Hall effect in organic layered conductors
The Hall effect in organic layered conductors with a multisheeted Fermi surfaces was considered. It is shown that the experimental study of Hall effect and magnetoresistance anisotropy at different orientations of current and a quantizing magnetic field relative to the layers makes it possible to...
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irk-123456789-1213082017-06-15T03:05:15Z Hall effect in organic layered conductors Hasan, R.A. Kartsovnik, M.V. Peschansky, V.G. The Hall effect in organic layered conductors with a multisheeted Fermi surfaces was considered. It is shown that the experimental study of Hall effect and magnetoresistance anisotropy at different orientations of current and a quantizing magnetic field relative to the layers makes it possible to determine the contribution of various charge carriers groups to the conductivity, and to find out the character of Fermi surface anisotropy in the plane of layers 2006 Article Hall effect in organic layered conductors / R.A. Hasan, M.V. Kartsovnik, V.G. Peschansky // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 145-150. — Бібліогр.: 13 назв. — англ. 1607-324X PACS: 72.15.Gd, 75.70.Cn DOI:10.5488/CMP.9.1.145 http://dspace.nbuv.gov.ua/handle/123456789/121308 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The Hall effect in organic layered conductors with a multisheeted Fermi surfaces was considered. It is shown
that the experimental study of Hall effect and magnetoresistance anisotropy at different orientations of current
and a quantizing magnetic field relative to the layers makes it possible to determine the contribution of various
charge carriers groups to the conductivity, and to find out the character of Fermi surface anisotropy in the
plane of layers |
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Article |
| author |
Hasan, R.A. Kartsovnik, M.V. Peschansky, V.G. |
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Hasan, R.A. Kartsovnik, M.V. Peschansky, V.G. Hall effect in organic layered conductors Condensed Matter Physics |
| author_facet |
Hasan, R.A. Kartsovnik, M.V. Peschansky, V.G. |
| author_sort |
Hasan, R.A. |
| title |
Hall effect in organic layered conductors |
| title_short |
Hall effect in organic layered conductors |
| title_full |
Hall effect in organic layered conductors |
| title_fullStr |
Hall effect in organic layered conductors |
| title_full_unstemmed |
Hall effect in organic layered conductors |
| title_sort |
hall effect in organic layered conductors |
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Інститут фізики конденсованих систем НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/121308 |
| citation_txt |
Hall effect in organic layered conductors / R.A. Hasan, M.V. Kartsovnik, V.G. Peschansky // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 145-150. — Бібліогр.: 13 назв. — англ. |
| series |
Condensed Matter Physics |
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AT hasanra halleffectinorganiclayeredconductors AT kartsovnikmv halleffectinorganiclayeredconductors AT peschanskyvg halleffectinorganiclayeredconductors |
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2025-07-08T19:37:29Z |
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2025-07-08T19:37:29Z |
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1837108780841566208 |
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Condensed Matter Physics 2006, Vol. 9, No 1(45), pp. 145–150
Hall effect in organic layered conductors
R.A.Hasan1, M.V.Kartsovnik2, V.G.Peschansky1,3
1 V.N.Karazin Kharkiv national university
Svoboda Sqr., 4, Kharkiv 61077, Ukraine
2 Walther-Meissner Institute,
Walther-Meissner Str. 8,
D–85748 Garching, Germany
3 B.I.Verkin Institute for Low Temperature
Physic and Engineering
Lenin Ave., 47, Kharkiv 61103, Ukraine
Received July 18, 2005, in final form February 13, 2006
The Hall effect in organic layered conductors with a multisheeted Fermi surfaces was considered. It is shown
that the experimental study of Hall effect and magnetoresistance anisotropy at different orientations of current
and a quantizing magnetic field relative to the layers makes it possible to determine the contribution of various
charge carriers groups to the conductivity, and to find out the character of Fermi surface anisotropy in the
plane of layers
Key words: Hall effect, organic layered conductors, multisheeted Fermi surface
PACS: 72.15.Gd, 75.70.Cn
Galvanomagnetic phenomena in degenerated conductors placed in a strong magnetic field B,
when cyclotron frequency of electrons ωB considerably exceeds their collision frequency 1/τ , are
very sensitive to the form of the electron energy spectrum [1,2]. Experimental studies of the mag-
netoresistance anisotropy were successfully utilized as a simple spectroscopic method for the re-
construction of the Fermi surface (FS) topology. Investigations of the Hall effect provide important
information about the charge carriers. In conductors with reduced dimensionality, the Hall effect
manifests itself in the manner essentially different from the conventional metals. For example, in
two-dimensional structures, the Hall field orthogonal to the current density on the magnitude of
quantizing magnetic field has got a step like form [3,4]. At the same time, in quasi-isotropic con-
ductors, at ωBτ � 1, the orbital quantization of electrons in a magnetic field being taken into
account does not essentially effect the magnitude of the Hall field.
