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Giant low bias negative differential resistance induced by nitrogen doping in graphene nanoribbon
Chemical Physics Letters 554 (2012) 172–176
Contents lists available at SciVerse ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Giant low bias negative differential resistance induced by nitrogen doping in graphene nanoribbon Peng Zhao a,⇑, De-Sheng Liu b,c, Shu-Juan Li a, Gang Chen a,⇑ a
School of Physics and Technology, University of Jinan, Jinan 250022, China School of Physics, State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, China c Department of Physics, Jining University, Qufu 273155, China b
a r t i c l e
i n f o
Article history: Received 13 September 2012 In final form 15 October 2012 Available online 23 October 2012
a b s t r a c t By applying nonequilibrium Green’s function formalism in combination with density functional theory, we have investigated the electronic transport properties of armchair graphene nanoribbon devices with periodic nitrogen-doping. Giant negative differential resistance behaviors with peak-to-valley ratio up to the order of 105 can be obtained in the mV bias regime by tuning the position and the concentration of the dopants. The negative differential resistance behavior is understood in terms of the evolution of the transmission spectrum and band structures with applied bias combined with the symmetry analyses of the Bloch wave functions of the corresponding subbands. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction One of the interesting quantum electronic transport phenomena is the negative differential resistance (NDR), which is characterized by an increasing followed by a deceasing current with increasing bias in some specific bias range. NDR is the basis for a number of device applications including high-frequency oscillators, analog-to-digital converters, memory, and logic gates [1–3]. In recent years, NDR has been observed in a variety of molecular devices [4–20], which typically occurs at a relatively high bias. However, for integrated circuits a low bias mV regime is desirable to reduce power consumption [21]. Moreover, most of the observed peak-to-valley ratio (PVR) is insufficient for practical application. Therefore, it is important and urgent to develop low bias NDR molecular devices with larger PVR. Very recently, we have demonstrated that Li-doping can significantly improve the NDR performance of fullerene dimer and shift the NDR peak position (Vpeak) from 0.7 V down to 0.1 V while the PVR increases from 2.1 to 30.5 [22]. Graphene, a two-dimensional lattice of carbon atoms, has drawn widespread interest recently due to its unique electronic transport properties and its potential technological applications [23,24]. The corresponding quasi-one-dimensional structure graphene nanoribbons (GNRs) can be obtained by conventional lithographic techniques in semiconductor industry. GNRs are recognized as the most promising materials in the carbon family for building new nanoscale electronic devices. Many interesting ⇑ Corresponding authors. Tel.: +86 531 82769008. E-mail addresses: [email protected] (P. Zhao), [email protected] (G. Chen). 0009-2614/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2012.10.045
physical properties based on GNRs, such as NDR [25,26], rectification [27,28], spin filtering [28,29], field-effect characteristic [30,31], etc., have been demonstrated. It has also been shown that the properties of GNRs can be tuned in many ways, such as doping [25,29], edge modification [32], adsorption [33], gate voltage [27], and stretch [34]. In particular, chemical doping, which is crucial for building logic functions and complex circuits [35], is one of the most frequently adopted ways to modify the properties of GNRs. Because it possess similar atomic radius to that of carbon, nitrogen (N) is the most commonly used dopants. More recently, the N-doped graphene has been synthesized in experiments [36–39]. In the present Letter, based on density functional theory (DFT) calculations and nonequilibrium Green’s function (NEGF) technique, we investigate the electronic transport properties of armchair GNR (AGNR) molecular devices with periodic N-doping. Our results show that low bias NDR behaviors are obtained in these devices, which are strongly dependent on the position (i.e., the dopants are placed either at the centers or at the edges) and the concentration of the dopants. Especially, giant NDR behaviors with PVR up to the order of 105 can be obtained in the mV bias regime by tuning the position and the concentration of the dopants. 2. Calculation methods and model As shown in the insert of Figure 1, in theoretical simulations, such a system is divided into three regions: a left electrode, a scattering region, and a right electrode. Each electrode marked by lime is modeled by a 7-AGNR with two repeated carbon unit cells along the transport direction (z), where 7 is the number of carbon dimer lines across the ribbon width [40]. The scattering region marked by pale green consists of a 7-AGNR of four unit cell length. All edge
P. Zhao et al. / Chemical Physics Letters 554 (2012) 172–176
Figure 1. Calculated I–V characteristics of M1C, M1E and GNR without doping. Schematic view of a periodic N-doped 7-AGNR molecular device is shown in the insert. The gray, blue and white spheres indicate the carbon, nitrogen and hydrogen atoms, respectively. The labels C and E stand for the doping site at the center and the edge, respectively. For clarity, the current through GNR without doping has been multiplied by 107. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.)
