Abstract
The effects of uniaxial tensile strain on the ultimate performance of a dualgated graphene nanoribbon fieldeffect transistor (GNRFET) 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 GNRFETs, 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 straininduced changes in both band gap and effective mass of the GNR. Intrinsic switching delay time, cutoff frequency, and I_{on}/I_{off} ratio are also calculated for various uniaxial strain values. The results indicate that the variation in both cutoff frequency and I_{on}/I_{off} 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.
Keywords:
Graphene nanoribbons FETs; Uniaxial strain; Analytic ballistic model; Device performance metricsBackground
Graphene is a promising material for nanoelectronics due to its high carrier mobility at room temperature and excellent mechanical properties [1,2]. However, the oncurrenttooffcurrent ratio of graphene channel fieldeffect 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, highfrequency applications [3]. 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. Zigzagedged 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. [6] 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 [6]. However, the band gap of the family N=3p+2 is significantly reduced, resulting in a closetometallic 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 [7]. 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 allgraphene electronics [8]. The effect of strain on the electronic structure and transport properties of graphene sheets and its ribbons have been studied both theoretically [911] and experimentally [1214]. Uniaxial strain can be applied by depositing a ribbon of graphene on transparent flexible polyethylene terephthalate (PET) and stretching the PET in one direction [12]. 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 [15]. To date, however, no experimental works on applying uniaxial strain to narrow GNRs (of sub10 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 [16]. 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 CC 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 firstprinciples calculations [1619]. 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 [16].
Although several studies have investigated the band structure of strained AGNRs, only a few have been focused on the performance of strained GNRFETs [2124]. These studies are based on firstprinciples quantum transport calculations and nonequilibrium Green’s function techniques. It is shown that the IV characteristics of GNRFETs 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 [22]. On the other hand, although semianalytical [25] or fully analytical models [26] for the IV characteristics of unstrained GNRsFETs have been proposed, no analytical model of GNRsFETs under strain has been reported.
In this work, using a fully analytical model, we investigate the effects of uniaxial tensile strain on the IV characteristics and the performance of doublegate GNRFETs. Compared to topgated GNRFET, a dualgated device has the advantage of better gate control and it is more favorable structure to overcome short channel effects [27]. Since significant performance improvement is expected for nanodevices in the quantum capacitance limit QCL [28], a doublegate AGNRFET 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 powerdelay time product are calculated and discussed.
Methods
Device model
Effective mass and band structure
The modeled GNRFET has a doublegate structure with gateinsulator HfO_{2} of thickness t_{ins}=1 nm and relative dielectric constant κ=16, as shown schematically in Figure 1a. The channel is taken to be intrinsic, and its length is supposed equal to the gate length L_{G}. The source and drain contacts are heavily doped regions with doping concentration value of 5×10^{−3} dopants per carbon atom. Since thin and high κ gate insulator is employed, we can expect excellent gate control to prevent sourcedrain direct tunneling. Moreover, the quantum capacitance limit (QCL), where the small quantum capacitance dominates the total gate capacitance, can be reached. The channel material is assumed to be a singlelayer AGNR of the family N=3p+1, as it is illustrated in Figure 1b. It is well known that this family of AGNR is semiconducting material with promising characteristics for switching applications [26]. The edge boundaries are passivated by hydrogen atoms. It has been demonstrated that hydrogen passivation promotes the transformation of indirect band gaps to direct ones resulting in improved carrier mobility [19]. Moreover, the edge of the GNR is assumed to be perfect without edge roughness for assessing optimum device performance. In what follows, a power supply voltage of V_{DD}=0.5 V and room temperature T=300 K are used.
Figure 1. Schematics of doublegate GNRFET and the atomic structure of AGNR.(a) Schematics of doublegate GNR FET where a semiconducting AGNR is used as channel material. (b) The atomic structure of AGNR. Hydrogen atoms are attached to the edge carbon atoms to terminate the dangling bonds. N is defined by counting the number of Catoms forming a zigzag chain in the transverse direction.
