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Angular magnetoresistance oscillations and the shape of the Fermi surface in β(ET)2IBr2
M. Kartsovnik, V. Laukhin, S. Pesotskii, I. Schegolev, V. Yakovenko
To cite this version:
M. Kartsovnik, V. Laukhin, S. Pesotskii, I. Schegolev, V. Yakovenko. Angular magnetoresistance
oscillations and the shape of the Fermi surface in β(ET)2IBr2. Journal de Physique I, EDP Sciences,
1992, 2 (1), pp.89-99. �10.1051/jp1:1992125�. �jpa-00246466�
J. Phys. I France 2
(1991)
89-99 JANUARY 1991, PAGE 89Classification Physics Abstracts
71.25H 74.70K 72.15G
Angular magnetoresistance oscillations and the shape of the Fernli surface in p-(ET)2IBr2
M-V-
Kartsovnik(~),
V-N-Laukhin(~),
S-I-Pesotskii(~),
I-F-Schegolev(~)
and V-M-
Yakovenko(~>*)
(~)
Institute of Solid State Physics, USSR Academy of Sciences, Chernogolovka, MD, 142432, U-S-S-R-(~) Chemogolovka Institute of Chemical Physics, USSR Academy of Sdences, Chemogolovka, MD, 142432, U-S-S-R-
(~) L-D- Landau Institute for Theoretical Physics, USSR Academy of Sciences, Kosygin St. 2, Moscow, 117940, U-S-S-R-
(Received
2 September 1991, accepted 1October1991)
Abstract. Angular magnetoresistance oscillations have been studied systematically for fl-
(ET)2
IBr2 in the magnetic field rotating in a series of planes perpendicular to the conducting
(a, b)-
plane. The oscillations have been found in all studied planes. The shape of the Fermi surfacetransverse cross-section has been reconstructed using the obtained data. Angular dependence of the slow Shubnikov-de Haas oscillations
frequency
and some fine features ofangular
magne-toresistance oscillations permit to discuss also the structure of the Fermi surface longitudinal cross-section. The Fermi surface consists most likely of main cylinders with inclined warping planes and small pockets or necks between them.
1. Introduction.
Recently, strong angular magnetoresistance oscillations, occurring
uponrotating
themagnetic
field in aplane perpendicular
to theconducting layers,
have been discovered [1 5] in a number oflayered organic
metals. Themagnetoresistance
was found to reach its local maximaperiodi- cally
in cot ~o, where~o is the
angle
between the field direction and theconducting (a, b)-plane.
The first
explanation
of thephenomenon
[6]employed
the fact that the distribution of thecyclotron
orbit areas for a Fermi surface in the form of aweakly corrugated cylinder degener-
ates
periodically
in cot ~o uponmagnetic
field rotation. It wassupposed
that thedegeneration
(*)
Present address: Serin Physics Laboratory, Department of Physics and Astronomy, RutgersUniversity, P-O- Box 849, Pixataway, NJ08855-0849, U-S-A-
should result in
increasing
themagnetoresistance. Later,
theangular magnetoresistance
oscil- lations wereexplicitly
obtained inresistivity
tensor calculations [7](see
also [8]) for aquasi
two-dimensional metal.
An alternative
explanation
of themagnetoresistance
increase for some field directionswas
proposed
in [2,8] asresulting
from the existence ofself-crossing trajectories
on the Fermi sur- face. In the case of aweakly corrugated cylindrical
Fermi surface theself-crossing trajectories
may be
associated,
forexample,
with the existence of some necksconnecting adjacent cylinders.
In this situation the oscillations should
disappear
for someplanes
ofmagnetic
field rotation.So,
apossible
way todistinguish
the differentexplanations might
consist in asystematic
in-vestigation
ofangular magnetoresistance
oscillations uponrotating magnetic
field in a series ofplanes perpendicular
to theconducting plane.
