PRL 96, 126601 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending31 MARCH 2006

Electric-Field Effect on the Angle-Dependent Magnetotransport Propertiesof Quasi-One-Dimensional Conductors

K. Kobayashi,* M. Saito, E. Ohmichi, and T. OsadaInstitute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan

(Received 23 April 2004; published 30 March 2006)

0031-9007=

We report a novel electric field effect on angular dependent magnetotransport in quasi-one-dimensionallayered conductors with a pair of sheetlike Fermi surfaces. Under tilted magnetic fields and additionalinterlayer electric fields, semiclassical electron orbits on two Fermi sheets become periodic at differentmagnetic field orientations. This causes double splitting of the Lebed’s commensurability resonance ininterlayer transport, and the amount of splitting allows us to estimate the Fermi velocity directly. We havesuccessfully demonstrated this effect in the organic conductor �-�BEDT-TTF�2KHg�SCN�4.

DOI: 10.1103/PhysRevLett.96.126601 PACS numbers: 72.15.Gd, 72.80.Le, 73.21.Ac

E

FIG. 1 (color online). Schematic sheetlike Fermi surfaces ofQ1D conductors and electron orbits. Under tilted magneticfields, electrons on different Fermi sheets move along dashedcurves with the same slope. When the electric field is applied,they move along solid curves with different slopes. The deviationof electron orbits from the Fermi sheets is ignored.

Interlayer magnetoresistance of layered conductorsshows remarkable angular dependence as a function ofmagnetic field orientations [1]. Particularly, quasi-one-dimensional (Q1D) layered organic conductors, such as�TMTSF�2ClO4, where TMTSF denotes tetramethyltetra-selenafulvalene, have shown very rich angular effects, theLebed resonance [2], the Danner-Chaikin oscillations [3],the third angular effect [4], and the peak effect [3].Although the magnetotransport shows nonclassical behav-iors in some cases [5], most features including these effectshave been explained as semiclassical Femi surface topo-logical effects [6,7]. They have been well reproduced byBoltzmann’s magnetotransport theory based on electronorbital motion on Fermi surfaces [8]. By using the angulareffects, we can study the ratio of band parameters.

In this Letter, we study the effect of electric field on themagnetoresistance angular effects, particularly on theLebed resonance, in Q1D conductors. Electric fields mod-ify the angular effects through the change of electronorbital motion. Using the electric field effect, we can obtainthe information not only on the ratio between band pa-rameters but also their absolute value.

First, we overview the semiclassical picture of magne-toresistance angular effects in Q1D multilayer systems [8].Let us consider the simplest band model for Q1D conduc-tors [3],

E�k� � @vF�jkxj � kF� � 2tb cosbky � 2tc cosckz: (1)

Here, orthogonal crystal axes are assumed with the x axisparallel to the conducting 1D chains and the z axis normalto the conducting 2D layers. vF, b, and c are the Fermivelocity along the 1D chain, the interchain distance insideeach layer, and the interlayer distance, respectively. Thetransfer energy tc between two neighboring layers is as-sumed to be much smaller than the transfer energy tbbetween two neighboring chains within the same layer(tc � tb). Such a system has a pair of sheetlike Fermisurfaces as shown in Fig. 1.

Under magnetic fields, electrons carry out orbital mo-tions on two Fermi sheets following the equations of

06=96(12)=126601(4)$23.00 12660

motion, @ _k � ��e�v� B, v � �1=@�@E�k�=@k. The inter-layer conductivity �zz�!� can be evaluated using theChambers formula which sums up the contributions fromall orbits [6,8]. When the interlayer transfer energy tc is sosmall as to satisfy jBz=Bxj � 2tcc=@vF, we can easilyobtain an analytic formula for �zz�!�.

��0�zz �!� � N�EF��etcc@

�2X�;�

J�

�2tbc@vF

BxBz

�2

��

1�!� vF

�� beBz

@�

ceBy@

��2�2: (2)

Here, N�EF� � 4=�2�bc@vF� is the density of states perunit volume including spin degeneracy. � is the scatteringtime, and J��z� is the �th Bessel function. Equation (2)corresponds to the lowest order contribution of interlayertransfer tc to the interlayer conduction [9]. Let us note thatthis formula is slightly changed from that in the quantummechanical picture [9].