Before dealing with the device performance under strain, we consider the effect of uniaxial strain on both band gap and effective mass of the AGNR. It has been verified that a 3NN tight binding model incorporating the edge bond relaxation can accurately predict the band structure of GNRs [29]. The 2NN interaction, which only shifts the dispersion relation in the energy axis but does not change the band structure, can be ignored. Any strain applied into the GNR modifies the CC bonds accordingly. As a result, each hopping parameters in the tightbinding Hamiltonian matrix of the unstrained GNR is assumed to be scaled in Harrison’s form [30]t_{i}=t_{0}(d_{i}/d_{0})^{2}, where d_{i} and d_{0} are the CC bond lengths with and without strain, respectively. Following the analysis of [16], where these changes are treated as small perturbations, we can express the energy dispersion of an AGNR under uniaxial strain in the form
with
and
where θ=π/(N+1), ± indicates the conduction band and valence band, respectively, N is the total number of Catoms in the zigzag direction of the ribbon, n denotes the subband index, and E_{C,n} is the band edge energy of the nth subband. The strain parameters are expressed as c_{1}=1+α, c_{2}=1+β, c_{3}=(γ_{3}c_{2}+Δγ_{1})/γ_{3}c_{2}(N+1) with α=−2ε+3ε^{2} and β=−(1−3ν)ε/2+(1−3ν)^{2}ε^{2}/4, where ε and ν are the strength of uniaxial strain and the Poissson ratio, respectively. Negative ε value corresponds to the compressive strain and positive ε value corresponds to the tensile strain. The first set of conduction and valence bands have band index s=−1. Due to the symmetric band structure of electrons and holes, one obtains for the energy gap E_{G,n}=2E_{C,n}. Also, γ_{1}=−3.2eV and γ_{3}=−0.3eV refer to the first and thirdnearest neighbor hopping parameters and Δγ_{1}=−0.2 eV is used for the correction to γ_{1} due to edge bond relaxation effect. A poisson’s ratio value of 0.165 is used in this study [31]. The electron effective mass of each conduction subband can be calculated by using the formula
and at the bottom of the conduction band is given by
Figure 2 illustrates the dependence of band gap E_{G,n} of the GNR’s family N=3p+1 on the uniaxial tensile strain ε. As it is seen, in the range of tensile strain 0%≤ε≤15%, E_{g} decreases first and then increases linearly. Therefore, there is a turning point, i.e., as the strain increases, there is an abrupt reversal in the sign of dE_{g}/dε, making the curves to display a V shape. The turning point moves toward smaller values of strain as the width of the AGNR increases. Moreover, the slope of E_{g}(ε) is almost identical for various N and the peak value decreases with increasing N. The above observations are in agrement with the main features revealed by using tightbinding or firstprinciples numerical calculations [17,20]. On the other hand, Figure 3 shows the variation of effective mass at the conduction band minimum with strain ε. As it is clearly seen, has similar strain dependence as E_{g} and a linear relation between and E_{g} is expected which could be correlated to an inverse relationship between mobility and band gap [32]. These effective mass variations is attributed to the change in the conduction band minimum position under various strain values.
Figure 2. Band gap variation versus uniaxial tensile strain for different (3p+1)GNRs with indicesp=3,4,5,6.
Figure 3. Effective mass variation versus uniaxial tensile strain for different (3p+1)GNRs with indicesp=3,4,5,6.
Device performance
Assuming a ballistic channel, the carriers with +k and −k states are in equilibrium with Fermi energies of the source (E_{FS}) and the drain (E_{FD}), respectively, with E_{FS}=E_{F} and E_{FD}=E_{F}−qV_{D}. The carrier density inside the channel can be obtained by employing the effectivemass approximation and integrating the density of states over all possible energies [26]
where F_{j} is the FermiDirac integral of order j defined as
and η_{n,S}=(E_{FS}−E_{C,n})/k_{B}T, η_{n,D}=(E_{FD}−E_{C,n})/k_{B}T.
