Article pubs.acs.org/JPCC
Electronic-Transport Properties of Single-Walled Zigzag SiGe Nanotubes Pabitra Narayan Samanta and Kalyan Kumar Das* Department of Chemistry, Physical Chemistry Section, Jadavpur University, 188, Raja S.C. Mallik Road, Kolkata 700 032, India ABSTRACT: Electronic-transport properties of zigzag singlewalled SiGe nanotubes of four different chiralities are studied by using density functional theory combined with nonequilibrium Green’s function formalism. Band structures of (n,0) SiGeNTs (n = 4−7) are computed using 1 × 1 × 100 kpoint sampling. Transmission coefficients of these devices are estimated at various positive and negative bias voltages within ±2.0 V. The (4,0) SiGeNT shows metallic character, while the high-chiral (7,0) nanotube is expected to be of the semiconducting type. The current−voltage (I−V) curves of these systems are constructed for different bias voltages. Except (6,0) nanotube, all SiGe nanotubes show negative differential resistance, which is explained from the appearance or disappearance of transmission peaks within the bias window. The dependence of the length of the central region on the I−V characteristic has been tested on (4,0) SiGeNT using three- and four-cell tubes. As the length of the central region increases, the peak-to-valley ratio of the I−V curve increases from 1.92 to 5.32; however, the nature of the curves remains almost the same. The rectifying performances of these devices are investigated by calculating the rectification ratio (I+/I−). al.19 have performed molecular dynamics simulations to demonstrate that free-standing nanofilm may bend into a nanotube with Ge as inner layer. It is also observed that rolled up nanotubes can accommodate a high level of strain. The existence and stability of single walled SiGeNTs are studied by Rathi and Ray20 using systematic hybrid density functional theory (DFT) calculations. The structure, energetic, and thermal evolution of SiGeNTs in both armchair and zigzag structures (n = 4−10) have been investigated by Liu et al.21 using the ab initio method and classical molecular dynamics simulations. These authors have also studied the effects of structure, temperature, and strain rate on mechanical properties of these nanotubes.22 It is predicted that by tuning and controlling these variables, SiGeNTs can be used in various fields like nanoelectronic devices, novel nanosolar cells, and photovoltaics, nanojunctions, and nanodevice components. The melting-like temperatures of Ge-substituted silicon nanotubes are found to decrease with the increase in Ge-concentration. Single-crystal Ge−Si core−shell nanowires have been synthesized23 inside a horizontal quartz-tube furnace in the chemical vapor deposition system with the use of gold nanocluster as a catalyst. On the basis of the electrically active nanotubular building blocks, electrical devices are described for future realization of the nanofluid FET. Song et al.24 have shown that Si/Ge double-layered nanotube can serve as an anode for lithium ion batteries to enable improvements in structural stability and electrochemical kinetics. Very recently, systematic ab initio studies of a Li
1. INTRODUCTION After the discovery of single-walled carbon nanotubes (SWCNTs) by Iijima,1 efforts are continuously been made to prepare 1-D nanostructural materials such as nanowires and nanotubes based on other group 14 elements due to their potential applications in semiconductor industries. With the advancement of nanotechnology and the progress in computational methodologies, there has been a strong urge to design the molecular devices and understand their transport properties, which are known to be related to quantum phenomena. An extensive review on the electronic and spin transport in molecular devices has been made by Kim et al.2 The role of molecular orbitals in quantum transport through molecular devices and the effects of electric and magnetic fields on these molecular orbitals are important to study.3 Over the past couple of decades, electronic/spintronic properties of single molecules, nanowires, nanotubes, and nanoribbons toward the design of nanodevices like switching devices, spin-valve, negative differential resistance (NDR) devices, and nanosensors are studied.2−7 Silicon nanotubes (SiNTs) have shown high mechanical strength, thermal and chemical stabilities, and excellent heat conduction.8,9 So, these are used as biochemical sensors and disease detectors and in electronics.10,11 Rathi and Ray12,13 have confirmed that both the armchair and zigzag SiNTs are stable and semiconducting in nature. Germanium being in the same group also produces germanium nanotubes (GeNTs), which are useful in nanomaterial-based technology.14,15 Silicon−germanium nanowires have been successfully used as high-performance field-effect transistors and sensors and in other electronic devices.16−18 Schmidt and Eberl15 were successful in preparing SiGe nanotubes of length 20 μm with a diameter of 530 nm by using two different methods. Zang et © 2014 American Chemical Society
Received: April 29, 2014 Revised: July 10, 2014 Published: July 15, 2014 18153
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where f is the Fermi function, μL,R are 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
atom and four types of SiGe (6,6) nanotubes of lengths about 20 Å have been carried out by Wanaguru and Ray25 to explore SiGeNT’s viability as anode material in Li-ion batteries. Following the importance of SiGeNTs as materials for devices, we urge to investigate the electronic properties of zigzag SiGeNTs and devices. The present study explores the electronic transport properties of (n,0) SiGeNTs for n = 4−7 using a combined DFT and nonequilibrium Green’s function (NEGF) formalism. Transmission energy spectrum, band structure, density of states (DOS), and projected density of states (PDOS) are computed. The I−V characteristics and the rectification ratio (I+/I−) of four SiGeNT devices are studied under different bias voltages.
