Article pubs.acs.org/JPCC
Chirality Dependence of Electron Transport Properties of SingleWalled GeC Nanotubes Pabitra Narayan Samanta and Kalyan Kumar Das* Department of Chemistry, Physical Chemistry Section, Jadavpur University, Kolkata 700 032, India S Supporting Information *
ABSTRACT: Electron transport properties of zigzag singlewalled GeC nanotubes (GeCNTs) of different chiralities have been studied by using a combined method of density functional theory and nonequilibrium Green’s function formalism. Transmission pathways at zero bias are analyzed for GeCNTs up to (8,0) chirality. The flow of electron is hindered for nanotubes of chiralities higher than (6,0), which leads to a drastic reduction in the conductance of these devices. Band structures of all five GeCNTs are calculated. The transmission coefficients are estimated for (n,0) GeCNTs (n = 4, 5, 6) at various positive and negative bias voltages within ±2.8 V. The current−voltage (I−V) curves of these systems are drawn for different bias voltages. The I−V characteristics of all three devices show negative differential resistance, which is analyzed from the transmission spectra and molecular projected self-consistent Hamiltonian states. The variations of the rectification ratio (I+/ I−) with the bias voltage for (4,0), (5,0), and (6,0) GeCNTs are also reported. electric and magnetic fields on these molecular orbitals are discussed by Kim and Kim.15 These authors have also highlighted the intuitive approach to design new electronic devices by appropriately tuning the molecular orbitals with external fields. Jia et al.16 have studied transport properties of heterojunctions composed of SW-SiCNTs of different lengths coupled to SW-CNTs using combined density functional theory (DFT) and nonequilibrium Green’s function (NEGF) formalism. The characteristics of the current−voltage (I−V) curve of the SW-CNT/SW-SiCNT/SW-CNT heterojunctions are also investigated by these authors. The effect of radial and axial deformation on electron transport properties in the semiconducting SiCNT has been studied by Choudhary and Qureshi.17 Negative differential resistance (NDR) is an important quantum transport phenomenon in electronic systems. It refers to the decrease in current when a voltage is applied, and in our model, where charging and electron− phonon interactions are neglected, it is due to a number of electronic factors including orbital alignment. It is a useful property in electronic devices having switching behavior. Kim et al.18 have shown that asymmetric couplings of CNTs with molecules like phenyl-ethyl oligomers and pyrrollo pyrrole give rise to NDR. The NDR effect has also been reported recently in the I−V characteristics of the armchair SW-SiCNT, which has originated from the changes of density of states by the applied bias voltage.19,20 The fluctuation of the transmission coefficient with the bias voltage is also found to be responsible for the NDR effect.21,22 Zhang et al.23 have observed the NDR
1. INTRODUCTION After the discovery of single-walled carbon nanotubes (SWCNTs) by Iijima,1 there was an explosion in their applications due to their unique structural and electronic properties. Several devices like field-effect transistors, diodes, oscillators, switches, and chemical sensors have been designed and fabricated using SW-CNTs in recent years.2−6 Hong et al.7 have devised a simple scheme for the controlled extraction of successive shells of multiwalled carbon nanotubes by using an atomic force microscope. Nanodevices made of hybrid atoms such as Si and C or Ge and C are the advanced topic of research in material sciences as a complementary of conventional electronic devices. Silicon carbide is generally a semiconductor with a wide band gap. Its physical and electronic properties have immense impact in the preparation of electronic devices used for hightemperature, high-power, and high-frequency applications.8 The single-walled SiC nanotubes (SW-SiCNTs) are known to have better thermal stability, high chemical reactivity, and uniform semiconducting behavior independent of the chirality.9−11 On the basis of controlled solid−solid reaction, hybrid nanostructures between SW-CNTs and SW-SiCNTs are fabricated by Zhang et al.12 Using the reaction of silicon obtained from a disproportion reaction of SiO with multiwalled carbon nanotubes, Sun et al.13 have prepared SiC nanotube. The intriguing transport properties of nanodevices are known to be related to quantum phenomena. Sometimes, these are very sensitive to the atomistic changes in the system. Kim et al.14 have made an extensive review on the electron and spin transport in molecular devices such as molecular wires, rectifiers, field effect transistors, electrical and optical switches, and so on. In another review, the role of molecular orbitals in quantum transport through molecular devices and the effects of © XXXX American Chemical Society
Received: July 2, 2012 Revised: December 1, 2012
