Download Electronic Transport through a M+@C60 (M=Cs, Li, Na) Molecular

CHINESE JOURNAL OF PHYSICS
VOL. 51, NO. 3
June 2013
Electronic Transport through a M+ @C60 (M=Cs, Li, Na) Molecular Bridge
Mojtaba Yaghobi∗
Islamic Azad University, Ayatollah Amoli Branch, Amol, Iran
(Received March 23, 2012; Revised July 17, 2012)
By using the nonequilibrium Green’s function technique and the Landauer theory, the
effects of displacement, movement direction and type of ion, and the gate and bias voltages
on the quantum transport through the M+ @C60 molecular junction are considered in calculations. The results compare the height and number of transmission peaks of Cs ion with
respect to that of Li and Na ions. The calculations indicate that the I − V characteristics
and the transmission coefficient behavior for M+ @C60 are dependent on the gate and bias
voltages, the ion position and movement direction inside the cage, and the ion type.
DOI: 10.6122/CJP.51.581
PACS numbers: 71.10.Fd, 73.40.Sx, 72.10.Bg
I. INTRODUCTION
Carbon materials occur in a rich variety of structures: diamond, graphite, amorphous
carbon, fullerenes, and nanotubes. Since the discovery of buckminsterfullerene in 1985 [1],
and especially after the macroscopic preparation of fullerenes in 1990 [2], many theoretical
and experimental investigations have been carried out on these novel molecules. An endohedral metallofullerene can be created by encapsulating metal atoms into the cage-like
structure of fullerenes. Jingnan et al. [3] used the SSH (Su-Schrieffer-Heeger) model to
calculate the structural properties of the endohedral complex Na+ @C60 . Also, Dong et
al. [4] used the same method to study the nonlinear optical properties of the substituted
fullerenes. Harneit [5], Suter and Lim [6], and Twamley [7] presented the concepts for
quantum computation using endohedral fullerenes as spin-qubits and a microwave pulse
controlled magnetic dipolar interaction between qubits. The effects of transferring charge
to the carbon cage of C60 , triggering the distortion and generating a low-energy surface for
the motion of the M atom, which occurs mainly off the carbon cage centre, can be seen for
these new series of materials [8–25]. The charge-transfer from dopant atoms to the fullerene
cage can be considered similar to the effect of optical excitation of the corresponding empty
cage fullerene. That is, electrons are placed into the lowest unoccupied molecular orbital
(LUMO), although there are no holes produced in the highest occupied molecular orbital
(HOMO) as there are in the case of optical excitation [26–28].
Using the density functional non-equilibrium Green’s function method, An et al.
found that the doping effect of the Li atom significantly changes the transport properties
of C20 fullerene [29]. Wang et al. investigated the effect of the center encapsulation of an Li
∗
Electronic address: [email protected]
http://PSROC.phys.ntu.edu.tw/cjp
581
c 2013 THE PHYSICAL SOCIETY
⃝
OF THE REPUBLIC OF CHINA
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ELECTRONIC TRANSPORT THROUGH A . . .
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atom on the electronic transport properties of a C20 F20 cage sandwiched between two bulk
gold electrodes [30]. Using density functional theory and quantum transport calculations
based on the nonequilibum Green’s function formalism, the charge transport properties of
endohedral M@C20 (M = Na and K) metallofullerenes were investigated by An et al. [31].
The results show that the conductance of C20 fullerene can be obviously improved by the
insertion of an alkali atom at its centre. Also, both linear and nonlinear sections were found
on the I-V curves of the Au-M@C20 -Au two-probe systems.
The paper is arranged as follows. The theoretical model and formalism are presented
in Section II. The numerical results of the coherent transport through the M+ @C60 molecular bridge in the presence of bias and gate voltage are given in Section III. A brief conclusion
is given in Section IV.
II. MODEL AND COMPUTATIONAL METHOD
In this research, we consider a system consisting of a M+ @C60 (M=Cs, Li, Na)
molecule which is contacted by two semi-infinite SWCNT (single wall carbon nanotube)
electrodes. The model of such a structure is shown schematically in Fig. 1. The Hamiltonian
of the system is expressed as follows:
Ĥ = ĤL + ĤR + ĤC + ĤT ,
(1)
where ĤL and ĤR are the Hamiltonians of the left (L) and right (R) electrodes, ĤC describes
the channel Hamiltonian, and ĤT is the coupling between the electrodes and the central
part. The Hamiltonian of the electrodes can be written as
∑
Ĥα =
(εα δiα ,jα − tiα ,jα ) c+
(α = L, R).
(2)
iα ,σ cjα ,σ ,
(iα ,jα ),σ
εα is the on-site energy of the electrodes and will be set to zero. The coupling Hamiltonian
is described as
(
)
∑
∑
ĤT =
(−tiα ,jα ) c+
(3)
iα ,σ djα ,σ + h.c. .