173
carbon atoms are saturated with hydrogen atoms to eliminate the dangling bonds. Large supercell dimensions (30 10 Å in the plane perpendicular to the transport direction are used to avoid the interaction between the AGNR and its neighboring images. We consider two typical doping configurations [41]: M1C and M1E, in which one center and one edge carbon atom per two carbon unit cells is substituted by an N atom, respectively. The quantum electronic transport properties are performed using the NEGF + DFT method as implemented in the ATOMISTIX TOOLKIT (ATK) package [42,43]. In our calculations, the exchange–correlation potential is described by the local-density approximation (LDA) [44]. The Troullier–Martins norm-conserving pseudopotentials [45] are adopted to represent the core electrons. The valence electrons are expanded in a single zeta plus polarization basis set (SZP) for all atoms and the cutoff energy is set to 150 Ry to achieve a balance between the calculation time and the accuracy. The electrode calculations are performed under periodical boundary conditions in all directions, and a 1 1 100 kpoint mesh in the x, y, and z direction is sampled in the Brillouin zone according to the Monkhorst–Pack scheme. Before calculating the electronic transport properties, all the structures are fully relaxed until the maximum absolute force is less tan 0.01 eV/Å. The nonlinear current through the device is calculated using the Landauer–Bütiker formula [46]:
Figure 2. Transmission spectra and band structures of both left and right electrodes for (A) M1C at 0, 0.3 and 0.7 V and (B) M1E at 0, 0.25, and 0.6 V. The Fermi level EF is set to be zero. The violet dashed lines represent the electrochemical potentials of the left and right electrodes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.)
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P. Zhao et al. / Chemical Physics Letters 554 (2012) 172–176
IðVÞ ¼ ð2e=hÞ
Z lR
TðE; VÞ½fL ðEÞ fR ðEÞdE
ð1Þ
lL
where e is the elementary electron charge, h is the Planck’s constant, fL,R are the Fermi distribution functions of electrons in the left and right electrodes, lL;R ðVÞ ¼ EF eV=2 are the electrochemical potentials of the left and right electrodes, EF is the Fermi level of the system which is set to be zero in our calculations. The energy region, ½lL ðVÞ; lR ðVÞ, that contributes to the total current integral is referred to as the bias window. TðE; VÞ is the transmission function defined as:
TðE; VÞ ¼ Tr½CL GR CR GA
ð2Þ
R;A
where G are the retarded and advanced Green’s functions of the scattering region, CL;R ¼ i½RRL;R RAL;R are the broadening functions, RRL;R are the corresponding self-energies of the scattering region, which contain all the effects of the electrodes. 3. Results and discussion 3.1. Effect of doping position After geometry optimizations, M1C and M1E still retain their planarity [47]. In contrast to the length of C–H bond in the pristine GNR (1.10 Å), the N–H bond shrinks to 1.02 Å in M1E due to the H atom moving inward from its original position. In M1C, the doped N atom rebonds with three adjacent C atoms. Only the local atomic structure of the GNR is affected by N doping. In Figure 1, we present the current–voltage (I–V) characteristics for M1C and M1E. It is clear that obvious NDR behaviors are observed in two curves. The currents in M1C and M1E increase linearly under low bias and then drop dramatically in the 0.3–0.7 V and 0.25–0.6 V range, respectively. Very interestingly, for M1C, the current is almost quenched to zero in the 0.7–1.0 V range. As a result, a giant PVR up to 104 is obtained in M1C, while the PVR in M1E is only about 18. For comparison, the I–V characteristic of GNR without doping is also presented and no NDR appears. It is reasonable to conclude that the observed NDR behaviors originate from N doping. To understand the mechanism of the observed NDR behavior and its doping-position-dependence, in Figure 2, we give the bias-dependent transmission spectra and band structures of both left and right electrodes for M1C at 0, 0.3, 0.7 V (Figure 2A) and M1E at 0, 0.25, 0.6 V (Figure 2B), respectively. From Figure 2, comparing with the band structures of the undoped case (which is not shown here), we can see that a roughly half-filled impuritysubband cutting the EF, labeled by N, appears in the band structures of two N-doped electrodes, respectively. Meantime, the p- and p⁄-subbands are located far away from the EF. This implies that the N-subband is the only subband near the EF and thus is responsible for the electron transmission under low bias range. Moreover, the steplike T(E, V = 0) curve can be perfectly correlated with the band structure. Since each subband is nondegenerate, T(E, V = 0) increases in steps of one unit. There is a large gap below the EF in T(E, V = 0) due to the absence of bands in this energy range. When the bias is applied, the subbands of the left electrode shift down, while those shift up in the right electrode. As a result, the overlap between N-subbands of two electrodes reduces significantly. Accordingly, the transmission T(E, V = 0.3 V) and T(E, V = 0.25 V) for M1C and M1E around the EF decreases, respectively. However, the currents increase from zero to their maximum values since the transmission integrals have reached their maximum values in the bias window of two cases. Here, we must point that the region of the bias window is actually [V/2, V/2] since the EF has been set to be zero. When the bias increases further, the overlap between N-subbands of two electrodes reduces again. As
Figure 3. Top and end isosurface plots of the C-point Bloch wave functions of Nand p ⁄-subbands of electrode for (A) M1C and (B) M1E. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.)
