Soc. Jpn.73, 3135 (2004). [22] A. Kobayashi, Y. Suzumuraand H. Fukuyama,J. Phys.
[7] S. Katayama, A. Kobayashi and Y. Suzumura, J. Phys. Soc. Jpn.77, 064718 (2008).
Soc. Jpn. 75, 054705 (2006). [23] H. Kinoand T. Miyazaki, J. Phys.Soc. Jpn. 75, 034704
[8] T. Ando,J. Phys.Soc. Jpn. 74, 777 (2005). (2006).
[9] N. Tajima, S. Sugawara, R.Kato, Y. Nishio and K. Ka- [24] K.Murata,M.Tokumoto,H.Anzai,H.Bando,G.Saito,
jita, Phys.Rev.Lett. 102, 176403 (2009). K.Jajimura,andT.Ishiguro,J.Phys.Soc.Jpn.54,1236
[10] N.Tajima,M.Sato,S.Sugawara,R.Kato,Y.Nishioand (1985).
K.Kajita, Phys.Rev.B 82, 121420R (2010). [25] B. Salameh, A. Nothardt, E. Balthes, W. Schmidt, D.
[11] S. Sugawara, M. Tamura, N. Tajima, R. Kato, M. Sato, Schweitzer, J. Strempfer, B. Hinrichsen, M. Jansen, and
Y. Nishio and K. Kajita, J. Phys. Soc. Jpn 79, 113704 D. K. Maude, Phys.Rev. B 75, 054509 (2007).
(2010). [26] N.F. Mott and E. A.Davis,”Elecron Processes in Non-
[12] T. Osada, J. Phys.Soc. Jpn.77, 084711 (2008). Crystalline Materials” Clarendon, Oxford 1979.
[13] T. Konoike (privatecommunication). [27] J. Chen, L. Li, W. G. Cullen, E. D. Williams and M. S.
[14] H.Fukuyama,J. Phys. Soc. Jpn. 76, 043711 (2008). Fuhrer, Nat.Phys. 7, 535 (2011).
[15] P.A. Wolff, J. Phys. Chem. Solids. 25, 1057 (1964). [28] K. Kanoda (privatecommunication).
5
[29] S.Iimori,K.Kajita,Y.NishioandY.Iye,Synth.Metals [30] M. Nakamura,Phys. Rev.B 76, 113301 (2007).
71, 1905 (1995).