Considering the electrostatics describing the structure, the following relation between the gate voltage and Fermi energy E_{F} can be obtained [33]
where q is the carrier charge, C_{ins} is the gateinsulator capacitance per unit length of the GNR and V_{FB} denotes the flatband voltage. The value of V_{FB} depends on the work function difference between the metalgate electrode and the GNR, and it can be set simply to zero as it is discussed in detail in [34]. The gateinsulator capacitance can be calculated by the simple expression [35]
where N_{G} is the number of gates (N_{G}=2 in our DGdevice), κ is the relative dielectric constant of the gate insulator, t_{ins} is the gateinsulator 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 [35]. The gate insulator capacitance increases linearly as the GNR width increases because the area of the GNR increases proportionally.
The biasdependent gate capacitance per unit length C_{g} can be modeled as a series combination of insulator capacitance per unit length C_{ins} and the quantum capacitance per unit length C_{Q}, that is,
The quantum capacitance describes the change in channel charge due to a given change in gate voltage and can be calculated by C_{Q}=q^{2}∂n_{1D}/∂E_{F} where q is the electron charge and n_{1D} is the onedimensional electron density [33]. Using Equation (6) and writing in terms of Fermi integrals of order (−3/2), we obtain [26]
Following Landauer’s formula and Natori’s ballistic theory [34,36], the device current is expressed by a product of the carrier flux injected to the channel and the transmission coefficient which is assumed to be unity at energies allowed for propagation along the channel. Contribution from the evanescent modes is neglected. Thus,
where f_{S,D}(E) are the FermiDirac probabilities defined as
After integrating, Equation (12) yields
For a welldesigned DGFET, we can assume that C_{ins}≫C_{D} and C_{ins}≫C_{S} which corresponds to perfect gate electrostatic control over the channel [28]. Moreover, carrier scattering by ionimpurities and electronhole puddle effect [37] 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
In this section, we firstly explore the calculated device characteristics. Figures 4 and 5 show the transfer I_{D}−V_{GS} and output I_{D}−V_{DS} characteristics, respectively, in the ballistic regime, for the DG AGNRFET of Figure 1 with N=16, which belongs in the family N=3p+1, for several increasing values of uniaxial tensile strain from 1% to 13%. The feasibility of the adopted range of tensile strain values can be verified by referring to a previous firstprinciples study [22,23]. As it is seen from the plots, the current first increases for strain values before the turning point ε≃7% in the band gap variation (see Figure 2) and then starts to decrease for strain values after the turning point. Moreover, the characteristics for ε=5% are very close to that of ε=9%, and the same can be observed when comparing the characteristics of ε=3% with that of ε=13%. Note that, in each region of strain values (region before the turning point and region after the turning point), there is an inverse relationship between the current and the band gap values. Similar features in the currentvoltage characteristics have been observed in the numerical modeling of [22,23] under uniaxial strain in the range 0≤ε≤11%. These features could be explained by the inverse relationship between mobility and energy gap which results to an increase in carrier’s velocity before the turning point and to the reduction in carrier’s velocity after the turning point [32]. It is also worth noting that, at the ballistic transport limit without electrostatic short channel effects, the characteristics in Figure 5 are independent on the channel length. This result is different from conventional FETs and can be explained by the fact that, under purely ballistic conditions (no optical phonon nor acoustic phonon scattering), the scattering mechanisms that cause the channel resistance to increase proportionally to channel length are neglected here.
Figure 4. Transfer characteristicsI_{D}−V_{GS} for various tensile strain values.
Figure 5. Output characteristicsI_{D}−V_{DS} for various tensile strain values.
