The distribution of field-induced charges in C₆₀ fullerite
The profile of injected charges in a C₆₀-based field-effect transistor (FET) is considered. A simple scheme for calculations of the charge distribution between the 2D layers of C₆₀ molecules is founded on a small magnitude of the interball electron hopping. Analytical solutions of the equations f...
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irk-123456789-1200552017-06-11T03:05:06Z The distribution of field-induced charges in C₆₀ fullerite Kuprievich, V.A. Kapitanchuk, O.L. Shramko, O.V. Kudritska, Z.G. Низкоразмерные и неупорядоченные системы The profile of injected charges in a C₆₀-based field-effect transistor (FET) is considered. A simple scheme for calculations of the charge distribution between the 2D layers of C₆₀ molecules is founded on a small magnitude of the interball electron hopping. Analytical solutions of the equations for the charge distributions are obtained in the limits of thick and thin crystals. The charge density is shown to drop exponentially with the crystal depth. The calculations predict the relative part of induced charges involved in the surface layer to be 3/4 and 2/3 in the cases of electron and hole injection, respectively. 2006 Article The distribution of field-induced charges in C₆₀ fullerite / V.A. Kuprievich, O.L. Kapitanchuk, O.V. Shramko, Z.G. Kudritska // Физика низких температур. — 2006. — Т. 32, № 1. — С. 125-128. — Бібліогр.: 16 назв. — рос. 0132-6414 PACS: 71.10.–w, 31.25.Eb, 73.90.+f http://dspace.nbuv.gov.ua/handle/123456789/120055 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы |
| spellingShingle |
Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы Kuprievich, V.A. Kapitanchuk, O.L. Shramko, O.V. Kudritska, Z.G. The distribution of field-induced charges in C₆₀ fullerite Физика низких температур |
| description |
The profile of injected charges in a C₆₀-based field-effect transistor (FET) is considered. A simple
scheme for calculations of the charge distribution between the 2D layers of C₆₀ molecules is
founded on a small magnitude of the interball electron hopping. Analytical solutions of the equations
for the charge distributions are obtained in the limits of thick and thin crystals. The charge
density is shown to drop exponentially with the crystal depth. The calculations predict the relative
part of induced charges involved in the surface layer to be 3/4 and 2/3 in the cases of electron
and hole injection, respectively. |
| format |
Article |
| author |
Kuprievich, V.A. Kapitanchuk, O.L. Shramko, O.V. Kudritska, Z.G. |
| author_facet |
Kuprievich, V.A. Kapitanchuk, O.L. Shramko, O.V. Kudritska, Z.G. |
| author_sort |
Kuprievich, V.A. |
| title |
The distribution of field-induced charges in C₆₀ fullerite |
| title_short |
The distribution of field-induced charges in C₆₀ fullerite |
| title_full |
The distribution of field-induced charges in C₆₀ fullerite |
| title_fullStr |
The distribution of field-induced charges in C₆₀ fullerite |
| title_full_unstemmed |
The distribution of field-induced charges in C₆₀ fullerite |
| title_sort |
distribution of field-induced charges in c₆₀ fullerite |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2006 |
| topic_facet |
Низкоразмерные и неупорядоченные системы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120055 |
| citation_txt |
The distribution of field-induced charges in C₆₀ fullerite / V.A. Kuprievich, O.L. Kapitanchuk, O.V. Shramko, Z.G. Kudritska // Физика низких температур. — 2006. — Т. 32, № 1. — С. 125-128. — Бібліогр.: 16 назв. — рос. |
| series |
Физика низких температур |
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2025-07-08T17:10:08Z |
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| fulltext |
Fizika Nizkikh Temperatur, 2006, v. 32, No. 1, p. 125–128
The distribution of field-induced charges in C60 fullerite
V.A. Kuprievich, O.L. Kapitanchuk, O.V. Shramko, and Z.G. Kudritska
Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine
14-B Metrologichna Str., Kiev, 03143, Ukraine
E-mail: kuprievich@bitp.kiev.ua
Received July 4, 2005, revised July 25, 2005
The profile of injected charges in a C60-based field-effect transistor (FET) is considered. A sim-
ple scheme for calculations of the charge distribution between the 2D layers of C60 molecules is
founded on a small magnitude of the interball electron hopping. Analytical solutions of the equa-
tions for the charge distributions are obtained in the limits of thick and thin crystals. The charge
density is shown to drop exponentially with the crystal depth. The calculations predict the rela-
tive part of induced charges involved in the surface layer to be 3/4 and 2/3 in the cases of elec-
tron and hole injection, respectively.
