Analytical modeling of uniaxial strain effects on the performance of double-gate graphene nanoribbon field-effect transistors
© Kliros; licensee Springer. 2014
Received: 29 September 2013
Accepted: 30 January 2014
Published: 8 February 2014
The effects of uniaxial tensile strain on the ultimate performance of a dual-gated graphene nanoribbon field-effect transistor (GNR-FET) are studied using a fully analytical model based on effective mass approximation and semiclassical ballistic transport. The model incorporates the effects of edge bond relaxation and third nearest neighbor (3NN) interaction. To calculate the performance metrics of GNR-FETs, analytical expressions are used for the charge density, quantum capacitance, and drain current as functions of both gate and drain voltages. It is found that the current under a fixed bias can change several times with applied uniaxial strain and these changes are strongly related to strain-induced changes in both band gap and effective mass of the GNR. Intrinsic switching delay time, cutoff frequency, and Ion/Ioff ratio are also calculated for various uniaxial strain values. The results indicate that the variation in both cutoff frequency and Ion/Ioff ratio versus applied tensile strain inversely corresponds to that of the band gap and effective mass. Although a significant high frequency and switching performance can be achieved by uniaxial strain engineering, tradeoff issues should be carefully considered.
Graphene is a promising material for nanoelectronics due to its high carrier mobility at room temperature and excellent mechanical properties [1, 2]. However, the on-current-to-off-current ratio of graphene channel field-effect transistors (FETs) is very small due to the lack of a band gap. As a result, monolayer graphene is not directly suitable for digital circuits but is very promising for analog, high-frequency applications . A sizeable band gap can be created by patterning the graphene sheet into a nanoribbon using planar technologies such as electron beam lithography and etching [4, 5]. The band gap of a GNR depends on its width and edge orientation. Zigzag-edged nanoribbons have a very small gap due to localized edge states. No such localized state appears in an armchair graphene nanoribbon (AGNR). Son et al.  have shown that the band gap of an armchair graphene nanoribbon (AGNR) arises from both the quantum confinement and the edge effects. In the presence of edge bond relaxation, all AGNRs are semiconducting with band gaps well separated into three different families N=3p, N=3p+1, and N=3p+2, with p an integer, and in each family, the gap decreases inversely to the ribbon width . However, the band gap of the family N=3p+2 is significantly reduced, resulting in a close-to-metallic channel. This classification has proved very helpful in the study of AGNRs since investigating AGNRs of various widths an equivalent behavior of ribbons of the same family is revealed.
Strain has important effects on the electronic properties of materials and has been successfully employed in the semiconductor technology to improve the mobility of FETs . For GNRs, it has been established that the band structure can be drastically modified by strain. As a result, it has been proposed that strain can be used to design various elements for all-graphene electronics . The effect of strain on the electronic structure and transport properties of graphene sheets and its ribbons have been studied both theoretically [9–11] and experimentally [12–14]. Uniaxial strain can be applied by depositing a ribbon of graphene on transparent flexible polyethylene terephthalate (PET) and stretching the PET in one direction . Moreover, local strain can be induced by placing the graphene sheet or ribbon on a substrate fabricated with patterns like trenches as it has been explored for achieving quantum Hall effect . To date, however, no experimental works on applying uniaxial strain to narrow GNRs (of sub-10 nm width) have been reported.
In comparison to a graphene sheet, whose band gap remains unaffected even under large strains of about 20%, the band gap of GNRs is very sensitive to strain . Since shear strain tends to reduce the band gap of AGNRs, most studies are concentrated to uniaxial strain. Uniaxial strain reduces the overlapping integral of C-C atoms and influences the interaction between electrons and nuclei. As a result, the energy band structure, especially the lowest conduction subbands and the highest valence subbands should be changed. Recently, the band structure and transport properties of strained GNRs have been theoretically explored using tight binding as well as density functional first-principles calculations [16–19]. It is found that uniaxial strain has little effect on the band structure of zigzag GNRs, while the energy gap of AGNRs is modified in a periodic way with a zigzag pattern and causes oscillatory transition between semiconducting and metallic states. Moreover, the band gaps of different GNR families show an opposite linear dependence on the strain which offers a way to distinguish the families. Tensile strain of more than 1% or compressive strain higher than 2% may be used to differentiate between the N=3p+1 and N=3p+2 families as their band gap versus strain relationship have opposite sign in these regions [18, 20]. However, shear strain has little influence on the band structure of AGNRs. On the other hand, neither uniaxial strain nor shear strain can open a band gap in zigzag GNRs due to the existence of edge states .
