Simulation of Graphene Piezoresistivity Based on Density Functional Calculations

The piezoresistive effect in graphene ribbon has been simulated based on the first-principles electronic-state calculation for the development of novel piezoresistive materials with special performances such as high flexibility and low fabriccation cost. We modified theoretical approach for piezoresistivity simulation from our original method for semiconductor systems to improved procedure applicable to conductor systems. The variations of carrier conductivity due to strain along with the graphene ribbon models (armchair model and zigzag model) have been calculated using band carrier densities and their corresponding effective masses derived from the one-dimensional electronic band diagram. We found that the armchair-type graphene nano-ribbon models have low conductivity with heavy effective mass. This is a totally different conductivity from two-dimensional graphene sheet. The variation of band energy diagrams of the zigzag-type graphene nano-ribbon models due to strain is much more sensitive than that of the armchair models. As a result, the longitudinal and transverse gauge factors are high in our calculation, and in particular, the zigzag-type graphene ribbon has an enormous potential material with high piezoresistivity. So, it will be one of the most important candidates that can be used as a high-performance piezoresistive material for fabricating a new high sensitive strain gauge sensor.


Introduction
One-atom-thick graphene sheet (see Figure 1) is gathering a lot of attention because of its unique electrical and mechanical properties such as high electron mobility and stiffness after its discovery by Novoselov et al. [1].The piezoresistance effect is defined as the electrical resistivity change under mechanical stress [2].With the ardent interest to merge graphene into piezoresistor applications, researchers are driven to study the piezoresistance effect of graphene sheets fabricated by various methods.Lee et al. reported that the gauge factor of graphene grown on Ni and Cu films by chemical vapor deposition was 6.1, with 1% applied strain [3].Chen et al. found that the gauge factor of mechanically exfoliated graphene was nearly 150 [4].Hosseinzadegan et al. reported that the gauge factor of graphene prepared by chemical vapor deposition on Si/SiO 2 wafer was 18,000 [5].A recent theoretical work predicted that the conductivity of gra- phene changed from the conductive state to the semiconductive state by the splitting of band gap due to strain [6], though single-layer graphene is generally impossible to have a band-gap [7].If a band-gap exists, the straininduced band-gap modulation could be one source of piezoresistivity in graphene [5].In our previous work [8][9][10][11][12][13], the simulation of piezoresistivity based on the first-principle band structure has been developed for semiconductor systems.In this work, the gauge factors of graphene models have been evaluated by our original simulation method based on the first-principles band structure with some modifications which can be applied to conductor systems.

Method of Calculation
First-principles calculations of the periodic boundary models for graphene have been carried out by FHI98MD program package [14] based on the density functional theory (DFT) [15].For the DFT exchange-correlation interaction, the generalized-gradient approximation (GGA) method was used by Perdew-Burke-Ernzerhof (PBE) functional [16].We adopted the three-dimensional supercell approximation technique with norm-conserving pseudopotentials prepared according to the Hamann method [17] and wave functions with plane-wave expansion.
The zigzag and armchair models have been defined as shown in Figure 2, where these models have been devi- sed by cutting out a fragment with a one dimensional periodic boundary, and all dangling bonds of C atoms were terminated with H atoms.The direction of the fragment which is parallel to the y direction can be defined as the longitudinal direction, while the parallel to the x direction can be considered as the transverse direction as illustrated in Figure 2(c).To represent graphene ribbon, vacuum space about 5 Å along x direction is considered.

Geometrical Optimization of Graphene Sheet
C-C length of the graphene sheet in the strain-free condition has been optimized by the first-principles calculation.We found that the optimized value of the C-C length of graphene is 1.422 Å which is corresponding to the C-C length for the minimum total energy.This value is completely corresponding to the experimental value (1.42 Å) by Heyrovska [18].C-C length in the strain condition can be evaluated by using Poisson's ratio as indicated in Equation (1).
where ε perp is the transverse strain (perpendicular to the applied load) and ε axial is the axial strain (in the direction of the applied load).The value of υ has been calculated by partial optimization of graphene sheet, where one lattice constant with 1% uniaxial tensile strain is fixed.We investigated two different strain directions: the first is the V strain while the second is the U strain as illustrated in Figure 1.The Poisson's ratio is determined as υ = 0.28 for the V strain and υ = 0.14 for the U strain.Experimental results of Poisson's ratio of graphene sheet have given a wide range of values from 0.12 to 0.3 [19][20][21][22][23][24].Actually, the results of partial optimization are sensitive and our results υ = 0.28 or 0.14 satisfy the experimental measurements.
We applied the optimized C-C length and Poisson's ratios to the graphene ribbon models.In this paper, we simulated the 9-ring armchair and 13-ring zigzag graphene ribbon models.The structural parameters of the models shown in Figure 2, number of C atoms (N C ), and numbers of H atoms (N H ) are tabulated in Table 1.