In uncompensated metals with closed FS ε(p) = εF, the Hall field orthogonal to the current J,
in collisionless limit (τ → ∞), has the following asymptotic form
EHall =
[J × B]
Nec
. (1)
The corrections to the Hall field, arising from the quantization of electron energy in the magnetic
field, occur only in the following approximations in the small parameter γ = 1/ωBτ � 1.
Here e is the electron charge, c is the velocity of light in vacuum, N = N1−N2, N1 is the number
of electrons in a unit volume, and N2 is the number of holes, whose velocity vector v = ∂ε(p)/∂p is
directed within the closed concavity of isoenergetic surface. In stationary electric E and magnetic
fields, the drift velocity of charge carriers in a plane, orthogonal to the magnetic field, is identical
to all conduction electrons
u = c
[E× B]
B2
. (2)
Since the directions of motion along the closed orbits of charge carriers ε(p) = const, pB =
pB/B = const in the electronic and hole state are different, the current density in collisionless
c© R.A.Hasan, M.V.Kartsovnik, V.G.Peschansky 145
R.A.Hasan, M.V.Kartsovnik, V.G.Peschansky
limit takes the form
J = N1eu− N2eu, (3)
which in view of the relation (2) gives an asymptotical expression (1) for the Hall field.
This result has been obtained by I.M. Lifshits [5] in one of the first theoretical papers devoted
to the analysis of galvanomagnetic phenomena in metals with arbitrary dispersion law of charge
carriers in the quantizing magnetic field.
The asymptote of Hall field (1) at γ � 1 is also valid in the case of the FS in the form
of corrugated cylinder even in the presence of open sections of this surface in a magnetic field,
orthogonal to the axis of the cylinder [6]. This is connected with the fact that the drift of charge
carriers along the open orbits in the plane, orthogonal to the magnetic field, turns out to be
compensated in the expression for the current density in collisionless limit. In the case of more
complicated open FS, for example a space or plane nets of cylinders, the asymptote of Hall field
cannot be described by the general expression (1) and essentially depends on the orientation of
the magnetic field. The reason is that in this case, depending on the magnetic field direction, the
same state of charge carriers can be either an electron or a hole [1].
FS of the organic conductors with layered structure is open and weakly corrugated along the
momentum projection pz = pn on the normal to layers n. Quasi-two-dimensional character of
electron energy spectrum of layered conductors essentially distinguishes them from the perfect
two- dimensional structures, as well as from quasi-isotropic metals.
Let us consider Galvanomagnetic phenomena in organic layered conductors with an arbitrary
dispersion law of charge carriers
ε(p) =
∑
∞
n=0 εn(px, py)cos(anpz
~
+ αn(px, py)); (4)
εn(−px,−py) = εn(px, py), αn(−px,−py = −αn(px, py).
It is assumed that the functions εn(px, py) and αn(px, py) are arbitrary, and εn(px, py) is rapidly
decreasing with the number n, so that the maximum value of function {ε(p) − ε0(px, py)}, which
is equal to ηεF on the Fermi surface, is much smaller than the Fermi energy εF
max {ε(p) − ε0(px, py)} = ηεF � εF . (5)
Here a is the spacing between adjacent layers, ~ is Planck’s constant.