α={L,R} (iα ,jα ),σ
The Hamiltonian of the central part can be extended as [8, 32]
∑
+ (0)
ĤC =
εi d+
i di Ĥc ,
(4)
i
Hc(0) =
∑
i
+
ξ i d+
i di
∑
⟨i,j⟩
(
) 1
2
+
[−t0 + αyi,j ] d+
i di HC + K (yij ) + V0i .
2
Here t0 are the hopping integrals for the C–C, α0 is the electron-phonon coupling constants
+
for the C–C bonds, yij is the change of the bond length between sites i and j, d+
i or ci (di
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MOJTABA YAGHOBI
583
FIG. 1: Geometric configuration of the carbon molecular junction. (a) A Cs+ @C60 molecule bridges
two (6, 0) SWCNTs. Schematic view of (b) the one (C1 ), (c) five (C5 ) and (d) six (C6 ) atomic
contacts of the M@C60 molecule. The symmetry axis is depicted by a cross inside a circle.
or ci ) is the creation (annihilation) operator of the π electron at the site i, K0 is the spring
constants for the C–C bonds, and the sum is taken over nearest-neighbor pairs ⟨i, j⟩. εi is
the on-site energy where it is equal to the gate voltage.
ξi is the orbital energy of the π electrons that describe the additional Coulomb interactions between the inner metal ion and the cage.
e
ξi = ε0 −
,
(5)
r0i
r0i is the distance between the sodium ion and the carbon atom at site i on the cage.
Also, V0i describes the Coulomb interaction between the sodium ion and the carbon
atoms of the cage:
V0i = A
e−βroi
.
r0i
(6)
Here A is a strength factor and β1 is the screening length. The SSH (Su-Schrieffer-Heeger)
method has been described in detail in Refs. [32, 33].
The transport calculations were carried out in the regime of low bias, using the multichannel formalism of Landauer-Büttiker based on the Green’s function [34]:
∫
2e ∞
I(Vb , Vg ) =
[f (ε − µL ) − f (ε − µR )]T (ε, Vb , Vg ) dε,
(7)
h −∞
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FIG. 2: Transmission probability as a function of energy at Vb = 0.5 V for (a) Cs+ @C60 , (b)
Li+ @C60 , and (c) Na+ @C60 in two cases of Vg = 0.5 (- -) and Vg = 0 (—) when the ions are located
in off-center positions.
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MOJTABA YAGHOBI
585
where T (ε, Vb , Vg ) is the transmission coefficient at energy ε, bias voltage Vb , and gate
voltage Vg , f is the Fermi function, µL,R are the electrochemical potential of the left and
right electrodes, h is Planck’s constant, and e is the electron charge.
The transmission coefficient T (ε, Vb , Vg ) is calculated using the Green’s function [35]
[
]
T (ε, Va , Vg ) = T r ΓL (ε − eVb /2) GR (ε, Vb , Vg ) ΓR (ε + eVb /2) GA (ε, Vb , Vg ) ,
(8)
where GR,A are the retarded and advanced Green’s function; the coupling functions ΓL,R
are the imaginary parts of the left and right self-energies, respectively.
We can write down the Green’s function of the system in the following form:
]−1
[
∑
∑
(ε, Vb )
.
(9)
GC (ε, Vb , Vg ) = εIˆ − HC (Vg ) −
(ε, Vb ) −
L
R
∑
+
Here α (ε, Vb ) = VC,α ĝα (ε, Vb ) VC,α
is the self-energy matrix, which contains the information of the electronic structure of the electrodes and their coupling to the molecule. The
semi-infinite leads do contribute to the scattering region, and this contribution is included
through the self-energies in the Green’s function of the scattering region. Correspondingly,
VCα defines the coupling matrix between the surface atomic orbitals of the lead α and the
channel. Also, gα is the surface Green’s function for the semi-infinite electrodes [35].
Using the iterative method for the calculation of the transfer matrices based on the
principal layers concept, the surface Green’s function of the left and right leads can be
obtained as [36]:
[(
ĝL (ε, Vb ) =
Vb
ε−
2
)
L
L
Iˆ − Ĥ00
− Ĥ01
T̂L
]−1
,
(10)
)
]−1
[(
Vb ˆ
R
R
I − Ĥ00 − Ĥ01 T̂R
,
ĝR (ε, Vb ) =
ε+
2
(11)
α and Ĥ α are matrices that correspond to an isolated principal layer and the
where Ĥ00
01
interaction between two nearest principle layers in the lead α, respectively. Also, T̂L and
T̂R are the transfer matrices; for the details of the calculations the reader can refer to
Ref. [34].