Modeling and Calculation of Armchair Models
The whole image of band energy diagram of the 9-ring armchair model is shown in Figure 3.The valence band (VB) maximizes and the conduction band (CB) minimizes at the Y point, respectively, and zero band gap was observed.Accordingly, the armchair graphene ribbon can be considered as a semi-metal or zero-band gap semi-  c lowest CB subband are in double degeneracy at the Y point, in the vicinity of the Fermi energy.This feature can be also derived by the simple Hückel method qualitatively for the armchair model, as shown in Figure 4.As the common characteristic of the armchair models, the π orbitals of all of armchair models are localized at the edges of graphene ribbon, and no conductance path exists in the center of graphene ribbon.The interaction due to π orbitals along the longitudinal direction is very small because of the non-bonding state due to the anti-symmetric relation for the translation with the phase factor i π e y y L , and accordingly, the band energy variation of egenerate subbands with respect to k point is quite small near the Y point.As a result, low conductivity with heavy effective mass can be explained by using the band orbital interaction.This is one of the most important effects due to miniaturization to armchair-type nanoribbon in the sense that a totally different conductivity from two dimensional graphene sheet can be given.
According to the longitudinal or transve

Mode
The whole image of band diagram of the 13-ring zigzag model is shown in Figure 6.The VB maximizes and the CB minimizes near the Γ point, respectively.Figure 7 represents the band energies diagrams for the strain-free and strain 13-ring zigzag models.As compared with the armchair models, the variation of band energy diagrams of the zigzag models by the strain is much more sensitive.In particular, the characteristics of VB top and CB bottom subbands near the Fermi energy are caused by both of the longitudinal and transverse strains.Actually, the π orbitals of VB top and CB bottom are delocalized, and the band orbital interaction is easy to be caused due to strains.

Evaluation of S
The electrical conductivity G or the electrical re can be represented in terms of carrier den effective mass by the conventional treatment [25].Variations of band structure will exert an influence on them, and frequently contribute to a sudden turn of the conductivity.In this paper, we have introduced the band carrier densities for conductive state.The conductivity has been represented as, 2 * e , where n j is the jth conduction band car density, m * is the band effective masse, relaxa-rier electron area j τ j is the tion time, and e 2 is the square of the absolute value of the elementary electric charge.The band carrier densities n j are defined with the Fermi energy and temperature T; (3 where S is the surface area of the graphene model (S = L × L ),   and ky , n j can be approximated as The value of E F can be solved easily with the total number of valence electrons in the model as follows, We have performed a sampling with 11 the Γ-Y path.The effective mass is generally a 3 × 3 tensor, a points along nd the reciprocal matrix of effective mass is defined as [26] ( ) where E is the band energy and ħ is equal to Planck's constant divided by 2π.The band energies of our graphene models remain constant along the transverse directions, namely, and therefore, the band effective ma the graphene models can be defined sim ss of the jth band for ply as a scalar, . In this paper, the second derivative on the right hand of Equation ( 8) has been estimated numerically as ) in nstant regardless of stress [8][9][10][11][12][13].This procedure seems to be rough to some extent, but the variation rate of carrier conductivity can be easily and adequately represented in relaxation times.
Gauge factor can be defined as, 0 α w resistances at applied strain ε α (α denotes longitudinal or transverse) and at no strain, respectively.From Equations gauge factor equation can be written as follows: here R α and R 0 are the graphene ( 2) and ( 10), the ( ) ( ) Using the data obtained from the energy diagrams which are tabulated in Table 2, we can easi the values of the gauge factors for both arm and zigzag model as shown in Table 3.

the longitudinal an basis calculation. The variations of carrier aphene ribbon have been calculated us ly calculate chair model
It was found that the carrier electron area densities of armchair model do not change due to any strains and the effective masses are very large regardless of strains.From the view point of piezoresistivity, d transverse gauge factors seem to be large because the variation ratios of inverse effective masses due to strain become large by high sensitivity with small values.As compared with the armchair model, drastic changes in the carrier electron area densities and effective masses due to strain can be observed in the zigzag model.

Conclusion
In this paper, the gauge factors in armchair and zigzag graphene ribbon models have been simulated on the of first-principles conductivity of gr ing band carrier densities and their corresponding effective masses.We found the different characteristics

Figure 3 .
Figure 3. Band energy diagram of 9-ring armchair model so much.These features are in good agreement ling and Calculation of Zigzag Models train Gauge Factors sistivity ρ sity and riations of the band structures of armchair models have been observed.As it is clear from the band energy diagram shown in Figure 5, the degenerate subbands in the vicinity of the Fermi energy are not lifted by the longitudinal or transverse strain effect, and the feature of conductivity based on the band energy diagram will not with quite small band orbital interaction as shown in Figure 4.
E t to the band di is the Heaviside step func spersion of oltzmann tion with respecjth subband E j (k y ), and k B is the B constant.By using the calculated band energy of the jth subband and k-point weight, E j,ky

Figure 4 .
Figure 4. Degenerate π orbitals at Y point near the Fermi energy for the 9-ring armchair models with 2 unit cells.
y = 0.05 × (2π/L y ).For the relaxation time graphene systems, we have introduced the approximation that all of the band relaxation times are equal and co consideration of the canceling of almost part of band

Table 1 . Structural parame
ters of graphene models.y