Although several studies have investigated the band structure of strained AGNRs, only a few have been focused on the performance of strained GNR-FETs [21–24]. These studies are based on first-principles quantum transport calculations and non-equilibrium Green’s function techniques. It is shown that the I-V characteristics of GNR-FETs are strongly modified by uniaxial strain, and in some cases, under a 10% strain, the current can change as much as 400% to 500%. However, the variation in current with strain is sample specific . On the other hand, although semi-analytical  or fully analytical models  for the I-V characteristics of unstrained GNRs-FETs have been proposed, no analytical model of GNRs-FETs under strain has been reported.
In this work, using a fully analytical model, we investigate the effects of uniaxial tensile strain on the I-V characteristics and the performance of double-gate GNR-FETs. Compared to top-gated GNR-FET, a dual-gated device has the advantage of better gate control and it is more favorable structure to overcome short channel effects . Since significant performance improvement is expected for nanodevices in the quantum capacitance limit QCL , a double-gate AGNR-FET operating close to QCL is considered. High frequency and switching performance metrics of the device under study, as transcoductance, cutoff frequency, switching delay time, and power-delay time product are calculated and discussed.
Effective mass and band structure
and ηn,S=(EFS−EC,n)/kBT, ηn,D=(EFD−EC,n)/kBT.
where NG is the number of gates (NG=2 in our DG-device), κ is the relative dielectric constant of the gate insulator, tins is the gate-insulator thickness and α is a dimensionless fitting parameter due to the electrostatic edge effect. In our numerical calculation, a value of α=1 is adopted following . The gate insulator capacitance increases linearly as the GNR width increases because the area of the GNR increases proportionally.
For a well-designed DG-FET, we can assume that Cins≫CD and Cins≫CS which corresponds to perfect gate electrostatic control over the channel . Moreover, carrier scattering by ion-impurities and electron-hole puddle effect  are not considered, assuming that such effects can be overcome by processing advancements in the future. In what follows, a representative AGNR with N=16 is considered.
Results and discussion
We investigated the uniaxial tensile strain effects on the ultimate performance of a dual-gated AGNR FET, based on a fully analytical model. The model incorporates the effects of edge bond relaxation and third nearest neighbor (3NN) interaction as well as thermal broadening. We have focused on the AGNRs family N=3p+1 which is suitable for device applications. The strong modulation of I-V characteristics due to the changes in the strain is directly related to the electronic structure of the GNR channel region, which is modified as a result of changes in atomic structure under strain. The on-state current, gate capacitance, and intrinsic unity gain frequency are steadily improved for tensile strain less than the ‘turning point’ value of the band gap V-type variation. The observed trends are in consistency with the recently reported results based on tight-binding quantum transport numerical calculations [21–23]. Switching delay times improves with the tensile strain that results in smaller band gap whereas degrades with the tensile strain that results in a larger band gap. However, when the Ion/Ioff ratio improves with the applied tensile strain, the Ion and switching performance degrade and vice versa. Therefore, although a significant performance can be achieved by strain engineering, tradeoff issues should be carefully considered.
It is worthy noting that since purely ballistic transport and negligible parasitic capacitances are assumed, our calculations give an upper limit of the device performance metrics. Moreover, when metal-graphene contacts are used, the on-current of the ARGN-FET are degraded  by lowering the voltage drop on the intrinsic part of the device by a factor of Rbal/(Rbal+2Rcont) where Rbal is the intrinsic resistance of the channel and Rcont is the contact resistances. Furthermore, in the presence of metal contacts, the cutoff frequency is degraded since the traversal time of carriers is significantly enhanced . On the other hand, our approach may underestimate the actual concentration of carriers in the channel, especially for large drain and gate biases, when parabolic band misses to match the exact dispersion relation. However, we believe that the present fully analytical study provides an easy way for technology benchmarking and performance projection. Our study can be extended to compressive strain allowing negative values of uniaxial strain ε in our model. However, as it has been demonstrated , narrow GNRs exhibit a maximum asymmetry in tensile versus compressive strain induced mechanical instability, that is, the critical compressive strain for bucking is several orders of magnitude smaller than the critical tensile strain for fracture. Such a large asymmetry implies that strain engineering of GNR-devices is only viable with application of tensile strain but difficult with compressive strain.
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