We have
performed
such measurements on a number ofhigh quality single crystals
offl-(ET)2 IBr~. Although
the oscillationsamplitude
have been found to vary for differentplanes
of the fieldrotation, they
do notdisappear
for anyplane. Therefore,
we conclude that theangular
oscillations are causedmainly by
the first of the two above mentioned mechanisms.Considerable enhancement of the Shubnikov-de Baas
(SdH)
oscillationsamplitude
observed for the field directionscorresponding
to local maxima of themagnetoresistance
[1,2] also supports this conclusion [6].Nevertheless, basing
on our results the existence of a morecomplicated
Fermi surface than a
simple warped cylinder
cannot becompletely
excluded.A substantial result of our
investigations
is the reconstruction of the Fermi surface cross- sectionshape
and dimensions in the(a,b). plane.
Thepossibility
of such a reconstruction on the basis of numerical calculations has beenpointed
out in [9]. We haveused, however,
for this purpose anexplicit analytic expression
derived in [9] which relates themagnetoresistance
maxima
angles
with the maximumprojection
of the Fermi momentum vector on the direction of themagnetic
field componentparallel
to the(a,b)-plane.
In section 2 we discuss the
physical
reasons for theangular magnetoresistance
oscillations and refine some theoretical ideasconcerning
theirexplanation.
In section 3experimental
results of themagnetoresistance
measurements for different field rotationplanes
arepresented
and theform of the Fermi surface is reconstructed.
2. Theoretical considerations.
Let us consider a
quasi-two-dimensional
metal with thefollowing
electrondispersion
law:E(pr,py,pz)
=e(p~,py)
2tcos(lpzd
+ p~u~ +pyuy]/h). (I)
Here
(p~,py)
+ pjj and pz are the momentum componentsparallel
andperpendicular
to theconducting planes;
t and d are,respectively,
thehopping integral
and thespacing
betweenadjacent conducting planes; e(pr,py)
=e(-p~, -py)
is anin-plane dispersion
law.Equation (I)
is written under theassumption
that electronshop
between theconducting planes along
vectors h and -h in the
(~,
y,z)
space, h =(ur,
up,d),
d > 0. We shall consider the case~ *
t/£F
<(2)
(eF
is the Fermienergy),
so that the Fermi surface is aweakly corrugated cylinder
with a closedconvex curve of an
arbitrary
form in a cross-section. Thecylinder
axis isparallel
toz-axis,
while the
planes
ofwarping
are inclined with respect to the(~,y)-plane being perpendicular
to the
hopping
vector h.N°1 ANGLULAR MAGNETORESISTANCE AND FERMISURFACE SHAPE 91
Magnetic
field direction is determinedby
twoangles,
~o and
b, according
to the formula H = H cos cos ~o, H sin b cos ~o, H sin ~o). Let us consider the case of strongmagnetic
field : 7 eI/war
«I,
where wc is thecyclotron frequency,
r the relaxation time. Electrons movealong
the orbits in theplanes perpendicular
toH;
each orbit may be labeledby
acoordinate, Pz,
of thepoint
where itsplane
intersects thepz-axis.
Forgeneric
values of~o
#
0 andPz
themean
velocity
componentparallel
to the fielddirection, @H(Pz),
is non-zero, whereas the meanperpendicular components
are zero. The bar over v means theaveraging
over the electron classical motionalong
the orbit.In the system of
coordinates,
related to the fielddirection,
thediagonal
component of theconductivity
tensoralong
themagnetic field,
aHH, may be written as [10]aHH =
a(~
+O(7~), (3)
and the ngular brackets mean
the veraging
ver
Pz.