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behavior of silicon monatomic chain encapsulated in carbon nanotubes. Electronic and transport properties of graphene nanoribbon device stacked optimally with DNA bases are studied by Min et al.24 using DFT-NEGF calculations. Germanium being in the same group and isoelectronic with C and Si, its compounds are also important for the nanoelectronics-based technology.25,26 GeC in the bulk form is a semiconductor with a wide band gap. The percentage of covalency in GeC is comparable to that of SiC. Germaniumfilled carbon nanotubes with an average diameter of 15−100 nm and Ge-filled carbon microtubes with an average diameter of 100−8000 nm have been prepared by Domrachev et al.27 using metal−organic chemical vapor deposition (MOCVD) techniques. Recently, Rathi and Ray28 have performed ab initio calculations for the electronic and geometric structures of three different types of armchair GeC nanotubes from (3,3) to (11,11) using hybrid DFT and the finite cluster approximation. These authors have predicted that all armchair GeC nanotubes are semiconducting with a wide spectrum of band gaps ranging between 0.549 and 3.016 eV. Electronic and magnetic properties of semiconducting (8,8) single-walled GeC nanotubes (SW-GeCNTs) filled with iron nanowires are studied by Wang et al.29 using first-principles projector-augmented-wave (PAW) potential method within DFT under generalized gradient approximation (GGA). In this article, we report the electrical transport properties of (n,0) SW-GeCNTs for n = 4−8 by studying transmission energy spectrum along with band structure, local transmission analysis, and I−V characteristics. The DFT and NEGF formalism are used to calculate the transmission coefficients and the amount of current passing through the devices under different applied voltages. We have investigated the nature of I−V curves for SW-GeCNTs of different chiralities. Rectification effects are also obtained by calculating the rectification ratio I+/I− which is the ratio between the absolute current under positive and negative bias. A value of 1 for I+/I− means the system has no rectification effect. Qiu et al.30 have studied the rectifying performance of a polyene-based molecular wire. Recently, the substitution effects on the rectification performance of azulene-like molecules in metal−molecule−metal junctions have been carried out by Zhou et al.31 Here we study the characteristics of the rectification ratio of (4,0), (5,0), and (6,0) GeCNTs under different bias voltages.
Figure 1. Optimized geometries of nanodevices: (a) (4, 0) GeCNT; (b) (5, 0) GeCNT; and (c) (6, 0) GeCNT at PBE/DZP level of theory using 1 × 1 × 100 k-point sampling.
The double-ζ polarized (DZP) basis sets for both Ge and C along with the norm-conserving Troullier−Martins pseudopotentials37 are used in the present calculations. The exchangecorrelation potential is approximated by the Perdew−Burke− Ernzerhof (PBE)38 parametrization of GGA function. The electrostatic potentials are computed on a real-space grid with a mesh cutoff energy of 150 Ry. The electrode temperature has been set to 300 K, and the 1 × 1 × 100 k-point sampling is employed throughout the calculation. The zero-bias transmission spectra are also calculated with 1 × 1 × 500 and 1 × 1 × 1000 k-point samplings. However, no appreciable changes are found in the transmission spectra upon the variation in kpoint sampling in the transport direction (z-direction), and the 1 × 1 × 100 k-point sampling has been found to be good enough. Within the NEGF formalism, the current (I) that passes through the central region (C) at a finite bias voltage (Vb) can be computed using the Landauer−Büttiker formula39 μR 2e I(Vb) = T (E , Vb)[f (E , μL ) − f (E , μR )] dE h μL (1)
2. COMPUTATIONAL DETAILS To investigate the chirality-dependent electron transport properties of (n,0) GeCNTs, we have adopted a two-probe model consisting of two semi-infinite electrodes (L and R) coupled to a central region (C). Atoms in both of the electrodes include parts of semi-infinite bulk electrodes, which interact with the atoms of the central region. As shown in Figure 1, in the two-probe model, the same GeCNT is used as electrodes, which minimizes the contact dependency with the central region.17 In the present study, the coupling effect as well as length dependency are also avoided by keeping a minimal length segment (10 Å) of GeCNT in the central region. Electronic structures and equilibrium geometries of the composite nanodevices are obtained from a first-principle total energy calculations based on pseudopotential DFT.32,33 Total energies and quantum transport calculations of the optimized devices are carried out by using TRANSIESTA code, as implemented in the SIESTA-3.1 package,34 which is based on the combination of DFT with Keldysh NEGF method.35,36
∫
where f is the Fermi function, μL,R are the chemical potentials of the left and right electrodes, respectively, and T(E,Vb) is the transmission coefficient at energy E and bias voltage Vb. The transmission coefficient, which determines the probability of electrons transferring between the two semi-infinite electrodes, can be evaluated by B
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Figure 2. Transmission spectra of (n,0) GeCNTs (n = 4−8) at zero bias.