Now, we focus on the effect of uniaxial strain on the gate capacitance C_{g} and transconductance g_{m}=∂I_{D}/∂V_{G} of the device under study. Uniaxial strain changes the density of states and hence changes the quantum capacitance C_{Q} of the channel which is directly proportional to the density of states. As a result, in the quantum capacitance limit, uniaxial strain changes considerably the intrinsic gate capacitance C_{g}. Figures 6 and 7 show C_{g} versus gate bias at drain bias V_{DS}=0.5 V and C_{g} in the onstate (where V_{GS}=V_{DS}=V_{DD}) versus strain ε, respectively. We clearly observe the nonmonotonicity of the C_{g}−V_{G} characteristics arising from the nonmonotonic behavior of the function F_{−3/2}(x) in Equation (11). A comparison of the curves in Figure 6 reveals that the gate bias V_{G} at which C_{g} peaks depends on the applied uniaxial strain. More specifically, the peak values of C_{g} are decreased and moved toward lower values of V_{G} as uniaxial strain is increased before the turning point and are increased and moved toward higher values of V_{G} as uniaxial strain is increased after the turning point. On the other hand, Figures 8 and 9 illustrate the effect of uniaxial strain on the transconductance g_{m} which describes the device’s switchingon behavior. As it is seen, g_{m} increases after threshold almost linearly with V_{GS} and does not peak at a certain gate voltage but gets saturated. Moreover, as uniaxial strain increases, g_{m} drastically increases from its value in the unstrainedGNR case, becomes maximum around the turning point ε≃7% and then decreases at a rate lower than that of the initial increase. This behavior follows the changes in carrier’s velocity with uniaxial strain, as explained earlier.
Figure 6. Gate capacitanceC_{g} versusV_{GS} for various tensile strains.
Figure 7. Gate capacitanceC_{g} versus uniaxial tensile strain in the ‘onstate’V_{GS}=V_{DS}=0.5 V.
Figure 8. Transconductanceg_{m} versusV_{GS} for various tensile strains.
Figure 9. Transconductanceg_{m} versus uniaxial tensile strain in the ‘onstate’V_{GS}=V_{DS}=0.5 V.
Next, we can assess the highfrequency performance potential of the device under strain. The cutoff frequency f_{T} is defined as the frequency at which the current gain becomes unity and indicates the maximum frequency at which signals can be propagated in the transistor. Once both gate capacitance and transconductance are calculated, f_{T} can be computed using the quasistatic approximation [38,39].
It should be noted that a rigorous treatment beyond quasistatic approximation requires the inclusion of capacitive, resistive, and inductive elements in the calculation. In Figure 5, the quantity f_{T}L_{G}, where L_{G} is the channel length, as function of V_{G}, for increasing values of uniaxial tensile stain, is depicted. Assuming a channel length of less than L_{G}=50 nm, f_{T} exceeds the THz barrier throughout the bias window, confirming the excellent highfrequency potential of GNRs. Furthermore, Figures 10 and 11 show the variation of cutoff frequency versus gate voltage and strain ε (in the onstate), respectively. We clearly observe that f_{T} increases rapidly until the turning point ε≃7% and then decreases with lower rate for higher strain values (ε>7%). This is a direct consequence of both transconductance and gate capacitance variations with strain. Therefore, the highfrequency performance of AGNRFETs improves with tensile uniaxial strain, before the ‘turning point’ of band gap variation but becomes worse after this point.
Figure 10. Dependence of (f_{T}L_{G}) onV_{GS} for various uniaxial strains. The drain voltage is held constant at 0.5 V.
Figure 11. Variation of () with uniaxial tensile strain in the ‘onstate’V_{GS}=V_{DS}=0.5 V.
Lastly, we study the effect of strain on the switching performance of the DGGNR FET. Figures 12, 13, and 14 show the dependence of I_{on}, I_{off} and I_{on}/I_{off} ratio on the uniaxial tensile strain, respectively. As it is clearly seen, the variation of both I_{on} and I_{off} is opposite to the variation of the band gap with strain whereas the ratio I_{on}/I_{off} changes with strain following the band gap variation. The oncurrent I_{on} changes almost linearly with strain whereas the I_{off} and the ratio I_{on}/I_{off} changes almost exponentially with strain. Note that the corresponding curves are not symmetric around the turning point, e.g., although for ε=12%, the GNR band gap returns to its unstrained value; the drain current at this stain value does not completely return to that of the unstrained GNR. This can be explained by the fact that although the band gap has returned its unstrained value, the carrier group velocity has been modified because, under tensile strain, some CC bonds of the AGNR have been elongated [9]. Figure 15 shows the I_{on} versus I_{on}/I_{off} plots for various strains which provides a useful guide for selecting device characteristics that can yield a desirable I_{on}/I_{off} under strain. As it is seen, increased tensile strain before the turning point of band gap variation leads to lower I_{on}/I_{off} ratio, whereas, increased tensile strain after the ‘turning point’ leads to higher I_{on}/I_{off} ratio albeit at lower oncurrent.