a result, the currents in M1C and M1E start to drop, respectively. Then NDR occurs. From the bottom panels of Figure 2, we can see that the N-subband of the right electrode rises to align with the p⁄-subband of the left electrode in two cases, respectively. However, as we can see, T(E, V = 0.7 V) for M1C is zero in the bias window, while T(E, V = 0.6 V) for M1E is nonzero but very small near the right electrochemical potential in the bias window. This implies that the electron tunneling from thep⁄-subband of the left electrode to the N-subband of the right electrode is forbidden in M1C, while there is very weak tunneling in M1E. This is the reason that the PVR of M1E is far less than that of M1C. In order to understand the abnormal tunneling phenomenon, in Figure 3, we give the isosurface plots of the C-point Bloch wave functions of the N- and p⁄-subbands of electrode for M1C and M1E, respectively. As shown in Figure 3A, for M1C, the N- and p⁄-subbands have even and odd parity under the yz midplane mirror operation, respectively. Since two subbands have an opposite symmetry, the tunneling between them is forbidden [48,49]. As a result, when 0.7 V bias is applied, the N- and p⁄-subbands cannot couple with each other to contribute to the transmission, and thus the current is essentially zero in the 0.7–1.0 V range. On the contrary, for M1E, there is no definite parity in the N- and p⁄-subbands since it does not exist yz midplane mirror due to the edge-doping, as shown in Figure 3B. As a result, there is a weak coupling between these two subbands, leading to the weak electron transmission, as shown in Figure 2f.
P. Zhao et al. / Chemical Physics Letters 554 (2012) 172–176
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Figure 4. (A) Calculated I–V characteristics and (B) band structures of electrode for M0C, M1C, M2C and M3C at equilibrium state. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.)
3.2. Effect of doping concentration Next, in order to realize NDR in mV bias regime, we investigate the effect of the concentration of dopants on NDR. Since center-doping can give rise to much larger PVR, we concentrate on this doping type. Limited by the computational sources, we just further calculate the transport properties of another three doping cases: M0C, M2C and M3C, in which one center carbon atom per one, four, and six carbon unit cells is substituted by an N atom, respectively. The corresponding I–V curves are shown in Figure 4A. As we can see, all systems show NDR and the peak position, Vpeak, strongly depend on the N-doping concentration. The Vpeak gradually moves down to
lower bias range when the doping concentration is decreased: M0C is 0.8, M1C is 0.3, M2C is 0.1 and M3C is 0.03 V, respectively. In addition, both the PVR values of the M2C and M3C are up to the order of 105, which are much bigger than those reported values [4–17]. Moreover, it is clear that the current drops more rapidly beyond the Vpeak with the decrease of doping concentration. This NDR fast response to the applied electric-field could be applied to fast switching in certain types of electronic devices [50]. To understand the doping-concentration-dependence of NDR, in Figure 4B, we give the band structures of electrode for M0C, M1C, M2C and M3C at equilibrium state, respectively. Two distinct features can be addressed as follows: (1) Except for M0C, the
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P. Zhao et al. / Chemical Physics Letters 554 (2012) 172–176
Foundation of Shandong Province of China (Grant No. ZR2009AL004) and the Doctoral Foundation of University of Jinan, China (Grant No. XBS1004). References
Figure 5. Calculated bandwidth (DE) of N-subband and predicted NDR peak position (Vpeak) for M1C, M2C, M3C, M4C and M5C. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.)