Allother omponents of the conductivity
tensor
go to zeroas
7 or 7~[10].
lYansformationto the
°(7~) °(7) °(~~7)
a;; =
-O(7)
ai$~cot~
~o +
O(7~) aj$~
cot ~o +O(~~7)
,
I,
j
= ~, y, z,(5) -°(~~7)
?~~~ Cot i~O(~~7)
?~$~ +O(~~7~
Inversion of
Pzz "
I/?zz>
Prr'- PVY " Po +
[°(~~)
+°(~~7~)]?zz~> (7)
where po is of the order of the
in-plane resistivity
withoutmagnetic
field and«zz =
«f~
+°(~~7~). (8)
The
origin
of theangular
oscillations may beexplained
now as follows. Whenmagnetic
field has ageneric direction, @z(Pz)
=@H(Pz)
sin §~ is non-2ero.So,
withincreasing magnetic
field theresistivity
grows and reaches the saturation valuegiven by equation (7)
with azz=
aj$~,
where aj$~ is determined
by equation (6).
But, as it will be shownbelow,
there existspecial
field directions at which @z(Pz nearly
vanishes for all values of Pz. For thesespecial
directions the second term inequation (8) plays
the main role andresistivity (7)
grows asH~
to saturate at veryhigh
fieldbeyond experimental capabilities. Thus,
when the constantmagnetic
field isrotated,
theresistivity
exhibitsstrong
maxima at theseparticular
directions. This result wasfirstly
obtained in [7],basing
on numerical andanalytical calculations,
and then was discussed also in [8].It follows from
equation (7)
that alldiagonal
components of theresistivity
tensor contain azz in denominator and thus should exhibit theangular oscillations, although
pm and pyycontain
azl
witha small coefficient. This is result of a
general
Boltzmannequation approach.
However,
in ther-approximation
for theparabolic dispersion
lawe(p~, py)
it was found bothnumerically
[7] andanalytically
[8] that the nominatorexactly
cancels the denominator inequation (7),
so that pr~ and pyy do not exhibitangular
oscillations.Experimentally,
all the components of theresistivity
tensor exhibit strong oscillationsii 3].
The values of the critical
angles
were foundanalytically
[6] for a modeldispersion
law andnumerically
[9] for thedispersion
law calculated inill].
Below we describe thegeneral
analytical
results of [8].According
to(I)
the meanvelocity
@z can be written as@z =
0E/0pz
= ~~~j d(
sin
[pz(f)d+ (pjj(()u)j /h
,
Th
(9)
~
where
(
represents the timevariable,
T is theperiod
ofrotation,
the timedependence p(()
has to be determined from classical
equations
of motion with Lorentz force. When an electronmoves in the
plane
tilted at theangle
~o with respect to the(~, y) plane,
thefollowing
relationholds:
Pz
(f)
" l~z PH(f)
C°~ §'>(1°)
Pi
- (maxi PV
ii,_
Fig. I. Schematic look at the transverse cross-section of a Fermi surface. Hjj is the magnetic field component parallel to the
(x,y)-plane,
p)j~~~~ is the in-plane component of the Fermi momentum whose projection on the direction of Hjj attains the maximal value
pj/~~~,
u is the in-plane part of the hopping vector.where Pz labels the
position
of theplane
and pH is theprojection
of the pjj vector on the direction of theiii-plane
component of themagnetic
field(see Fig-I). Following
[8] letus consider
relatively
small ~o, where cot ~o » I and theintegrand
in(9)
oscillatesrapidly
due to relation(10).
Theintegral
can be taken in thevicinity
of the twopoints, p(~~~~
=-p)j~~~~,
wherepH(()
takes themaximum, pj/~~
>0,
and theminimum, -pj/~~~
<0, values,
respectively, (see Fig, I):
fiz(Pz)
« sin~~)
cos~~~~~~
~~~~~~~~~~~~~
i)
4 IIN°i ANGLULAR MAGNETORESISTANCE AND FERMI SURFACE SHAPE 93
It follows from
(II)
that the criticalangles,
where@z(Pz)
= 0,obey
the relationwhere N is an
integer,
thesigns
+correspond
topositive
andnegative
values of ~oc,respectively.
Relation
(12) permits
to determineexperimentally
the form of the Fermi surface cross-sectionby rotating magnetic
field in differentplanes perpendicular
to the(~, y)-plane.
For every value of b one can deducepjf"~(b)
from theslope
of the cot ~oc[ vs. N lineardependence (12).