T (E , Vb) = Tr[ΓL(E , Vb)G(E , Vb)ΓR (E , Vb)G†(E , Vb)] (2)
where G(E,Vb) is the Green’s function of the two-probe model and ΓL/R are the coupling matrices. The Green’s functions are constructed using the DFT Hamiltonian obtained from a given electron density. The DFT Hamiltonian can be expressed in the following form ⎡ HL + Σ L VL ⎤ 0 ⎢ ⎥ ⎢ VL† HC VR ⎥ ⎢ ⎥ † ⎢⎣ 0 VR HR + Σ R ⎥⎦
2. COMPUTATIONAL DETAILS A two-probe model consisting of two semi-infinite electrodes (L and R) coupled to a central region (C) has been adopted to investigate the chirality-dependent electronic-transport properties of zigzag SiGeNTs. Geometries of the perfect (n,0) SiGeNTs are optimized first; then, the optimized structures are used to construct the two-probe model through repetition of the (n,0) SiGeNT unit cell. In the present study, the lengths of the central region and left/right electrodes are taken as three periods (20.88 Å along z) and one period (6.96 Å along z), respectively. Structural relaxations and total energies of (n,0) SiGeNTs are calculated using the SIESTA-3.126 package based on pseudopotential DFT calculations.27,28 The double-ζ polarized (DZP) basis sets for both Si and Ge along with the norm conserving Troullier−Martins pseudopotentials29 are used in the present calculations. The valence electrons are described by localized pseudo-atomic orbitals (PAOs). The exchange correlation potential is approximated by the Perdew− Burke−Ernzerhof (PBE) parametrization of GGA function.30 Electrostatic potentials are computed on a real-space grid with a mesh cutoff energy of 150 Ry. Relaxed geometries are obtained by minimizing the total energy using Hellmann−Feynman forces including Pulay-like corrections. In the full course of geometry optimization, the relaxation is performed by conjugate gradient until the forces are smaller than 0.05 eV/Å. Transport properties of the above two-probe model systems are explored using the TranSIESTA code as implemented in SIESTA-3.1 program, which is based on the combination of DFT and Keldysh NEGF method.31,32 The contact dependency is minimized by using SiGeNT similar to the central region in the two-probe model. 33 Moreover, the junction is so constructed that the periodic replicas of SiGeNT along the direction parallel to the electrode edge are separated by nanometer range. 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 k-point sampling in the transport direction (z-direction), and the 1 × 1 × 100 kpoint 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 Landauer−Büttiker formula34 I(Vb) =
2e h
∫μ
μR L
T (E , Vb)[f (E , μL ) − f (E , μR )] dE
At first, the SCF calculations of the SiGeNT electrodes in periodic boundary conditions are performed to obtain the density matrix, the real-space Hamiltonian (HL and HR), and the self-energies (ΣL and ΣR). These results are used in the subsequent calculation to compute the surface Green’s function of the electrodes. In the next step, the previous procedure is repeated for the whole system (L/R and C) to obtain the nonequilibrium electron density matrix (under open boundary conditions) of the system and the remaining parts of the Hamiltonian (HC, VL, and VR). VL/R are the interaction terms of the electrodes with the central region. Finally, the electron transmission function and the current flowing through the (n,0) SiGeNT nanodevices at finite bias (Vb) are calculated using the postprocessing tool TBTrans, as included in the TranSIESTA package.