Figure 12. Variation of the oncurrentI_{on} versus uniaxial strain.
Figure 13. Variation of the offcurrentI_{off} versus uniaxial strain.
Figure 14. Variation of the ratioI_{on}/I_{off} versus uniaxial strain.
Figure 15. Variation ofI_{on} versusI_{on}/I_{off} ratio for various strain values.
Intrinsic delay time τ_{s} is also an important performance metric that characterizes the limitations on switching speed and AC operation of a transistor. Once the gate capacitance is calculated, τ_{s} is given by [28].
where the oncurrent is the drain current at V_{G}= V_{D}=V_{DD}. Apparently, the switching delay time τ_{s} has similar variation as the gate capacitance has with strain, as it is depicted in Figure 16. Moreover, as it is seen from Figure 17, the switching delay time abruptly decreases with strain before the ‘turning point’ of band gap variation but increases rapidly after this point. We can say that switching performance improves with the tensile strain that results in smaller band gap whereas degrades with the tensile strain that results in a larger band gap. It is worth noting that the switching delay time for the unstrained case (ε=0%) is found to be τ_{s}∼23 fs/nm, that is at least three times larger than the corresponding delay time in uniaxially strainedGNR case. Figures 18 and 19 show the switching delay time τ_{s} as a function of oncurrent I_{on} and I_{on}/I_{off} ratio, respectively. For digital applications, high I_{on}/I_{off} ratio and low switching time delay are required. However, when the I_{on}/I_{off} ratio improves with the applied tensile strain, the I_{on} and switching performance degrade and vice versa. Another key parameter in the switching performance of the device is the powerdelay product Pτ_{s}=(V_{DD}I_{on})τ_{s} that represents the energy consumed per switching event of the device. Figures 20 and 21 illustrate the dependence o of powertime delay product Pτ_{s} on strain and on I_{on}/I_{off} ratio, respectively, where similar behavior to that of switching delaytime can be observed.
Figure 16. Switching delay timeτ_{s}/L_{G} versus gate voltage for various uniaxial strains.
Figure 17. Switching delay timeτ_{s}/L_{G} versus uniaxial strain in the onstateV_{GS}=V_{DS}=0.5 V. The delay time τ_{s}/L_{G} for the unstrained case (ε=0%) (not shown) is found to be approximately 23 fs/nm.
Figure 18. Switching delay timeτ_{s}/L_{G} versus on currentI_{on} for various uniaxial strains.
Figure 19. Switching delay timeτ_{s}/L_{G} versusI_{on}/I_{off}ratio for various uniaxial strains.
Figure 20. Powerdelay time productPτ_{s}/L_{G} versus uniaxial strain in the onstateV_{GS}=V_{DS}=0.5 V for various uniaxial strains.
Figure 21. Powerdelay time productPτ_{s}/L_{G} versusI_{on}/I_{off}ratio for various uniaxial strains.
Conclusions
We investigated the uniaxial tensile strain effects on the ultimate performance of a dualgated 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 IV 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 onstate current, gate capacitance, and intrinsic unity gain frequency are steadily improved for tensile strain less than the ‘turning point’ value of the band gap Vtype variation. The observed trends are in consistency with the recently reported results based on tightbinding quantum transport numerical calculations [2123]. 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 I_{on}/I_{off} ratio improves with the applied tensile strain, the I_{on} 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 metalgraphene contacts are used, the oncurrent of the ARGNFET are degraded [40] by lowering the voltage drop on the intrinsic part of the device by a factor of R_{bal}/(R_{bal}+2R_{cont}) where R_{bal} is the intrinsic resistance of the channel and R_{cont} 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 [41]. 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 [42], 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 GNRdevices is only viable with application of tensile strain but difficult with compressive strain.
Competing interests
The author declares that he has no competing interests.
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