N-subband is the only one in the vicinity of the EF. For M0C, the Nand p⁄-subbands cross at the EF. Therefore, the electron can tunnel from the N-subband (p⁄-subband) of the left electrode to the corresponding N-subband (p⁄-subband) of the right electrode in a wide energy range. As a result, NDR can only occur at a relatively high bias (Vpeak = 0.8 V) in M0C. (2) The bandwidth (DE) of N-subband decreases gradually with the decrease of doping concentration. Since the NDR is originated from the matching and mismatching of N-subband between two electrodes (M0C is an exception, in which the p⁄-subband also play an important role), the Vpeak is determined by the corresponding DE of N-subband: V peak ¼ DE=ð2eÞ. The calculated DE of N-subband is 0.64, 0.19, and 0.06 eV from M1C to M3C, respectively. Therefore, the predicted Vpeak is 0.32, 0.10, and 0.03 V, which is in good agreement with the calculated values. When the doping concentration further decreases, it is reasonable to conclude that the DE of N-subband will continue to drop, and the Vpeak will enter the mV bias regime. As shown in Figure 5, when one center carbon atom per eight (M4C) and ten (M5C) carbon unit cells is substituted by an N atom, NDR occurs at 8.2 and 2.5 mV, respectively. 4. Conclusion In summary, we have investigated the electronic transport properties of periodic N-doped AGNR molecular devices by applying NEGF + DFT. The I–V characteristics exhibit robust NDR behaviors, which are strongly dependent on the position and the concentration of the dopants. The center-doping gives rise to better NDR performance than the edge-doping. The reason can be attribute to the inhibition of tunneling between the N-subband of the right electrode and the p⁄-subband of the left electrode, which have an opposite symmetry. Moreover, giant NDR behaviors with PVR up to the order of 105 can be obtained in the mV bias regime by tuning the position and the concentration of the dopants. The present findings might be useful for the application of the GNRs in the field of low bias NDR molecular devices. Acknowledgments This Letter was supported by the National Natural Science Foundation of China (Grant No. 11104115), the Natural Science
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Contents lists available at SciVerse ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Giant low bias negative differential resistance induced by nitrogen doping in graphene nanoribbon Peng Zhao a,⇑, De-Sheng Liu b,c, Shu-Juan Li a, Gang Chen a,⇑ a
School of Physics and Technology, University of Jinan, Jinan 250022, China School of Physics, State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, China c Department of Physics, Jining University, Qufu 273155, China b
a r t i c l e
i n f o
Article history: Received 13 September 2012 In final form 15 October 2012 Available online 23 October 2012
a b s t r a c t By applying nonequilibrium Green’s function formalism in combination with density functional theory, we have investigated the electronic transport properties of armchair graphene nanoribbon devices with periodic nitrogen-doping. Giant negative differential resistance behaviors with peak-to-valley ratio up to the order of 105 can be obtained in the mV bias regime by tuning the position and the concentration of the dopants. The negative differential resistance behavior is understood in terms of the evolution of the transmission spectrum and band structures with applied bias combined with the symmetry analyses of the Bloch wave functions of the corresponding subbands. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction One of the interesting quantum electronic transport phenomena is the negative differential resistance (NDR), which is characterized by an increasing followed by a deceasing current with increasing bias in some specific bias range. NDR is the basis for a number of device applications including high-frequency oscillators, analog-to-digital converters, memory, and logic gates [1–3]. In recent years, NDR has been observed in a variety of molecular devices [4–20], which typically occurs at a relatively high bias. However, for integrated circuits a low bias mV regime is desirable to reduce power consumption [21]. Moreover, most of the observed peak-to-valley ratio (PVR) is insufficient for practical application. Therefore, it is important and urgent to develop low bias NDR molecular devices with larger PVR. Very recently, we have demonstrated that Li-doping can significantly improve the NDR performance of fullerene dimer and shift the NDR peak position (Vpeak) from 0.7 V down to 0.1 V while the PVR increases from 2.1 to 30.5 [22]. Graphene, a two-dimensional lattice of carbon atoms, has drawn widespread interest recently due to its unique electronic transport properties and its potential technological applications [23,24]. The corresponding quasi-one-dimensional structure graphene nanoribbons (GNRs) can be obtained by conventional lithographic techniques in semiconductor industry. GNRs are recognized as the most promising materials in the carbon family for building new nanoscale electronic devices. Many interesting ⇑ Corresponding authors. Tel.: +86 531 82769008. E-mail addresses: [email protected] (P. Zhao), [email protected] (G. Chen). 0009-2614/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2012.10.045