3. RESULTS AND DISCUSSION SiGeNT devices of chiralities (4,0)−(7,0) are optimized at the PBE/DZP level of theory using 1 × 1 × 100 k-point sampling. Figure 1 shows the two-probe model for a representative (7,0)
Figure 1. Two-probe model of the (7,0) SiGeNT device.
SiGeNT device. In the zigzag nanotube lattice, alternating Si and Ge atoms have only Ge or Si as nearest neighbors. Upon optimization, the tube surface gets buckled because the more electronegative Si atoms move radially outward, while Ge atoms shift inward of the curved hexagonal layer. Each Ge atom is bonded to two symmetrically equivalent Si atoms having an average Ge−Si bond length of 2.38 Å, and the third Ge−Si
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bond is slightly longer by ∼0.015 Å. For all models, the Si−Ge bond length varies from 2.36 to 2.41 Å, which is in accordance with the plane-wave DFT based calculations of Liu et al.21 using 1 × 1 × 5 k-point mesh and PW91 functional. Because the hybridizations of Si and Ge are different, there is a variation in Si−Ge−Si and Ge−Si−Ge bond angles between 95 and 121°, which results in slight distortions in SiGeNT compared with the pure carbon nanotube. Again, these bond angles do not vary much with the chirality of the nanotube. The degree of buckling reduces with the increase in radius of the tube. Equilibrium transport properties under zero bias voltage for different zigzag SiGeNTs with chiralities n = 4−7 are investigated first. The probability for electrons with certain energy transferring from left electrode to the right electrode, when no bias voltage is applied, is proportional to the transmission coefficient, T(E,0). The computed transmission coefficients for each of the (n,0) SiGeNTs (n = 4−7) are plotted in Figure 2 as a function of energy. The Fermi level has
semiconducting character for the high-chiral SiGeNT. A steplike behavior is observed in each of (n,0) SiGe nanotubes. This feature is related to the available conductance channels from bands. To examine the conductance behavior of SiGeNTs, we have carried out band-structure calculations of all four (n,0) SiGeNTs by performing the Brillouin zone integration within the Gamma (Γ)-centered Monkhorst−Pack scheme using 1 × 1 × 100 k points. Figure 3 shows that there are almost no band
Figure 3. Band structures of (n,0) SiGeNTs (n = 4−7).
gaps in (4,0), (5,0), and (6,0) nanotubes, while in (7,0) SiGeNT, there is no crossing at the Fermi level and a gap appears. At the Γ point, the band gap is estimated to be 0.12 eV, and at the Z point it is ∼0.46 eV for (7,0) SiGeNT. This compares well with the average HOMO−LUMO gap computed for zigzag SiGeNTs by Rathi and Ray.20 As reported by these authors, SiGe nanotubes in the zigzag form have lower band gaps compared with these in the armchair conformation possibly due to delocalized nature of the electrons. It may be mentioned that SiCNTs, GeCNTs, and SnCNTs have different band structures,35,36 and their dependences on chirality are also somewhat different. Electronic density of states (DOS) has been calculated to confirm the nature of transmission through SiGeNTs of four different chiralities. Figure 4a shows the zero bias DOS spectra for (n,0) SiGeNTs (n = 4−7), while the PDOS of Ge and Si in the nanotubes are displayed in Figure 4b. In the case of (4,0) SiGeNT, there is a steady electron distribution near the Fermi level, which corresponds to a transmission coefficient of four in the transmission spectrum. With the increase in chirality, this distribution decreases, and consequently the transmission coefficient value reduces as revealed from the transmission spectrum. However, in all SiGeNTs there is a significant contribution of both the atoms around the Fermi level, which is in accordance with the symmetric nature of the PDOS spectrum of Ge and Si atoms. In (7,0) SiGeNT, the energy gap between the highest occupied and the lowest unoccupied band states is nonzero, which corroborates with the energy range of zero transmission, as shown in Figure 2. NDR is a significant quantum transport phenomenon in electronic systems like Esaki and resonant tunneling diode, which has been proposed for the low-noise amplification, highfrequency oscillations, analog-to-digital conversion, and digital logic.37,38 NDR is generally characterized by a discontinuity in the monotonic increase in the current with the applied bias voltage. In the present model, where charging and electron− phonon couplings are neglected, it arises due to various electronic features including orbital alignment. It has been
Figure 2. Zero bias transmission spectra of (n,0) SiGeNTs (n = 4−7).