Equation (2) explains most of magnetotransport featuresobserved in Q1D conductors. dc interlayer conductivity

1-1 © 2006 The American Physical Society

z

x

F

b

B

B

v

ct2-3

-33

0

0

3

z

y

B

B

b

cp=2

p=1

p=0

p=-1

p=-2

ϕ=60deg

0909- 0

ε =0

ε =0.01

ε =0.03

ε =0.05

Bv

E

F

z=ε

deg60=ϕ

Polar Angle θ (deg)

-410

-310

-210

-510

-610

-610

-610

-610

Inte

rlaye

r C

urre

nt j

z/Ez

p=0p=-1 p=1

p=2p=-2

x (1D)

y

zθ B

ϕ

E,J

(a) (b)

FIG. 2. Calculated interlayer conduction in Q1D conductorsunder the electric field as a function of magnetic field orienta-tion. (a) Interlayer current for several electric fields when themagnetic field is rotated in a plane normal to conducting layers.(b) Angular dependent pattern of interlayer current for generalmagnetic field orientations.

PRL 96, 126601 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending31 MARCH 2006

�zz�! � 0� shows resonant increase when pbeBz=@�ceBy=@ � 0 (p is an integer). This is the Lebed resonanceat the pth magic angle By=Bz � p�b=c�. In general fieldorientations, an electron trajectory sweeps all over theFermi sheet, so that the interlayer velocity averaged alongthe orbit, hvzi, is zero. On the other hand, at the magicangles, the electron motion on the Fermi sheets becomesperiodic, and averaged velocity hvzi becomes finite causingresonant increase of interlayer conduction. The amplitudeof the pth Lebed resonance, Jp�2tbc=@vF��Bx=Bz��2, os-cillates when the field is tilted to the 1D axis. This causesthe Danner-Chaikin oscillations. The pth Lebed resonanceshows a maximum peak at �2tbc=@vF��Bx=Bz� � p, so thatall the Lebed resonance maxima satisfy By=Bx �2tbb=@vF. These maxima converge to a single peak ofthe third angular effect at the limit of p! 1. However,the peak effect cannot be deduced from the lowest ordercontribution represented by Eq. (2).

On the other hand, ac interlayer conductivity �zz�!�shows open orbit cyclotron resonances at ! �vF�pbeBz=@� ceBy=@�. These resonances have been ob-served as the ‘‘Fermi surface traversing resonance (FTR)’’[10] or the ‘‘periodic orbit resonances (POR)’’ [11] inmicrowave absorption.

Now, let us consider the effect of electric fields appliedalong the stacking direction. The equation of motionchanges to @ _k � ��e�v� B ��e�E. Roughly speaking,in Q1D systems with two Fermi sheets, the electric fieldE � �0; 0; Ez� tends to tilt electron open orbits as shown inFig. 1. Here, we have to note that the electron orbits deviatefrom the Fermi surfaces under electric fields. However, thisdeviation is less than 4tc=@vF in the k space, so that theelectron orbits can be treated to exist almost on Fermisheets when the warping of the Fermi sheets is small.

If the warping is ignored, electrons on the kx > 0 Fermisheet have a group velocity v � �vF; 0; 0�, and electrons onthe kx < 0 sheet have v � ��vF; 0; 0�. In this case, theelectric force ��e�E can be replaced by the virtualLorentz force ��e�v� Beff , where the effective magneticfield is defined as Beff � �0; Ez=vF; 0� for the kx > 0 Fermisheet and as Beff � �0;�Ez=vF; 0� for the kx < 0 sheet.Since electrons on the Fermi sheet then move as if they areunder the magnetic field B Beff , the direction of electronopen orbits is tilted from those in absence of electric field.The Lebed resonance condition is accordingly modified to�By � Ez=vF�=Bz � p�b=c� or

ByBz� tan� sin’ � p

bc

EzvFBz

: (3)

Here, � is the polar angle of B from the z axis, and ’ is theazimuthal angle of B around the z axis measured from the xaxis. We have to note that the Lebed resonance condition(3) is different for two Fermi sheets. In other words, underelectric fields, a single Lebed resonance splits into doubleresonances, each of which originates from opposite Fermisheet.

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Equation (2) can be extended for finite interlayer electricfields Ez if we could employ the similar calculation as in(2). The interlayer dc current jz�! � 0� can be approxi-mately represented as

Re�jz�! � 0�

Ez�! � 0�

�� N�EF�

�etcc@

�2X�;�

J�

�2tbc@vF

BxBz

�2

��

1�!B � vF

�� beBz

@�

ceBy@

��2�2:

(4)

Here, !B � ceEz=@ is the Bloch frequency. We can seethat the interlayer current jz�! � 0� shows resonant in-crease when the generalized Lebed resonance condition (3)is satisfied.