PACS: 71.10.–w, 31.25.Eb, 73.90.+f
Keywords: C60 crystal; FET; ñharge distribution
Introduction
The phenomenon of charge injection controlled by
electric field is observed in the field-effect transistor
(FET) structure with both pristine and doped
fullerene C60 [1–4]. Because fullerene C60 with its
unique properties is considered as a prospective mate-
rial for FET devices, the important problems deter-
mining the FET characteristics are connected with the
shape of the distribution of gate-induced charges in
the C60 crystal.
A rough estimation of the charge distribution is ob-
tained from the observed dependence of the conductiv-
ity on the gate voltage at different film thicknesses
[5]. The authors have concluded that all of the in-
duced charges are located on the one or two top
monolayers.
The problem of the charge profile in the field-doped
C60 crystal is considered theoretically in connection
with the possibility of superconductivity in organic
FETs. Though reports on its observation have now
been retracted [6], the calculation of the charge distri-
butions in the FET is important by its own rights. The
study [7] of this problem within the tight-binding
model taking into account electron repulsion via mean
field shows that induced charges of a significant
amount are distributed nearly completely in the top
fullerite monolayer. Moreover, the calculations by the
modified Thomas–Fermi approach show that the
field-induced charge is concentrated in a layer near the
interface even smaller than the C60 diameter [8].
However, in view of the C60 electronic characteristics
these results should be taken with care. Specifically,
in the fullerite crystal the Hubbard repulsion U on a
C60 molecule is estimated to be 1 eV or more [9],
whereas the maximal magnitude of the interball hop-
ping t is below 50 meV. Thus, with a ratio U/t of
more than 20, the C60 crystals have to be classified as
systems with strong electron correlation which has not
taken into account properly within both the above ap-
proaches.
The present study considers the charge-profile
problem in fullerite from other positions which are in
better correspondence with the correlation characte-
ristics of this material. We apply the approach used
earlier by us for the calculations of charge transfer in
the fulleride crystals A3C60. It is just the smallness of
the hopping in comparison with electrostatic energies
that enables us to develop the simple scheme in the
present case also. Correspondingly, the scheme expli-
citly takes into account only electrostatic interactions
between the C60 molecules, neglecting the intermo-
lecular hopping. More accurately, some minor hop-
ping is formally present, allowing an equilibrium elec-
tron distribution to be reached.
© V.A. Kuprievich, O.L. Kapitanchuk, O.V. Shramko, and Z.G. Kudritska, 2006
In principle, our scheme is based on the known
Hohenberg–Kohn theorems [10] stating that (i) the
electron density is in one-to-one correspondence with
the electric field, and (ii) with the outer potential
fixed, the electron distribution is determined by the
minimum condition of energy considered as a func-
tional of this distribution with the electron number
fixed. In the preliminary brief report [11] the equa-
tions for the charge distribution are solved numeri-
cally. Considering that approach in more detail here,
we present analytical solutions of the equations defin-
ing the charge distribution in the C60-based FET
structure in the limits of thin and thick crystals.
Model and method
As in Ref. 7, we consider the fcc fullerite lattice
with a (001) plane parallel to the gate. We are inte-
rested in the distribution of injected charges between
the parallel (001) layers of C60 molecules that form
square lattices with the side length b = 10 �.
Let �n be the total number of extra electrons per
molecule in layer n, so their charge is �e n� (the case
of injected holes is represented by negative �n). We
consider a system with a total number of layers N+1,
with the surface layer labeled by zero.
The basic assumption of our model that was pro-
posed in Ref. 11 is neglecting the overlap between C60
molecules. Accordingly, the total energy of the crystal
is the sum of the energies of the separate layers of mo-
lecules that interact electrostatically. The energy En
per molecule in a layer n with the potential Un is
E E eUn n n n� �0( )� � , (1)
where E0(�n) is the energy of a free molecule with �n
extra electrons.
This expression is accurate for integer �n, so that
E0(�n) can be treated as the energies of C60 ions with
corresponding charges. For noninteger �n, it is natu-
rally assumed that E0(�n) can be interpolated by the
quadratic fit,
E E E En n n n( ) ( )� � �� � �0 1 2
20
1
2
. (2)
Then the minimum condition of total energy with re-
spect to �n leads to equation
E E eUn n1 2� � �� �, (3)
where � is a Lagrange multiplier taking into account
the additional restriction of fixing total electron num-
ber,
� �n
n
� � tot . (4)
On the other hand, the layer potentials are, in turn,
related to the charges by the Poisson equations.