Experimental observation of Shubnikov-de Haas oscillations (SdH) in the magnetoresistance of
practically all layered conductors of an organic origin over a wide interval of the angles θ between the
vectors B and n proves that at least one cavity of their FS is a slightly corrugated cylinder. Velocity
distributions of charge carriers in the plane of layers for such FS have no essential anisotropy, and
for the sake of convenience of evaluations we suppose that the function ε0(px, py) is isotropic and
of the form
ε0(px, py) =
p2
x + p2
y
2m
, (6)
moreover, all the rest functions εn(px, py) with n � 1 are equal to stationary values.
By making use of the equation of motion of a charge in the magnetic field
B = (B cosϕ sin θ, B sin ϕ sin θ, B cos θ),
∂px
∂t
=
eB cos θ
c
(vy − vz sinϕ tan θ), (7)
∂py
∂t
=
eB cos θ
c
(vz cosϕ tan θ − vy), (8)
it is easy to find the contribution to the components of conductivity tensor σij from a group of
charge carriers whose states belong to a sheet of the FS having the form of a slightly corrugated
cylinder
σxx =
γ2σ0
1 + γ2
+ σzz
tan2 θ
(1 + γ2)
2 (cos2 ϕ − γ2 sin2 ϕ); (9)
146
Hall effect in organic layered conductors
σyy =
γ2σ0
1 + γ2
+ σzz
tan2 θ
(1 + γ2)
2 (sin2 ϕ − γ2 cos2 ϕ); (10)
σxy =
γσ0
1 + γ2
+ σzz
tan2 θ
1 + γ2
(sin ϕ cosϕ −
γ
1 + γ2
); (11)
σyz = σzz
tan θ
1 + γ2
(sin ϕ − γ cosϕ); (12)
σzx = σzz
tan θ
1 + γ2
(cos ϕ − γ sin ϕ); (13)
where γ = 1/ωBτ , ωB = eB cos θ/mc coincides with the cyclotron frequency of electrons in the
magnetic field in the leading approximation in the energy spectrum quasi-two-dimensionality pa-
rameter η.
When the FS of a layered conductor consists only of one cylinder, the Hall field linearly grows
with the magnetic field [6,7] and quantum oscillations in the strong magnetic field arise only in the
following approximations in parameter γ, the same as in uncompensated metals.
A considerable group of layered organic conductors have multisheeted FS, and therefore con-
sist of various topological elements such as corrugated cylinders and planes slightly corrugat-
ed along pz axis. For example, such is the FS of organic conductors based on tetrathiafulvalene
(BEDT − TTF)2M(SCN)4 , where M is one of the metals Na, K, Tl, or NH3.
One can choose the coordinate system so that the plane adjoining the slightly corrugated planes
of the FS is parallel to the pypx plane. At any orientation of a magnetic field there are open sections
of corrugated plane pB = const, and charge carriers drift along the open trajectories, generally,
along x axis with the velocity
vx =
c(py(0) − py(T ))
TeB cos θ
. (14)
Time T can be interpreted as the time of an electron motion along the open trajectory in the
momentum space for a distance about the period along py axis or pz axis.
After averaging the equation of motion of charge (8) over a sufficiently long time interval about
the order of the mean free time of electron τ , one can obtain the following relation:
vy = vz sin ϕ tan θ, (15)
therefore, at angles θ noticeably different from π/2, the drift of charge carriers along y axis is
rather small.
When the current flows along the x axis, the main contribution to the total electric conductivity
σij = σ
(1)
ij + σ
(2)
ij , (16)
is made by charge carriers, whose states belong to the corrugated plane of FS. Here, the matrix
σ(2)ij takes into account the contribution to electric conductivity tensor from charge carriers whose
states belong to the corrugated cylinder, and the components of conductivity tensor σ(1)ij are due
to the presence of additional group of charge carriers belonging to the FS in the form of slightly
corrugated planes. The asymptote of the component in a strong magnetic field is equal to σ1. And,
as follows from relation (14), it has the same order as in the absence of magnetic field.