T (E , Vb) = Tr[ΓL(E , Vb)G(E , Vb)ΓR (E , Vb)G†(E , Vb)]
CNT. Again, these bond angles vary with the chirality of the nanotube. It has been found that the variation is minimum in the case of (6,0) GeCNT. The equilibrium transport properties under zero bias voltage for different zigzag SW-GeCNTs with chiralities n = 4−8 are determined first. The transmission coefficient, T(E,0) refers to the probability for electrons with certain energy transferring from one electrode to the other when no bias voltage is applied. The computed transmission coefficients for each of (n,0) GeCNTs (n = 4−8) are plotted in Figure 2 as a function of energy. The Fermi level has been set to EF = 0 for better understanding. At the Fermi level of (4,0) GeCNT, the transmission coefficient is about 1.0, as shown in the transmission spectrum, and it remains constant for 0.58 eV above the Fermi energy. It may be noted that T(E,0) at the Fermi level decreases with the increase in the chirality of
(2)
where G(E,Vb) is the Green’s function of the two-probe model and ΓL/R are the coupling matrices.
3. RESULTS AND DISCUSSION SW-GeCNT devices of chiralities (4,0), (5,0), and (6,0), which are optimized at the PBE/DZP level of theory using 1 × 1 × 100 k-point sampling, are shown in Figure 1. In the zigzag nanotube lattice, each Ge is bonded to two symmetrically equivalent carbon atoms having a shorter Ge−C bond length, and the third Ge−C bond is longer by 0.011 to 0.025 Å. The representative Ge−C bond lengths are also shown in Figure 1. Because of different hybridization of C and Ge, there is a wide variation in C−Ge−C and Ge−C−Ge bond angles, which results in some distortions in GeCNT compared with the pure C
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Figure 3. Transmission pathways for (n,0) GeCNTs (n = 4−8) at the Fermi level. (“Magnitude” refers to the local bond contribution of the transmission coefficient and the red colored arrows correspond to a positive value of the local transmission coefficient in the direction of net current flow, as described in the text in detail.)
GeCNT. The calculations show that it becomes 0.9 for (5,0) and 0.82 for (6,0). At the same time, the width of the constant T(E,0) above Fermi level increases from (4,0) to (6,0) GeCNT. There are almost no transmissions at the Fermi level for both (7,0) and (8,0) GeCNTs. As seen in Figure 2, the widths of the zero transmission region for (7,0) and (8,0) are 0.43 and 0.52 eV, respectively and they are almost symmetrically distributed on both sides of the Fermi level. It implies that (4,0)−(6,0) GeCNTs exhibit metallic character, whereas (7,0) and (8,0) GeCNTs are semiconducting in nature. We have estimated transmission pathways for all five (n,0) GeCNTs at the Fermi level to gain insight into the local transmission.40 In the transmission pathway analysis, the total transmission coefficient is split into local bond contributions, Tij so that
T (E ) =
∑ Tij(E) i,j
where Ti,j can take up either positive or negative value. A positive value is visualized as flow of electrons from atom i to atom j, whereas the negative value is from j to i. Therefore, a negative value corresponds to the fact that the electron is backscattered along the bond. In Figure 3, the red -colored arrows are used to show the components of the transmission that are in the direction of net current flow; otherwise the arrows are blue- or violet-colored. Arrows are colored red only when Tij has a positive value. The local transmission plots clearly indicate that for (4,0), (5,0), and (6,0) GeCNTs the electron propagation from left to right electrode is smooth. Moreover, in the case of (4,0) GeCNT, the electron transport follows not D
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Figure 4. Band structures of (n,0) GeCNTs (n = 4−8).