Because of negligible hopping, as it is adopted in our
model, the C60 molecules interact only electrostati-
cally and can be represented by point charges due to
their near spherical form. According to Ref. 7, the po-
tential Un of layer n is determined by the equation
� � �
�
�eU m nn m
m
N
n� � ��min( , )
0
, (5)
where the coefficients � and � are expressed in terms
of the C60 dielectric constant � and the distance b be-
tween neighboring molecules:
�
�
�
�
� �
4
2
3 92 2e
b
e
b
,
. (6)
Excluding Un from (3) and (5), one obtains a system
of linear inhomogeneous equations that determine, to-
gether with the condition (4), the charge densities �n,
�� � �n m
m
n
m
m n
N
m n C n N� � � �
� � �
� �
0 1
0 1, , , , ,� (7)
where the parameter � is defined by the equality
� � �� �(E /2 ) (8)
and C E /� �( )� �1 is a new constant. As can be seen
from Eq. (7) at n = 0, this constant is in close rela-
tionship to the charge density in the surface layer,
C = ��0.
The parameter � can be treated as a localization pa-
rameter. Indeed, the solutions of the system (7) in the
limiting cases
� � �n n� tot 0 at � � 0, (9)
� � �n / N� � �0 1tot ( ) at �
� (10)
show that at � = 0 all of the injected charge is concen-
trated in the surface layer, but in the opposite case of
great � the charge turns to be uniformly distributed
between the layers.
For a finite � > 0 and a thick crystal (N
�), the
set (7) is found to have a solution in the form of a de-
scending geometric progression:
�n
nax� . (11)
For the proof, consider the dependence on x of the
left-hand side of Eq. (7) with �n in the form (11) di-
vided by a normalization constant a,
R x x S x n S x S xnN
n
n N n( ) ( ) [ ( ) ( )]� � � �� 1 0 0 ,
(12)
126 Fizika Nizkikh Temperatur, 2006, v. 32, No. 1
V.A. Kuprievich, O.L. Kapitanchuk, O.V. Shramko, and Z.G. Kudritska
where S0n(x) and S1n(x) are the sums
S x x S x mxn
m
m
n
n
m
m
n
0
0
1
0
( ) , ( ) .� �
� �
� � (13)
Expressing the sums in explicit forms
S
x
xn
n
0
11
1
�
�
�
�
,
S x x
dS x
dx
x x n nx
x
n
n
n
1
0
1
2
1
1
( )
( ) ( )
( )
� �
� � �
�
�
(14)
we arrive at the following functional form of RnN:
R x x
x x
x
nx
xnN
n
n N
( )
( )
( )
.� �
�
�
�
�
�
�
1
1 12
1
(15)
For N
�, the last term in (15) diminishes at x < 1.
So, all the functions RnN(x) have the same, inde-
pendent of n, value at x satisfying by the equation
� � �x/ x( )1 2. (16)
From its two roots, both positive, just the least one
x t t t t /� � � � �1 2 1 22 , � (17)
obeys the inequality x < 1, thus providing the valid-
ity of the solution (7) for N
�.
Results and discussion
The solution obtained shows that the shape of
charge distribution among the layers is independent of
total charge �tot that determines only the normaliza-
tion constant a in (12), a = �tot(1 – x). Finally, one
obtains from (11)
� �n
nx x� �tot ( )1 . (18)
Due to a relation that follows from (18),
x /� �1 0� �tot , (19)
the value of x can be treated as the relative part of
the injected charge in the deep layers of the crystal.
It is of interest to compare the results in the limit
N
� with those for the cases of a few layers. Solving
(7) for N = 1 and N = 2, one can evaluate x defined by
(19),
x /� �� �2 1( ) at N � 1, (20)
x /� � � �� � � �( ) ( )2 1 3 4 12 at N � 2. (21)
The dependencies of x on � tabulated by the expres-
sions (20) and (21) are depicted in Fig. 1. The plots
show that the charge density, which at � = 0 is com-
pletely localized in the surface molecules, sharply ex-
pands into deeper layers as � rises. However, further
increase of x becomes slower, so that even at moderate
� the surface charge remains dominant.