been set to EF = 0 for simplicity. Transmission spectra show that at the Fermi level the transmission coefficient for (4,0) SiGeNT is considerably large, showing its metallic character. However, it decreases with the increase in chirality. The (7,0) nanotube shows zero transmission, although the width of the zero transmission is only 0.1 eV. Therefore, one may expect 18155
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Figure 4. (a) DOS and (b) PDOS spectra of (n,0) SiGeNTs (n = 4−7) at zero bias.
The origin of the NDR effect may be related to the changes in the transmission spectra of the respective SiGeNTs at different bias voltages. The electronic states, which are present in the bias window [μL(Vb), μR(Vb)] region, contribute to the total current through the nanotube. The average Fermi level, which is the average of μL and μR, is set to zero. It is therefore sufficient to analyze the finite part of the transmission spectra within the bias window. Figure 6 shows the transmission spectra of (4,0) SiGeNT at three bias voltages such as +0.2 (current peak), +0.8 (current valley), and +1.6 V. At the bias voltage 0.2 V, there is a transmission peak within the window. The transmission area within the bias window reduces at 0.8 V, thereby lowering the current flow through the nanotube. However, at the higher voltage 1.6 V, a couple of broad peaks appear and the current increases steadily. The observed NDR effect is, therefore responsible for the suppression of conduction channel at a certain bias, which is in accordance with the NDR mechanism, as shown by other nanodevices like GeCNT,35 SnCNT,36 and C60 molecular devices.44 To gain a further insight into the origin of NDR, we also calculated the PDOS of Ge and Si atoms for (4,0) SiGeNT under +0.2, +0.8, and +1.6 V, shown in Figure 7. At +0.2 V bias voltage, the contribution of Si atoms within the bias window is more than that of Ge atoms. As the bias voltage reaches +0.8 V, the PDOS peaks of Si and Ge atoms inside the bias window are significantly depressed. Consequently, the degree of tunneling for the electron transfer from one electrode to another reduces, resulting in the decrease in current with the increase in the bias
found that the asymmetric couplings of CNTs with phenylethyl oligomers and pyrrollo pyrrole are responsible for NDR.39 The origin of the NDR effect in the I−V curve of the armchair single-walled SiCNT has been identified as being due to changes of DOS by the applied bias voltage.40,41 In some cases, the fluctuation of the transmission coefficient with the bias voltage gives rise to the NDR effect.42,43 The computed I−V curves of (4,0)−(7,0) SiGeNTs are given in Figure 5. The bias voltage has been varied from −2.0 to +2.0 V. Except for (6,0) SiGeNT, all curves show NDR effects, and they are not always symmetric in the positive and negative bias voltage regions. With the increase in the applied voltage in either direction, the current flowing through these three SiGeNT devices increases steadily and reaches a maximum value. The maximum for (4,0) SiGeNT is located at the bias voltage of ±0.2 V, while those for (5,0) and (7,0) are obtained at ±0.5 and ±0.6 V, respectively. The maximum is found to be broadening with the chirality of the nanotube. Beyond the previously mentioned voltages, the current flow through the nanotube drops to a minimum value with the increase in the bias voltage. Such a drop of current due to increase in the applied voltage results in the onset of the occurrence of the NDR effect. The largest drop is noted in (4,0) SiGeNT, where the minimum current becomes 3.4 μA at the negative bias voltage of 1.4 V. However, the I−V curve of the (6,0) nanotube does not show any maximum or minimum; only the slope gets reduced in the range of voltage 0.5 to 0.9 V. 18156
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Figure 6. Transmission spectra of the (4,0) SiGeNT device under bias voltage 0.2 (current peak), 0.8 (current valley), and 1.6 V. The vertical dotted lines represent bias windows.