physical properties based on GNRs, such as NDR [25,26], rectification [27,28], spin filtering [28,29], field-effect characteristic [30,31], etc., have been demonstrated. It has also been shown that the properties of GNRs can be tuned in many ways, such as doping [25,29], edge modification [32], adsorption [33], gate voltage [27], and stretch [34]. In particular, chemical doping, which is crucial for building logic functions and complex circuits [35], is one of the most frequently adopted ways to modify the properties of GNRs. Because it possess similar atomic radius to that of carbon, nitrogen (N) is the most commonly used dopants. More recently, the N-doped graphene has been synthesized in experiments [36–39]. In the present Letter, based on density functional theory (DFT) calculations and nonequilibrium Green’s function (NEGF) technique, we investigate the electronic transport properties of armchair GNR (AGNR) molecular devices with periodic N-doping. Our results show that low bias NDR behaviors are obtained in these devices, which are strongly dependent on the position (i.e., the dopants are placed either at the centers or at the edges) and the concentration of the dopants. Especially, giant NDR behaviors with PVR up to the order of 105 can be obtained in the mV bias regime by tuning the position and the concentration of the dopants. 2. Calculation methods and model As shown in the insert of Figure 1, in theoretical simulations, such a system is divided into three regions: a left electrode, a scattering region, and a right electrode. Each electrode marked by lime is modeled by a 7-AGNR with two repeated carbon unit cells along the transport direction (z), where 7 is the number of carbon dimer lines across the ribbon width [40]. The scattering region marked by pale green consists of a 7-AGNR of four unit cell length. All edge
P. Zhao et al. / Chemical Physics Letters 554 (2012) 172–176
Figure 1. Calculated I–V characteristics of M1C, M1E and GNR without doping. Schematic view of a periodic N-doped 7-AGNR molecular device is shown in the insert. The gray, blue and white spheres indicate the carbon, nitrogen and hydrogen atoms, respectively. The labels C and E stand for the doping site at the center and the edge, respectively. For clarity, the current through GNR without doping has been multiplied by 107. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.)
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carbon atoms are saturated with hydrogen atoms to eliminate the dangling bonds. Large supercell dimensions (30 10 Å in the plane perpendicular to the transport direction are used to avoid the interaction between the AGNR and its neighboring images. We consider two typical doping configurations [41]: M1C and M1E, in which one center and one edge carbon atom per two carbon unit cells is substituted by an N atom, respectively. The quantum electronic transport properties are performed using the NEGF + DFT method as implemented in the ATOMISTIX TOOLKIT (ATK) package [42,43]. In our calculations, the exchange–correlation potential is described by the local-density approximation (LDA) [44]. The Troullier–Martins norm-conserving pseudopotentials [45] are adopted to represent the core electrons. The valence electrons are expanded in a single zeta plus polarization basis set (SZP) for all atoms and the cutoff energy is set to 150 Ry to achieve a balance between the calculation time and the accuracy. The electrode calculations are performed under periodical boundary conditions in all directions, and a 1 1 100 kpoint mesh in the x, y, and z direction is sampled in the Brillouin zone according to the Monkhorst–Pack scheme. Before calculating the electronic transport properties, all the structures are fully relaxed until the maximum absolute force is less tan 0.01 eV/Å. The nonlinear current through the device is calculated using the Landauer–Bütiker formula [46]:
Figure 2. Transmission spectra and band structures of both left and right electrodes for (A) M1C at 0, 0.3 and 0.7 V and (B) M1E at 0, 0.25, and 0.6 V. The Fermi level EF is set to be zero. The violet dashed lines represent the electrochemical potentials of the left and right electrodes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.)