Keeping in mind the relation (15), it is not hard to show that
σ(1)
yy = σ(1)
zz sin2 ϕ tan2 θ + γ2
1σ2 , (17)
where σ2 is the magnitude of the same order as σ1, and γ1 = T/2πτ . Hereinafter, we shall not
distinguish γ and γ1 in the expression for σij in a sufficiently high magnetic field.
The non-diagonal component σ
(1)
xy is much smaller than σ
(2)
xy . And its asymptote in the high
magnetic field
σ(1)
xy =
2e2
(2π~)3
∫ 2π~ cos θ/a
0
dpH
c
eH cos2 θ
∫ py(T )−py(0)
0
(px − px)dpy , (18)
147
R.A.Hasan, M.V.Kartsovnik, V.G.Peschansky
is determined by the degree of corrugation of the FS plane pxpy. Here px is the average value of
px(t) on the electronic orbit, and so the integrand is sign-variable.
There are sufficient grounds to consider that the energy spectrum of charge carriers with a FS
in the form of corrugated planes in the above mentioned salts is quasi-one-dimensional [8,9]. In
this case σ2 � σ1 and Hall components of the magnetoresistance tensor takes the form
ρxy =
H cos θ
Nec(1 + σ1/σ0)
, ρzx =
H sin θ sin ϕ
Nec(1 + σ1/σ0)
, (19)
ρyz =
H cosϕ sin θ
Nec
−
H2 sin ϕ sin 2θ
2(Nec)2
σ1
σ1 + σ0
, (20)
where N is the density of carrier group whose states belong to a slightly corrugated cylinder.
The studies of Hall field together with the dissipation components of the magnetoresistance
tensor [10]
ρxx =
1
σ1 + σ0
; ρyy =
1
σ0
+
H2σ1 cos2 θ
(Nec)2(1 + σ1/σ0)
(21)
allow us to separately determine the contributions to electric conductivity from various groups of
charge carriers.
At rather low temperatures when the distance between quantized Landau levels exceeds the
temperature smearing of the Fermi distribution function of charge carriers, the collision frequency
of electrons with FS in the form of cylinder exhibits a quantum oscillation
1
τ
=
1
τ0
(1 + ∆osc), (22)
where ∆osc is the quantum correction to the collision frequency of the electrons, oscillating as a
function of 1/H [11–13]. At the same time, the charge carriers whose states belong to the corrugated
plane do not give any contribution to quantum oscillation effect as their energy distribution is
essentially continuous.
Although these charge carriers do not participate in forming SdH oscillations, their presence
results in a substantial growth of the amplitude of the Hall field quantum oscillations. In conductors
with multi-sheeted FS, the relative amplitude of the Hall field oscillations
Eosc
Hall
Emon
Hall
= ∆osc
σ1
σ1 + σ0
(23)
is of the same order of magnitude as the magnitoresistanse oscillations.
In magnetic fields substantially accessible and satisfying the requirement
η � γ � 1 (24)
it is easy to perform quite informative formulas for the ratio of Hall field and the electric field along
the current under the most general assumptions concerning the dispersion law of charge carriers.
In the expansion of the magnetoresistance tensor components ratio Eβ/Eα = ρβα/ραα in terms
of γ, in this work we will neglect the terms proportional to γ2 as small corrections. Therefore, we
shall keep just the first two summands.
In this approximation, when current flows along the x axis, i.e. α = x, the ratio of non-diagonal
Hall components of the electroresistance tensor ρβx and ρxx take the form:
ρyx
ρxx
= −
σxy
σyy
;
ρzx
ρxx
= − cosϕ tan θ +
σxy
σyy
. (25)
Besides, the linear growth in the magnetic field Ez contains the terms that depend just on the
orientation of this magnetic field.