The effect of length of the central region on the transport properties has been investigated by constructing a similar twoprobe model for (4,0) SiGeNT, which consists of four periodic unit cells (32 Ge/32 Si atoms) instead of three-cell (24 Ge/24 Si atoms) in the central region. Figure 8 compares the I−V characteristics for these two three- and four-cell (4,0) SiGeNTs in the positive bias region only. It may be noted that nanotubes of both lengths show the NDR effect. In the case of three-cell SiGeNT, the NDR behavior is observed in the bias range of 0.8
Figure 5. Current−voltage (I−V) curves for (n,0) SiGeNT (n = 4−7) devices.
voltage. When the bias voltage is reached at +1.6 V, the PDOS peaks extend into the bias window and the electrons can transfer easily through tunneling, which leads to the higher current through the SiGeNT. Therefore, the variation of density of states is another source of NDR for these nanotubes. 18157
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peak-to-valley ratios for three- and four-cell SiGeNTs are 1.93 and 5.32, respectively, which indicates a strong dependency on the transport properties of these nanotubes. One can thus tune the NDR by choosing different lengths of SiGeNTs. Figure 9 displays the variation of the rectification ratio with the bias voltage for all four SiGeNTs. The I+/I− values for (6,0)
Figure 9. Rectification ratio (I+/I−) for (n,0) SiGeNT (n = 4−7) devices at different bias voltages.
and (7,0) nanotubes remain close to unity in the whole bias voltage region, which signifies that there is almost no rectification effect. However, in the case of (4,0) and (5,0) SiGeNT devices, such rectification ratios are above unity in the bias range below 2.0 V.
4. CONCLUSIONS First-principle quantum-transport calculations based on a combined method of DFT and NEGF are used to investigate the characteristics of the electron transport through devices made of zigzag (4,0)−(7,0) SiGe nanotubes. A certain degree of buckling has been noted in all (n,0) SiGeNTs upon optimization of the tube geometry, and the magnitude of the buckling decreases with the increase in the diameter. For all models, the computed Si−Ge bond length varies from 2.36 to 2.41 Å, which agrees well with the plane-wave DFT calculations using 1 × 1 × 500 k-point mesh and PW91 functional. There is a nonzero transmission coefficient at the Fermi level for the first three SiGeNTs showing their metallic characteristics when there is no bias voltage. (7,0) SiGeNT shows no transmission probability of electrons at the Fermi level having a width of 0.1 eV. A semiconducting behavior of the high-chiral device is expected. These features are supported by the computed band structures and DOS spectra. The band gap computed at the PBE/DZP level of theory and 1 × 1 × 100 Monkhorst−Pack kpoint sampling compares well with the average HOMO− LUMO gap obtained at the B3LYP/LANL2DZ level of theory. The current−voltage curves show strong nonlinearities. Except (6,0) SiGeNT, all other nanotubes exhibit NDR effects over a range of bias voltage within ±2.0 V, which originate from the suppression of conduction channel within the bias window. However, the NDR effect is also observed with increasing the length of the central region, and the peak-to-valley ratio is increased two to three times, thereby showing a strong length dependency on the transport properties of SiGeNTs. The asymmetric nature of the rectification ratio (I+/I−) is observed for (4,0) and (5,0) SiGeNTs, while for the nanotubes of bigger
Figure 7. Bias-dependent PDOS spectra of the (4,0) SiGeNT device.
Figure 8. Current−voltage (I−V) curve for the (4,0) SiGeNT device with four-cell in the central region. The inset shows the I−V curve for the (4,0) SiGeNT device with three-cell.
to 1.4 V. When the length is enhanced to four-cell (27.84 Å), the system shows NDR in the bias range 1.2 to 2.6 V. The 18158
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size, the ratio remains close to unity. The present findings could be helpful to design the NDR based nanodevices.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interests.
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dx.doi.org/10.1021/jp504169t | J. Phys. Chem. C 2014, 118, 18153−18159