the curve becomes flat around the maximum. Beyond the applied voltages mentioned above, the current-flow through the GeCNT drops significantly with the increase in bias voltage. Such a reduction of current due to increase in the bias voltage shows the occurrence of the NDR, which is an important property used in molecular switches and other devices. The NDR effect continues up to a certain bias voltage. In the (4,0) GeCNT device, the current drops to a small value of 1.66 μA at the bias voltage ±1.8 V. A similar NDR behavior prevails at the same bias voltage for (5,0) GeCNT, but the amount of current is slightly larger, at 4.84 μA. The I−V curve of (6,0) GeCNT shows that the NDR effect disappears at a smaller bias voltage of ±1.2 V. Therefore, as the chirality of the nanotube increases, the magnitude of the drop in current decreases. To understand the origin of the NDR effect, we analyzed the transmission spectra of (4,0), (5,0), and (6,0) GeCNTs at different bias voltages, which are shown in Figures S1−S6 (Supporting Information). It is evident from eq 2 that only electronic states in the bias window [μL(Vb), μR(Vb)] contribute to the current. In the present calculation, the average Fermi level, which is the average of μL and μR, is set as zero. It is therefore sufficient to analyze the finite part of the transmission spectra within the bias window. To understand the NDR behavior, we consider the transmission spectra of (4,0) GeCNT at the bias voltages 0, 0.8 (current peak), and 1.8 V (current valley). The zero-bias transmission spectrum consists of some wide and narrow peaks around EF. As the bias voltage increases, these peaks shift downward, and at 0.8 V the transmission area within the bias window becomes large, although the peak height reduces compared with the zero-bias spectrum. The combined effects of the increased transmission area and the appearance of resonance peak into the bias window result in the enhancement of current. When the voltage is ∼1.8 V, all peak heights within the bias window are significantly reduced. After 1.8 V, the peak height gradually rises and the current starts to increase. The suppression of the conductance channel is thus one of the factors for the NDR effect. A similar mechanism has been suggested in previous work on carbon nanotube heterojunctions44 and C60 molecular device.45
only the bond between the nearest atoms but also the bonds between the non-neighbored atoms in a particular GeC ring of the central region. For (7,0) and (8,0) GeCNTs, local transmissions are blocked, thereby reducing the conductance significantly. These results apparently suggest that (4,0), (5,0), and (6,0) GeCNTs are more metallic than (7,0) and (8,0) nanotubes. To check the chirality dependence of the conductance of GeCNTs further, we have carried out band structure calculations of all five (n,0) GeCNTs by performing the Brillouin zone integration within the Gamma (Γ)-centered Monkhorst-Pack scheme using 1 × 1 × 100 k-points. As seen in Figure 4, the band gap gradually increases from (4,0) to (8,0) GeCNT. The (4,0) nanotube has relatively a narrow band gap compared to others. This may be due to the curvature-induced σ−π hybridization, which is common in small-radius tubes.41,42 The zero-bias transmission spectrum, local transmission analysis, and band structure support the metallic character of GeCNTs in low chirality and semiconducting character in high chirality. A similar feature has been reported43 for SiCNTs in which (n,0) nanotubes with n = 7−9 are found to be semiconductors, whereas those with n = 5 and 6 have metallic characters. The current−voltage characteristics are studied for three metallic GeCNTs, namely, (4,0), (5,0), and (6,0) nanotubes. The bias voltage has been varied from −2.8 to +2.8 V, and the variations of the computed current are shown in Figure 5. All three I−V curves are similar in nature and look slightly asymmetric in the positive and negative external bias voltage regions. The flow of current from left to the right electrode refers to the positive bias, and that from right to the left is reflected by the negative bias. A point of inflection around the zero bias voltage is observed in each of three I−V curves. As the bias voltage increases in either direction, the current flowing through the GeCNT increases steadily and reaches a maximum value. For (4,0) GeCNT, the maximum value appears at the bias voltage of ±0.8 V, whereas the same is obtained at ±1.0 V for (5,0) GeCNT. In the case of (6,0) GeCNT, the current becomes maximum at −0.8 and +0.4 V, respectively, in the negative and positive bias voltage. As the chirality is increased, E
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Figure 6. MPSH states corresponding to HOMO and LUMO of (4,0) GeCNT at the bias voltages 0.0, 0.8, and 1.8 V.