Using Eq. (4), we can simulate the angular dependentinterlayer conduction under electric fields. Figure 2(a)shows the ratio of interlayer current jz and electric fieldEz traced as a function of magnetic field orientation forseveral interlayer electric fields. The field configuration isshown in the inset of Fig. 2(a). Magnetic field with a fixedstrength B is rotated in a plane including the z axis with afixed azimuthal angle ’ � 60�. The parameter " �Ez=vFB indicates electric field strength. In this calculation,we have assumed a set of band parameters a:b:c �1:0:2:0:3:7, ta:tb:tc � 1:0:0:1:0:003, and vF �

���2ptaa=@

so as to simulate TMTSF compounds. A relaxation time� � 20=�vFbeB=@� is employed. The ordinate jz=Ez isnormalized by the factor N�EF��etcc=@�2=�vFbeB=@�. Atthe limit of zero electric field ("! 0), jz=Ez�! �zz�shows several single-peak structures labeled by the indexp corresponding to the conventional Lebed resonances.Under finite electric fields, each single peak splits todouble peaks of which the width increases with the appliedelectric field.

Figure 2(b) shows the density plot of jz=Ez as a functionof magnetic field orientations. Darkness indicates the value

1-2

30 60 90 120

Polar Angle θ (deg)

0

10

20

Inte

rlaye

r C

urre

nt (

mA

)

V=20.0V

15.0V

10.5V

9.0V

α-(BEDT-TTF)2KHg(SCN)4

p=-1-2-3

5.0V3.5V2.0V

p=1 3p=2 4 5

a

θ BE,J

c'

b*

1D

φ0 ϕ φ p=3p=2p=1p=0p=-1

p=-2

p=-3

ϕ0φ

φ22

4

4-2

-2-4

-4

0

1D

*b

c'

BB

b*

a

BB

FIG. 3. Angular dependence of interlayer current for severalvoltages when the magnetic field is rotated in a plane normal toconducting layers in �-�BEDT-TTF�2KHg�SCN�4. Inset: theobserved Lebed resonance positions and the 1D axis directionof the electronic system.

PRL 96, 126601 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending31 MARCH 2006

of jz=Ez. Here, the magnetic field strength B is assumed tobe constant, and the electric field is chosen as " � 0:03.The abscissa Bx=Bz� tan�cos’ is normalized by�2tbc=@vF�

�1 so as to be the operand of the Bessel functionin Eq. (4). On the other hand, the ordinate By=Bz �tan� sin’ is normalized by b=c so as to give the Lebedresonance at interger values at zero electric field as indi-cated by dashed horizontal lines. Under the electric field,each Lebed resonance splits into two branches obeying (3),which appear as a pair of dark hyperbola in the figure. Theamplitude of each branch, that is, the darkness of eachhyperbola, is modulated as Bx=Bz is changed..

The tilted white line indicates the trace of magnetic fieldrotation with fixed B and ’, which corresponds to the ex-perimental scan as simulated in Fig. 2(a). When a Lebedresonance splits into double peaks at � and �� in the fieldrotation, the difference � tan� � j tan� � tan��j satis-fies the following relation,

� tan� �2V

dvFB sin’

��������������������1 tan2�

p: (5)

Here, d and V � Ezd are the sample thickness and theapplied voltage along stacking direction, respectively. Inthe right-hand side, tan� means the center position ofdouble peaks �tan� tan���=2. Since B, V, d, and ’can be measured, we can experimentally determine theFermi velocity vF from the split of the Lebed resonance.

Another experimental method to determine the Fermivelocity vF in Q1D conductors uses the POR in microwaveabsorption measurement [11]. The present method usingjust dc transport gives the same information as POR sinceEq. (4) is obtained by substituting the Bloch frequency !Bfor the microwave frequency ! in Eq. (2).

In order to demonstrate the above electric field effect inreal systems, we have performed angular magnetotransportmeasurements of layered organic conductors under pulsedhigh electric fields. First, we chose one of the most typicalQ1D organic conductors �TMTSF�2ClO4, where all ofmagnetoresistance angular effects in Q1D systems havebeen well established. Although we observed unclearshift of Lebed resonances, we could not see clear split-ting [12]. In the case of �TMTSF�2ClO4, interlayer resis-tivity is too small to apply high electric fields. Too muchcurrent easily heats up electron temperature and blurs outthe splitting. Then, we selected the organic conductor �-�BEDT-TTF�2KHg�SCN�4, where BEDT-TTF denotes bis-ethylenedithia-tetrathiafulvalene. This layered compoundhas thick polymeric insulating layers so that it shows largeinterlayer resistivity compared to �TMTSF�2ClO4.