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P. Zhao et al. / Chemical Physics Letters 554 (2012) 172–176
IðVÞ ¼ ð2e=hÞ
Z lR
TðE; VÞ½fL ðEÞ fR ðEÞdE
ð1Þ
lL
where e is the elementary electron charge, h is the Planck’s constant, fL,R are the Fermi distribution functions of electrons in the left and right electrodes, lL;R ðVÞ ¼ EF eV=2 are the electrochemical potentials of the left and right electrodes, EF is the Fermi level of the system which is set to be zero in our calculations. The energy region, ½lL ðVÞ; lR ðVÞ, that contributes to the total current integral is referred to as the bias window. TðE; VÞ is the transmission function defined as:
TðE; VÞ ¼ Tr½CL GR CR GA
ð2Þ
R;A
where G are the retarded and advanced Green’s functions of the scattering region, CL;R ¼ i½RRL;R RAL;R are the broadening functions, RRL;R are the corresponding self-energies of the scattering region, which contain all the effects of the electrodes. 3. Results and discussion 3.1. Effect of doping position After geometry optimizations, M1C and M1E still retain their planarity [47]. In contrast to the length of C–H bond in the pristine GNR (1.10 Å), the N–H bond shrinks to 1.02 Å in M1E due to the H atom moving inward from its original position. In M1C, the doped N atom rebonds with three adjacent C atoms. Only the local atomic structure of the GNR is affected by N doping. In Figure 1, we present the current–voltage (I–V) characteristics for M1C and M1E. It is clear that obvious NDR behaviors are observed in two curves. The currents in M1C and M1E increase linearly under low bias and then drop dramatically in the 0.3–0.7 V and 0.25–0.6 V range, respectively. Very interestingly, for M1C, the current is almost quenched to zero in the 0.7–1.0 V range. As a result, a giant PVR up to 104 is obtained in M1C, while the PVR in M1E is only about 18. For comparison, the I–V characteristic of GNR without doping is also presented and no NDR appears. It is reasonable to conclude that the observed NDR behaviors originate from N doping. To understand the mechanism of the observed NDR behavior and its doping-position-dependence, in Figure 2, we give the bias-dependent transmission spectra and band structures of both left and right electrodes for M1C at 0, 0.3, 0.7 V (Figure 2A) and M1E at 0, 0.25, 0.6 V (Figure 2B), respectively. From Figure 2, comparing with the band structures of the undoped case (which is not shown here), we can see that a roughly half-filled impuritysubband cutting the EF, labeled by N, appears in the band structures of two N-doped electrodes, respectively. Meantime, the p- and p⁄-subbands are located far away from the EF. This implies that the N-subband is the only subband near the EF and thus is responsible for the electron transmission under low bias range. Moreover, the steplike T(E, V = 0) curve can be perfectly correlated with the band structure. Since each subband is nondegenerate, T(E, V = 0) increases in steps of one unit. There is a large gap below the EF in T(E, V = 0) due to the absence of bands in this energy range. When the bias is applied, the subbands of the left electrode shift down, while those shift up in the right electrode. As a result, the overlap between N-subbands of two electrodes reduces significantly. Accordingly, the transmission T(E, V = 0.3 V) and T(E, V = 0.25 V) for M1C and M1E around the EF decreases, respectively. However, the currents increase from zero to their maximum values since the transmission integrals have reached their maximum values in the bias window of two cases. Here, we must point that the region of the bias window is actually [V/2, V/2] since the EF has been set to be zero. When the bias increases further, the overlap between N-subbands of two electrodes reduces again. As
Figure 3. Top and end isosurface plots of the C-point Bloch wave functions of Nand p ⁄-subbands of electrode for (A) M1C and (B) M1E. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.)
a result, the currents in M1C and M1E start to drop, respectively. Then NDR occurs. From the bottom panels of Figure 2, we can see that the N-subband of the right electrode rises to align with the p⁄-subband of the left electrode in two cases, respectively. However, as we can see, T(E, V = 0.7 V) for M1C is zero in the bias window, while T(E, V = 0.6 V) for M1E is nonzero but very small near the right electrochemical potential in the bias window. This implies that the electron tunneling from thep⁄-subband of the left electrode to the N-subband of the right electrode is forbidden in M1C, while there is very weak tunneling in M1E. This is the reason that the PVR of M1E is far less than that of M1C. In order to understand the abnormal tunneling phenomenon, in Figure 3, we give the isosurface plots of the C-point Bloch wave functions of the N- and p⁄-subbands of electrode for M1C and M1E, respectively. As shown in Figure 3A, for M1C, the N- and p⁄-subbands have even and odd parity under the yz midplane mirror operation, respectively. Since two subbands have an opposite symmetry, the tunneling between them is forbidden [48,49]. As a result, when 0.7 V bias is applied, the N- and p⁄-subbands cannot couple with each other to contribute to the transmission, and thus the current is essentially zero in the 0.7–1.0 V range. On the contrary, for M1E, there is no definite parity in the N- and p⁄-subbands since it does not exist yz midplane mirror due to the edge-doping, as shown in Figure 3B. As a result, there is a weak coupling between these two subbands, leading to the weak electron transmission, as shown in Figure 2f.