between the central region and the right electrode. This, in turn, leads to the fact that the electrons entering the central region from left cannot be carried to the other end. The MPSH-LUMO state is distributed in the left electrode in the absence of any bias voltage. At the bias voltage of 1.8 V, the LUMO state is localized more at the left contact region than at the right. A similar explanation is applicable for (5,0) and (6,0) GeCNTs. The NDR behavior here may thus be attributed to the suppression of transmission peaks and different alterations of HOMO−LUMO levels under different bias voltages. Figure 7 displays the variation of the rectification ratio with the bias
Figure 5. Current−voltage (I−V) curves for (n,0) GeCNTs (n = 4− 6).
The NDR feature is further analyzed from the nature of the frontier molecular orbitals of the molecular-projected selfconsistent Hamiltonian (MPSH) of (4,0) GeCNT as a representative case. This is calculated by projecting the selfconsistent Hamiltonian onto the Hilbert space spanned by the basis functions of the atoms in the central region. However, the transmission peaks are determined by the eigenstates of the MPSH, which are linked to the poles of the Green’s function.46 The MPSH-HOMO and MPSH-LUMO of (4,0) GeCNT for the bias voltages 0.0, 0.8, and 1.8 V are shown in Figure 6. The MPSH-HOMO state is found to be linearly distributed along the central region under zero bias. When the bias voltage is raised to 0.8 V, the corresponding HOMO state shifts to the right contact. However, at the bias of 1.8 V, the MPSH-HOMO state is completely localized around the left contact. So, there is a reduction of the transmission peak due to the weak coupling
Figure 7. Rectification ratio (I+/I−) for (n,0) GeCNTs (n = 4−6) at different bias voltages.
voltage for all three GeCNTs. The I+/I− values for (4,0) and (5,0) nanotubes remain close to unity in the low-bias voltage region (0.2 to 1.2 V), which signifies that there is almost no rectification effect. However, in the case of (6,0) GeCNT device, such rectification ratios are above unity in the bias range below 1.0 V. In the bias region 1.0 to 1.8 V, the ratios for all three nanotubes drop nonlinearly to smaller values, implying larger current in the negative bias than in the other direction. F
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4. CONCLUSIONS Transmission coefficients computed from the combined DFT and NEGF methods reveal that (4,0), (5,0), and (6,0) GeCNTs are metallic, whereas (7,0) and (8,0) GeCNTs have semiconducting characters. These are supported by the band structure calculations. The transmission pathways of the electron propagation confirm that the probability of electron transferring through (7,0) and (8,0) GeCNTs is negligible, thereby reducing their conductances drastically. The I−V curves of three metallic GeCNTs are nonlinear and show NDR effects. The curves are found to be asymmetric with respect to the direction of the applied bias voltages. In (4,0) GeCNT, the drop in current is very sharp near ±0.8 V bias, whereas for (5,0) and (6,0) nanotubes the onsets of NDR are comparatively slow and I−V curves are relatively broad. The current through (4,0) GeCNT at +1.8 V bias voltage reduces drastically to near-zero value. For the (6,0) GeCNT, the NDR effect disappears when the bias voltage increases from ±1.2 V. The NDR behavior is manifested due to the suppression of transmission peaks in the bias window and the localization of MPSH states. The rectification ratio reduces nonlinearly from unity in the bias voltage region 1.0 to1.8 V for all three nanotubes.
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ASSOCIATED CONTENT
* Supporting Information S
Transmission spectra of (4,0), (5,0), and (6,0) GeCNTs at different bias voltages. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interests.
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ACKNOWLEDGMENTS The authors are grateful to one of the reviewers for some clarifications and suggestions. REFERENCES
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