Thenit is necessary to put a
point
at the distancepjf"~
from theorigin
in the directiongiven by
thecqrresponding angle
and draw a linethrough
thispoint
in theperpendicular
direction(see Fig, I).
Theenvelope
of the lines drawn for allangles gives
the Fermi surface cross-section.It follows from
(12)
that for agiven
N the values of cot ~oc[ forpositive
andnegative
~oc differ
by 2(p(~~~u)/pj/"~d. Upon
variation off this difference vanisheschanging
itssign
when
pf~~
becomesperpendicular
to u which is thein-plane
component of the electron inter-plane hopping
vector. Due to this effect anexperimental study
of the differencedependence
vs.
angle permits
to determine thehopping
vector h and thus the orientation of the Fermi surfacecylinder warping.
To establish a relation with the results of reference [6] one can use an
easily
provenidentity:
~ ~~ ~
os(p~)/op~ os(p~)/op~
~z
z = Pz =
os/oE
= 2xm ' ~~~where S is the area enclosed
by
electronorbit,
m thecyclotron
mass.According
toequation (3),
@z may vanish if the nominator vanishesor the denominator goes to
infinity.
The latter case, considered in details in [8], takesplace
in the presence ofself-crossing trajectories.
We shall not consider it here. Due to smallness of ~,approximately
m =mo/cos~o,
where mo is thecyclotron
mass at ~o =x/2,
so, the denominator does not have§ingularities.
On the otherhand,
it was shown in [6] that there exist criticalangles
~oc where the maximum and minimumvalues of S as function of Pz
nearly coincide,
so thatS(Pz) practically
does not vary with Pz.It follows from
equation (3)
that@z(Pz) nearly
vanishes for theseangles.
Of course,they
areexactly
theangles given by equation (12).
For more accurate formulation let us
expand
theperiodic dependence S(Pz)
in a Fourier series. At ageneric angle
~9 the firstgarmonic
dominates and furthergarmonics
are minor corrections due to condition(2).
At the criticalangles
~oc theamplitude
of the firstgarmonic exactly
vanishes(see (11))
and@z is determined
by
the secondgarmonic
of theS(Pz)
function which is ~ times smaller than the first one.Beside,
thehopping
between the next nearestplanes
with theamplitude
t' may also contribute to the second harmonic a term of the relativemagnitude tilt
[8]. Hence, due toequation (6)
the ratio of the saturation values of pzz(7)
atgeneric
and criticalangles
is verysmall, being
of the order of(Citlef
+C2tllt)~,
whereCl
andC2
are some coefficients.As the
amplitude
of SdH oscillations isproportional
to(0~S/0Pf)~~/~
[10], it should have maximaexactly
at the sameangles,
where theresistivity
does [6]. This statement is confirmedexperimentally
[1, 2].3.
Experinlental
results and discussion.For the
experiment,
fourfl-(ET)2IBr2 single crystals
were taken with the residual resistance ratioR(300)/R(4.2)
= 2000 to 3000. Measurements of the
resistance, R, perpendicular
to theJOURNAL DE PHYSI~LI -1 2 ' J'NVIFR ltJ<J2 4
highly conducting (a,b)-plane
were carried outby
means of a standard acfour-probe
method withpasted
contacts inmagnetic
field up to 16 T at thetemperature
T = IA K.The
samples
were mounted into a cell which could be rotated in twoplanes
with respect to the externalmagnetic
field direction. Thisarrangement provided
measurements of the resistanceangular dependences
for the fieldrotating
in anyplane perpendicular
to thecrystal (a,b)- plane. Thus,
aftercooling
thesample,
one couldperform
the whole set ofmagnetoresistance
measurements
avoiding repeated thermocycling
harmful for thecrystal quality.