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Figure 4. (A) Calculated I–V characteristics and (B) band structures of electrode for M0C, M1C, M2C and M3C at equilibrium state. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.)
3.2. Effect of doping concentration Next, in order to realize NDR in mV bias regime, we investigate the effect of the concentration of dopants on NDR. Since center-doping can give rise to much larger PVR, we concentrate on this doping type. Limited by the computational sources, we just further calculate the transport properties of another three doping cases: M0C, M2C and M3C, in which one center carbon atom per one, four, and six carbon unit cells is substituted by an N atom, respectively. The corresponding I–V curves are shown in Figure 4A. As we can see, all systems show NDR and the peak position, Vpeak, strongly depend on the N-doping concentration. The Vpeak gradually moves down to
lower bias range when the doping concentration is decreased: M0C is 0.8, M1C is 0.3, M2C is 0.1 and M3C is 0.03 V, respectively. In addition, both the PVR values of the M2C and M3C are up to the order of 105, which are much bigger than those reported values [4–17]. Moreover, it is clear that the current drops more rapidly beyond the Vpeak with the decrease of doping concentration. This NDR fast response to the applied electric-field could be applied to fast switching in certain types of electronic devices [50]. To understand the doping-concentration-dependence of NDR, in Figure 4B, we give the band structures of electrode for M0C, M1C, M2C and M3C at equilibrium state, respectively. Two distinct features can be addressed as follows: (1) Except for M0C, the
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Foundation of Shandong Province of China (Grant No. ZR2009AL004) and the Doctoral Foundation of University of Jinan, China (Grant No. XBS1004). References
Figure 5. Calculated bandwidth (DE) of N-subband and predicted NDR peak position (Vpeak) for M1C, M2C, M3C, M4C and M5C. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.)
N-subband is the only one in the vicinity of the EF. For M0C, the Nand p⁄-subbands cross at the EF. Therefore, the electron can tunnel from the N-subband (p⁄-subband) of the left electrode to the corresponding N-subband (p⁄-subband) of the right electrode in a wide energy range. As a result, NDR can only occur at a relatively high bias (Vpeak = 0.8 V) in M0C. (2) The bandwidth (DE) of N-subband decreases gradually with the decrease of doping concentration. Since the NDR is originated from the matching and mismatching of N-subband between two electrodes (M0C is an exception, in which the p⁄-subband also play an important role), the Vpeak is determined by the corresponding DE of N-subband: V peak ¼ DE=ð2eÞ. The calculated DE of N-subband is 0.64, 0.19, and 0.06 eV from M1C to M3C, respectively. Therefore, the predicted Vpeak is 0.32, 0.10, and 0.03 V, which is in good agreement with the calculated values. When the doping concentration further decreases, it is reasonable to conclude that the DE of N-subband will continue to drop, and the Vpeak will enter the mV bias regime. As shown in Figure 5, when one center carbon atom per eight (M4C) and ten (M5C) carbon unit cells is substituted by an N atom, NDR occurs at 8.2 and 2.5 mV, respectively. 4. Conclusion In summary, we have investigated the electronic transport properties of periodic N-doped AGNR molecular devices by applying NEGF + DFT. The I–V characteristics exhibit robust NDR behaviors, which are strongly dependent on the position and the concentration of the dopants. The center-doping gives rise to better NDR performance than the edge-doping. The reason can be attribute to the inhibition of tunneling between the N-subband of the right electrode and the p⁄-subband of the left electrode, which have an opposite symmetry. Moreover, giant NDR behaviors with PVR up to the order of 105 can be obtained in the mV bias regime by tuning the position and the concentration of the dopants. The present findings might be useful for the application of the GNRs in the field of low bias NDR molecular devices. Acknowledgments This Letter was supported by the National Natural Science Foundation of China (Grant No. 11104115), the Natural Science
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