It has been known that �-�BEDT-TTF�2KHg�SCN�4undergoes a phase transition to the charge-density-wave(CDW) phase at about 10 K [13]. Clear observation ofLebed resonances [14,15] and POR (or FTR) [10,11] in thisCDW phase strongly suggests that sheetlike Fermi surfacesexist in the electronic structure reconstructed by CDWformation [16,17]. The Lebed resonances appear at mag-

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netic field orientations satisfying tan� � �1:27p� 0:55�=cos����0�, where � and � are the polar angle from thecrystal b� axis (normal to the conducting ac plane) and theazimuthal angle around the b� axis measured from the aaxis, respectively [15]. p is an integer, and �0 � 27�. Thisfact shows that the 1D axis normal to the Fermi sheets istilted by �0 � �=2 � �63� from the a axis in the acplane in the CDW phase.

Single crystals of �-�BEDT-TTF�2KHg�SCN�4 havebeen grown by the conventional electrochemical method.The interlayer conduction under strong electric fields wasmeasured using pulsed electric fields to avoid sampleheating. Electric contacts were formed by gold evaporationon the top and bottom flat surfaces. Typical crystal thick-ness is d � 0:1–0:2 mm. Pulsed voltage V was appliedbetween the top and bottom surfaces, and the respondinginterlayer current was measured. The waveforms of bothvoltage and current pulses were recorded in the 12-bitdigitizer. The pulse width of 5 �s and the duty ratio of1=3000 were chosen so as to avoid sample heating.Although the measured voltage-current characteristicswere nonlinear, they showed no clear threshold behaviorfor the sliding motion of CDW. The samples were mountedon the rotating holder with 0.1� resolution, set in the 13 Tsuperconducting magnet.

Figure 3 shows the interlayer current as a function ofmagnetic field orientations for several interlayer voltages.The magnetic field strength was fixed at 13 T, and thetemperature was held at 1.8 K so that the system was inthe CDW phase. When the interlayer voltage is lowenough, for example, V � 2 V, the interlayer currentshows a series of peaks due to the Lebed resonance as

1-3

1.5 3.00

2.0

1.0

4.5 6.0

∆tan

θ

VB

θ2tan1+

x

p=1p=2p=3p=-2p=-1

∆tanθ

tanθ

FIG. 4. Plot of split width vs normalized voltage for five Lebedresonances. The solid line indicates a fit to Eq. (5).

PRL 96, 126601 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending31 MARCH 2006

indicated by arrows. As shown in the inset of Fig. 3, theirpositions satisfy the resonance condition tan� � �1:27p�0:55�= cos����0� represented by tilted dashed lines, ifwe assume that magnetic fields were rotated in a planenormal to the ac plane with the fixed azimuthal angle � ��9� measured from the a axis. In other words, the rotationplane was tilted by ’ � 54� from the 1D axis.

At large interlayer electric fields, the Lebed resonancepeaks clearly split into double peaks, and their split widthincreases as the applied electric field is increased. The splitwidth becomes smaller at Lebed resonances with higherindices p. Experimental results in Fig. 3 qualitativelyreproduce those of Fig. 2(a) calculated for an ideal Q1Dsystem (1).

In Fig. 4, the split width is plotted as a function of thevoltage for several Lebed resonances. The abscissa isnormalized by the normal component of magnetic fieldBz � B�1 tan2���1=2. Here, the split width and the cen-ter position are determined by � tan� � j tan� � tan��jand tan� � �tan� tan���=2, where � and �� arepositions of split double peaks. We can see that the splitwidth � tan� for different Lebed resonances with differentindices p almost exist along a common straight line pass-ing through the origin. This means that the split width isproportional to the electric field Ez and scaled by thenormal component of the magnetic field as expected inEq. (5). Therefore, we could conclude that the observedsplit originates from the electric field effect discussedabove. From the slope of the line in Fig. 4, which is equalto 2=dvF sin’, we can estimate the Fermi velocity vF as�9:0� 1:5� � 104 m=s using d � 0:1 mm and ’ � 54�. Itis agreeable with the value 6:5� 104 m=s obtained fromthe POR measurements [11].

In summary, we have discussed the Lebed resonance ofinterlayer conduction in Q1D layered conductors understrong electric fields. Under strong electric fields, twoFermi sheets show the Lebed resonances at different fieldorientations, of which the difference can be used to esti-

12660

mate the Fermi velocity. We have experimentally demon-strated the above features with the low-dimensionalorganic conductor �-�BEDT-TTF�2KHg�SCN�4.

The authors are grateful to Professor M. J. Naughtonfor valuable suggestion on the pulse measurement. Theygreatly thank Professor K. Kanoda and Dr. K. Miyagawafor teaching us crystal growth. They also thankProfessor W. Kang for critical reading. This work is sup-ported by the Grant-in-Aid for Scientific Research (B)(No. 12440083 and No. 15073206) from Japan Societyfor the Promotion of Science and the Toray ScienceFoundation.

1-4

*Present address: Aoyama Gakuin University, Fuchinobe,Sagamihara 229-8558, Japan.

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