The error in determination of the
crystal
orientation with respect to the field direction wasless than 1° for the
~o and did not exceed 5° for the S, where ~o and are the
angles
shown in the inset infigure
2.In
figure
2 fourdependences
of themagnetoresistance
on theangle
~o are
exemplified
for differentangles
S.Strong angular
oscillations are observed for everyplane
of fieldrotation, being
the mostpronounced
when is close to 90°. This fact forces us to attribute them to the mechanism discussed in section 2 rather than to the appearance ofself-crossing
orbits on the Fermi surface.The
plots
of N versus cot ~9c, where ~9c and N are,respectively,
theangle
and the number of a local maximum in the R vs. ~9dependence, clearly
demonstrate theperiodicity
of theoscillations in cot ~9c
(figure3):
N=
A~(S)
+B~(S)
cot~9),
where thesigns
+correspond
topositive
andnegative
values of ~9c,respectively.
We note that theexperimental
data are fitted wellby straight
lines for everyN,
while thetheory predicts
the oscillationsperiodicity only
for
big
N andsufficiently
small ~9c when cot ~9c »(12).
For agiven
thestraight
linescorresponding
topositive
andnegative
values of ~9c intersect theN-axis, generally,
at differentpoints, although
theirslopes
coincide within +5$lo, I-e-AA(S)
eA+(S) A~(S) #
0,B+(S)
m-B~(S).
The cut-oftdifference, AA, changes
itssign
in thevicinity
of S m 70° + 5° and atangles
farenough
from thisangle approaches
to the value +0.3.According
to the considerations of section 2~ it means that electronhopping
takesplace along
a direction tilted away from thec*-axis in the
plane forming
theangle approximately
70° + 90°= 160° with the a-axis. The direction of the
in-plane
component u of thehopping
vector h is shown infigure
4 and is close to the-a) direction,
whereaj
is the coniponent of the inversed latice vector a*perpendicular
to the c*.
Using
the cited above value of AA andequation (12)
one can evaluate anangle
abetween the
hopping
vector h an(I the c" direction: a earctan(u/d)
m 6° " 8°,u/d
m 0.ll.Using equation (12)
one can determine the form of the Fermi surface transverse cross-sectionas described in Sec. 2. The results are shown in
figure4.
The Fermi surface cross-section areaturns out to amount to m 3.9 x 10~~ cm~~ that is close to the
value,
obtained from the SdHoscillations
frequency Scsc
m 3.7 x10'~ cm~~,
whichcorresponds
to 55% of the Brillouinzone area
SBz Ill.
The form of the cross-section is inqualitative agreement
with that calculated in[I Ii.
It should be noted that the
sharp peak
at ~9 = 0° in the R vs.~9
dependences
diminishes when theangle
Sapproaches
to~-
60°,
and becomes undetectable in theangle
interval S= 65°
to 75°
(see Fig.2).
As mentionedabove,
in the sameangle
interval the cut-OR difference inN vs. cot ~9c
dependencies
is small andchanges
itssign.
With furtherincreasing
the S thepeak sharply
restores.Special
efforts have been undertaken to prove that the effect is not a consequence of a non-controlledmisalignment
of thesample.
At present, it is not clear
why
thepeak disappears.
Its existence seems to be associated with the appearance of the open orbitsalong
the Fermi surfacecylinder
at ~9 = 0.So,
it isnatural to suppose that at S m 70° + 5° the contribution of the open orbits to the
conductivity
suddenly
decreases. Thismight happen,
forexample,
if thecylinders
are connectedby
necks in theplane perpendicular
to the direction S m70°,
so that closed instead of open orbits are formed at ~9 = 0. The existence of such necks should not influencemarkedly
theapplicability
N°i ANGLULAR MAGNETORESISTANCE AND FERMI SURFACE SHAPE 95
R.~
50 Q a5
I.5
I-o
e=25 05
o
-OJ -OO -~O -20 o 20 40 6O bO W
Pa b
O-Q I.