III. RESULTS AND DISCUSSION
Using the non-equilibrium Green’s function (NEGF) method, we have studied the
electronic transmission spectra and currents of the M+ @C60 molecular bridge in which a
carbon atom of the M+ @C60 molecule is coupled to the two single-wall metallic (6, 0)
carbon nanotubes (CNT) [see Fig. 1]. In the Numerical calculation we take: α = 4.85,
4.7, and 3.99 eV/Å, β1 = 0.55, 0.46, 0.45 Å for Cs, Li, and Na ions, respectively, and T =
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300 K [8]. We also used the parameters in the SSH model of M+ @C60 from Refs. [8, 28].
The calculations indicated that the energy of Li and Na ions at the cage center is a local
maximum, and the ions are unstable [8, 9]. When the Li (Na) ion moves to a 0.7 (0.4) Å
distance from the cage center, it reaches a minimum energy [8, 9]. The coupling between
the nearest sites in the electrodes is taken to be t0 , and that between the molecule and the
electrodes is taken to be t′ . The hopping integral between the electrode and the carbon
atom is taken as t′ = 0.5t0 , since it is much weaker than those in the electrodes and in
the molecule [37]. It should be noted that when the ion (M+ ) is in the cage center, the
M+ @C60 properties are similar to pristine C60 , so we compare our results in this state with
the C60 properties.
Figs. 2 (a)–(c) indicate the transmission coefficients as a function of the energy of the
electron which is emitted from the left electrode into the M+ @C60 molecule, with respect to
Vb = 0.5 V and Vg = 0 or 0.5 V and when ions locate in the off-center position along towards
atom 1. Since in the process of electronic transport, the electrons with Fermi energy carry
most of the current, we have shown the transmission coefficients around the Fermi level. It
is clear that, in most of the energy ranges, the value of transmission is small. The results
show that the peak height and number of the transmission of Cs ion is smaller than that
of Li and Na ions. This is because the band gaps between the HOMO and the LUMO of
the endohedrals dramatically depend on the ion type [8], which is an important factor in
the transport properties of molecular junctions. Figs. 2 (a)–(c) show that the change of the
gate voltage shifts the position of the transport coefficient peaks against the electron energy
and the transport coefficients vary around the Fermi level. The transmission coefficients of
Cs+ @C60 are similar to the theoretical calculations for the C60 molecule [38] when the Cs
ion is in the cage center.
FIG. 3: Density of states of the Cs+ @C60 molecule are presented at Vb = 0.5 V and Vg = 0.
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FIG. 4: Transmission probability as a function of energy at Vb = 0.5 V and Vg = 0 for Cs+ @C60
when the Cs ion moves a 1 Å distance from the centre.
FIG. 5: Current vs bias voltage characteristics for Cs+ @C60 (⃝) , Li+ @C60 (▽), and Na+ @C60 at
Vg = 0.
Figure 3 presents the DOS near the Fermi energy against the energy of the electron
with respect to Vb = 0.5 V and Vg = 0 when Cs+ @C60 is coupled to a single atom of every
electrode. The DOS of a system describes the number of states per interval of energy at
each energy level that are available to be occupied. The comparison of Figures 2(a) and
3 indicates that the peaks of the transmission functions are near the molecular levels of
Cs+ @C60 , where a resonance occurs between the incident electron and the molecular level.
Equations (8) and (9) indicate that the transmission coefficient depends on two factors of
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ELECTRONIC TRANSPORT THROUGH A . . .
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FIG. 6: Current vs gate voltage characteristics for Cs+ @C60 (⃝), Li+ @C60 (▽), and Na+ @C60 ()
at Vb = 0.5 V.
FIG. 7: Current vs the position of Cs (⃝), Li (▽) and Na () ions at Vb = 0.5 V and Vg = 0.
the molecule electronic energy and the electrode/molecule coupling strength. Therefore,
the Green’s function, and hence the density of states of the coupled molecule and the
transmission spectrum, vary on changing the electronic energy of the coupled molecule.
Therefore, the large values of the transmission functions are near the molecular levels of
M+ @C60 . In coherent transport, the electron wave function is assumed to extend coherently
across the whole system. Accordingly, a large transmission arises when the electron energy
nearly coincides with the molecular levels, and the electron resonantly transmits through
the molecule.
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FIG. 8: The total current-voltage characteristics of Li+ @C60 against the bias voltage (Vb ) for transport through the one atomic (♢), five atomic (*) and six atomic (×) contacts when the Li ion is
located in the off-center position along the atom 1.
The effect of the ion position on the transmission coefficients of Cs+ @C60 is indicated
in Fig. 4, with respect to Vb = 0.5 V and Vg = 0 when the Cs ion moves off the center along
the towards atom 1. Fig. 4 in comparison with Fig. 2 (a) shows that the peak numbers
(height) of the transmission of Cs+ @C60 reduce (increase) when the Cs ion moves off the
center. Using our model, we can give a clear physical explanation of this phenomenon.