So far the quantities �n are treated as electron den-
sities. However, the above consideration is equally ap-
plicable to the case of injected holes, the densities of
which are characterized by negative �n. As can be easi-
ly seen, replacing �n by �� does not change the rela-
tive density distributions derived. Thus, the obtained
dependencies x(�) presented by Eqs. (20), (21) and
the plots in Fig. 1 are valid both for electrons and
holes.
To obtain numerical estimates in real C60 crystals
one should evaluate only E2 in the approximation (2).
Considering the latter as a fit to the ground-state ener-
gies of C60 molecule and its ions, one obtains
E E E E A A2 60 600 2 2 1� � � � � �( ) ( ) ( ) ( ) ( )C C =
= 2.7 eV (for electrons),
E E E E I A2 60 600 2 2 1� � � � � � � � �( ) ( ) ( ) ( ) ( )C C =
= 3.8 eV (for holes),
where A are electron affinities and I are ionization po-
tentials with known experimental values (in eV)
[12,13]:
A A( ) . , ( )C C60 602 7 0� �� ,
I I( ) . , ( ) . . .C C60 607 6 19 0 7 6 11 4� � � �� .
The distribution of field-induced charges in C60 fullerite
Fizika Nizkikh Temperatur, 2006, v. 32, No. 1 127
0 0.5 1 1.5 2.0
0.2
0.4
0.6
x
holes
electrons
N =
N = 2
N = 1
Fig. 1. Relative occupation x of subsurface layers as a
function of the localization parameter � in Eq. (8) and in
the C60-based FET structure with N+1 layers.
Using these data we obtain the estimates presented in
Table for injected charges of both signs.
Table. Curvature E2 of energy-harge fit, localization pa-
rameter � and relative occupation x of subsurface layers in
the C60-based FET structure
Carriers E
2
, eV � x
Electrons 2.7 0.492 0.266
Holes 3.8 0.871 0.359
It can be seen that injected charges, both electrons
and holes, are located predominantly in the surface
layer. However, their localization is far from com-
plete: no less than a quarter of the charges come from
the top layer, mainly to the next one. Note that the
relative surface occupation in the case of electrons
markedly exceeds that of holes: one third of the holes
turn to be beneath the surface.
The charge profile is formed as the result of compe-
tition between the effect of the gate field and charge
interactions inside the C60 crystal. The latter prevents
charge concentration, firstly, because of charge repul-
sion and, secondly, because too many molecules could
appear to be in higher-ionic states that are unfavorable
by energy, especially, in the case of great E2. Electro-
static charge repulsion is treated identically in our ap-
proach and in Ref. 7, but the factor of higher ionicity
is taken into account in different manners. In [7] the
on-site repulsion of extra charges is represented by the
parameter U0 in the electrostatic potential. Our ap-
proach directly uses the ion energies determining E2
and thus avoids the separate estimations of U0.
Finally, it should be noticed that the possibility of
our simple analytic consideration of the charge-profile
problem is essentially based on the quadratic approxi-
mation (2) of the charge-energy dependence for ions.
An analysis of the results of quantum mechanical cal-
culations both semiempirical and ab initio [14–16]
shows that the approximation (2) is rather accurate
for several electrons or holes. The higher-order terms
can violate the independence of the charge-profile
shape on the concentration of injected charges. How-
ever, a pronounced effect of these terms should be ex-
pected for much greater charges corresponding to the
gate voltage beyond the physically admitted values.
Conclusion
To conclude, we summarize the main differences of
our results from those obtained in Ref. 7. (i) The
charge profile in [7] depends substantially on the
charge concentration, whereas our relative distribu-
tion is independent of it. (ii) According to [7] the to-
tal charge is confined almost completely to the surface
layer, contrary to our calculations showing a relative
surface-layer population of less than three quarters.
The approximation used here takes into account
properly the electronic characteristics of the C60 crys-
tal, which is treated as a strongly correlated system.
Thus, our model, though simple, is believed to de-
scribe, at least semiqualitatively, the main features of
the charge distribution in the C60-based FET system.
The polarization and interball hopping effect ne-
glected in our study, apparently should be taken into
account to get quantitative results. The question on
the role of non-negligible occupation of subsurface
layers in the electronic properties of fullerite in FET
structures needs a separate consideration.
The authors are thankful to Prof. V. Loktev and
Prof. E. Petrov for useful discussions. This work was
partially supported under contract of the Ukrainian
Scientific Program «Nanostructures, Nanomaterials
and Nanotechnologies» (project No. 8/05—H).
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