,,
Oh $ "~
O? " 'E
i
04 03
02 6m?3°
o-I
o
-6o -do 4o -20 O 20 40 do 8O lf°
n-Q
c
023
om
o<o
e.95°
oos
O
8O 6O 40 20 O 2o AC OO dO it°
Fig. 2. Examples of magnetoresistance dependences vs, angle ~ between the magnetic field direction and the
(a, b).plane
for different angles 8; the angles ~ and 8 are shown in the inset.RQ
d
ozo
025
o2o
ois
oio
S .145°
OOS
8O dO 40 ZO O 20 40 6O 8O ~°
Fig. 2.
(continued)
«25
9 8
? 6 5 4 a
2
o
o -~~
Fig. 3. Dependences of the magnetoresistance local maximum number N on cot~ac, where ~c are angles at which the local maxima are observed, for the field rotation planes presented in figure 2.
of the theoretical model considered in Section 2
provided
the necks aresufficiently
small and do not influence very much the Fourier expansion coefficients of theS(Pz)
function at ~9#
0(see equation (3)).
As it was noted in
[12,2,1],
the necks may be alsoresponsible
for slow SdH oscillations observed incrystals
studied. The areacorresponding
to these oscillations issmall,
S~- 5 x
10~~cm~~
m
0.6%SBz.
Thedependence
of the oscillationsfrequency
vs.~9 at = 0° is shown in
N°i ANGLULAR MAGNETORESISTANCE AND FERMISURFACE SHAPE 97
Q
(51-I()/2
1 /
/ / / / / /
, /
£Cni' /
/ °
/ i i /
b '
' '
I I
I '
' '
I I
I I
Fig. 4. Transverse cross-section of the Fermi surface deduced from the
pjf~~(8)
dependence as described in section 2(thick line).
Thin line interpolating experimental points represents thepj/"~
vs. b dependence. Different symbols denote four different samples. Vector u shows the direction of the in-plane component of the hopping vector. The
magnitude
of u is arbitrary.Framing
lines indicate the Brillouin zone boundaries, the c*-axis is directed upward with respect to the figure plane.figure
S. It may beconjectured
from thisdependence
that the area of an electron orbitsharply
diminishes with
tilting
themagnetic
field direction from the c*-axis toward the -a-axis. This fact can beexplained
if the necks goalong
a direction tilted out of the(a, b)-plane.
It is not excluded also that somepockets, lying
between the Fermi surfacecylinders
andseparated
from themby
a small gap, exit instead of the continuous necks. In this case amagnetic
breakdownmight
occur at theparticular
field directionsgiving
rise to a closed electron orbits.In
conclusion,
we have studied theangular magnetoresistance
oscillations infl~(ET)2IBr2
under the
magnetic
field rotation in differentplanes perpendicular
to thecrystal (a, b)-plane.
The existence of the oscillations for every rotation
plane
evidences thatthey
are associated withquasi-two-dimensional
character of the el,I iron system,namely,
withperiodic vanishing
of the mean electron
velocity
componentalong
themagnetic
fielddirection,
vH> in the case of aweakly corrugated cylindrical
Fermi surface [6 9].Using
the obtainedexperimental
data we have reconstructed the Fermi surface transverse cross-section offl-(ET)2IBr2.
Theanalysis
of the oscillations hasprovided
an information about the electroninterplane hopping
vector. Aiii?"
~iii
al O-i O o-i O.2
fi MG
Fig. 5. Dependence of the slow SdH oscillations frequency F vs. ~ at 8 = 0°.
sharp
decrease ofthe contribution from open orbits to thein-plane magnetoresistance
has been observed for the field directionperpendicular
to thehopping
vector. Itmight
be associated with necks orpockets lying
between the main Fermi surfacecylinders.
Acknowledgements.
We express our thanks to N.E.Alekseevskii and T.Palevskii for kind
permission
toperform
a part of the measurements in the International
Laboratory
ofStrong Magnetic
Fields and LowTemperatures
inWroclaw, E.B.Yagubskii
and E.E.Laukhina for thecrystal synthesis, V.G.Peschansky
for usefuldiscussions, S.M.Zhemchugin
and N.A.Belov for technical assistance.References
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