When the M atom moves off the center, the additional Coulomb interaction term will
break the cage symmetry, and therewith changes the electronic structure. Then, when the
atom lies far from the cage center, the attractive long-rang Coulomb interaction between
the M atom and the π electrons on the cage, and thereby the physical properties of the
M@C60 molecules, change.
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The current-voltage (I − V ) characteristics are indicated in Fig. 5 for M+ @C60 when
the ions are located in the off-center position along towards atom 1. The calculations show
that the total current values with respect to the ion type can be arranged as I(Cs)> I(Li)>
I(NA). Also, a threshold voltage is needed to generate a current through the device, which
shows that at a low applied voltage the device is in its off state. This is due to the fact that
there is no peak at about the Fermi energy in Figs. 2 (a)–(c). The I − V characteristics
of the M+ @C60 molecules show a steplike behavior which is independent of the ion type.
In Fig. 5, the I − V characteristics of the M+ @C60 molecule are qualitatively in agreement
with the experimental and theoretical results for the C60 molecule in Refs. [31, 38–41].
The steplike behavior of the I − V characteristics can be explained by the curve of the
transmission coefficients [see Figure 2 (a)]. If we set EF = 0, the region of the bias window
or integral window in the total current integral is actually [−Vb /2, Vb /2]. Furthermore,
the total current is equal to the area under the curve of transmission in the region of the
bias window (see Eq. (7)). It is evident that the current value increases remarkably when
transmission of a new peak moves into the bias window with the increase of the bias voltage
(it opened a new channel).
The current curves as a function of gate voltage have been shown in Figure 6 for
the M@C60 molecules when the ions are located in an off-center position along towards
atom 1. The effects of the type of ion on the transport properties have been studied at
the bias voltage 0.5 V. The curves in Figure 6 indicate that the M+ @C60 molecules behave
as a semiconductor for the values of the gate voltages such that the total current becomes
zero, and for the other values they act as a metal. The present results for the current as a
function of the gate voltage are qualitatively in agreement with the results in Refs. [41–43].
Figure 7 shows the calculated current-voltage (I − V ) characteristics against the
position of ions for the M+ @C60 molecular junction with respect to Vb = 0.5 V and Vg = 0
and when the M ion moves off the center along towards atom 1. This indicates that when
an ion moves a distance from the center, the current values reach a minimum, which is lower
than that at the center for M+ @C60 . It should be pointed out that the I − V characteristics
behavior for M+ @C60 is dependent on the gate and bias voltages and the ion position at
the cage.
The depicted calculations in Figs. 8(a) and (b) show the effects of type of contact
and movement direction of the Li+ ion inside the C60 cage on the current values against
the bias voltage of the molecule. In this study, the coupling through two facing atoms
(Fig. 1(b)) or pentagons (Fig. 1(c)) or hexagons (Fig. 1(d)) of a molecule to a single atom
of each electrode will be considered. The position of the ion is the displacement of the Li+
ion off the center of the cage along towards atom 1 and 2 (see Fig. 1(a)). The gate voltage
is selected to be 0.0 V in Figs. 8(a) and (b). A comparison between Fig. 8(a) and fig. 8(b)
indicates that the contact type and movement direction of the ion inside the cage affects
the current values. The movement of carbon atoms, the distortion of the cluster structure,
and the changes of bond lengths are the effects of ion movement. It is evidence that the
M+ @C60 cluster structure and symmetry properties are dependent on the ion movement
direction inside the cage [32].
The total current values of Li+ @C60 with respect to the contact types can be arranged
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591
as I(C5 ) > I(C1 ) > I(C6 ), which is similar to the C60 molecular junction [38, 44]. The effect
of the contacts enters by the self-energy matrices in the Green’s function (see Eq. (10)). The
one, 25, or 36 elements of the self-energy matrices are nonzero for the transport through
the opposite C1 , C5 , or C6 contacts, respectively. Therefore, the Green’s function, and
hence the density of states of the coupled molecule and the transmission spectrum, vary on
changing the number of contact points.
IV. CONCLUSION
We have applied the nonequilibrium Green’s function technique and the Landauer
theory for studying the effects of the encapsulation of an M atom on the electronic transport
properties of a C60 cage sandwiched between two one-dimensional metallic electrodes. Our
results indicate that the effects of the displacement and type of ion and the gate and bias
voltages are important factors in the electron conduction. The calculations show that total
current values with respect to the ion type can be arranged as I(Cs)> I(Li)> I(Na), and
the presence of the gate voltage shifts the peaks of the transmission coefficient. The results
suggest that M+ @C60 molecules are interesting candidates for the operation of devices with
a nanoscale current.
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