148
Hall effect in organic layered conductors
When the current flow along the Hall field is directed predominantly along the normal to the
layers and quadratically grows in the magnetic field as well as the electric field along the current
Ey, until sinϕ � γ
ρzy
ρyy
= − sinϕ tan θ −
σxy
σ1
cosϕ tan θ ;
ρxy
ρyy
=
σzz
σ1
tan2 θ sin ϕ cosϕ +
σxy
σ1
. (26)
The magnetoresistance across the layers ρzz, with a sufficient order of accuracy, is equal to 1/σzz.
Therefore, knowing ρzz, it is possible to determine σ1 using the component of the expression (26)
independent of B for ρxy/ρyy.
The main contribution to the component of conductivity tensor σxy = σ
(1)
xy + σ
(2)
xy is made by
charge carriers, whose states belong to the corrugated cylinder of FS,
σ(2)
xy =
Nec
H cos θ
, (27)
and the density of charge carriers of this group N = (Smax + Smin)/2a(2π~)2 can be found from
measuring SdH oscillations, whose period is determined by the extremal cross-sections of the cyli-
nder Smax and Smin.
If the number N of charge carriers whose states belong to the weakly corrugated cylinder is
known, one can determine the contributions of both groups of conduction electrons in the absence
of a magnetic field by solving the equations (21) with respect to σ0 and σ1. Therefore, having
experimentally determined the constant of proportionality γ = Nec/ /σ0H cos θ in the expression
(26) for the ratio of the Hall field and the electric field Ey, it is possible to determine the fraction of
participation of electrons on the corrugated plane of FS in the Hall component of the conductivity
tensor σxy. In the case of quasi-one-dimensional character of the energy spectrum of these electrons,
the component of the conductivity tensor should be neglected.
Thus, experimental study of magnetoresistance anisotropy and Hall effect at different orienta-
tions of current and magnetic field relative to the layers enables us to determine the contribution
of various charge carriers groups to the conductivity, as well as to find out the character of FS
anisotropy in the plane of layers.
149
R.A.Hasan, M.V.Kartsovnik, V.G.Peschansky
References
1. Lifshits I.M., Peschansky V.G., Zh. Eksp. Teor. Fiz., 1958, 35, 1251 (in Russian); Sov. Phys. JETP,
1959, 8, 857.
2. Lifshits I.M., Peschansky V.G., Zh. Eksp. Teor. Fiz., 1960, 38, 188 (in Rusian); Sov. Phys. JETP,
1960, 11, 137.
3. von Klizing K., Droda G., Pepper M., Phys. Rev. Lett., 1980, 45, 494.
4. Tsui D.C. , Stromer H.L., Gossard A., Phys. Rev. B, 1983, 28, 2274.
5. Lifshits I.M., Zh. Eksp. Teor. Fiz., 1957, 32, 1509 (in Russian); Sov. Phys. JETP, 1957, 5, 1227.
6. Peschansky V.G., Fiz. Nizk. Temp., 1997, 23, 47 (in Russian); Low Temp. Phys., 1997, 23, 35.
7. Peschansky V.G., Kartsovnik M.V., J. Low Temp. Phys., 1999, 117, 1717.
8. Rousseau R. et al., J. Phys. I (France), 1996, 6, 1527.
9. Kartsovnik M. V. et al., Synth. Metals, 1995, 70, 811.
10. Peschansky V.G., Hasan R.A., Savel’eva S.N., Fizika Metalov i Metalovidenie, 2002, 94, 14
(in Russian).
11. Kartsovnik M. V., Peschansky V.G., Fiz. Nizk. Temp., 2005, 31, 249 (in Russian); Low Temp. Phys.,
2005, 31, 185.
12. Singelton J., Rep. Prog. Phys., 2000, 63, 1111–1207.
13. Kartsovnik M.V., Chem. Rev., 2001, 104, 5737.
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PACS: 72.15.Gd, 75.70.Cn
150
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