Quantum confinement effect in armchair graphene nanoribbons: Effect of strain on band gap modulation studied using first-principles calculations

The quantum confinement effect may play an important role in the gap modulation of armchair graphene nanoribbons (AGNRs) under strain. Using the phase accumulation model, we have investigated the energy-dependent phase shift $\varphi \left(\varepsilon \right)$ at the $\Gamma$ point of AGNRs under various strains using first-principles calculation. The calculation results show that although the energy dispersion of the phase shift is modified by strain, the phase shift near the Fermi level is close to $0.75\pi$, indicating that strain has little effect near that energy level. We can approximate the energy-dependent phase shift by a constant, $\varphi \left(\varepsilon \right)=0.75\pi$, for AGNRs under various $x$ strains. Due to the structural similarity between AGNRs and zigzag carbon nanotubes (ZCNTs), the electronic properties of AGNRs should be similar to those of ZCNTs. The quantization condition of the wave vector of ZCNTs governed by the periodic boundary condition along the circumference direction is similar to that of AGNRs except that the phase shift is equal to zero, $\varphi \left(\varepsilon \right)=0$. Using the zone-folding (ZF) method, we can calculate the band gap of any strained AGNR (ZCNT) from the phase shift $\varphi =0.75\pi$ $(\varphi =0)$ and the electronic structure of the strained graphene. The AGNR shows a zigzag behavior of the dependence of the band gap on strain which is very similar to the ZCNT. The zigzag patterns are significantly shifted by different phase shifts. The peak value of the band gap and the period of the pattern decreases as the width of the ribbon increases. For a given AGNR, the peak value and the period of the pattern increase as the strain increases. A flattening of the peaks appears at the strain where the maximum band gap occurs due to large compressive strain. All these observations can be understood easily from our ZF calculations. The agreement between our model and real local-density approximation calculations indicates that our model can provide an efficient and accurate method to estimate the band gap of AGNRs and ZCNTs under strain, and therefore can provide a better understanding of the effect of quantum confinement on the electronic properties of AGNRs.

Figure 1

(a) Schematic of a 10-AGNR. The solid blue circles denote hydrogen atoms passivating the edge carbon atoms, and open circles represent carbon atoms. The one-dimensional unit cell distance and ribbon width are represented by $L_x$ and $L_y$, respectively. (b) The induced strain along the width direction ${\sigma }_y$ in various AGNRs as function of applied strain along the ribbon length ${\sigma }_x$. The curve for graphene is used to represent the case where the width of the ribbon approaches infinity.

Figure 2

(a) Brillouin zone of graphene. (b) Band structure of strained graphene with strain ${\sigma }_x=4.7\%$ and ${\sigma }_y=-0.8\%$. The first $\pi$ band (black) and the second $\pi$ band (green) are indicated by ${\pi }_1$ and ${\pi }_2$. (c) The ${\pi }_1$ band of the graphene with strain ${\sigma }_x=4.7\%$ and ${\sigma }_y=-0.8\%$ (left) and the energy at the $\Gamma$ point of 10-AGNR with strain ${\sigma }_x=4.7\%$ (right). The vacuum energies of the graphene and the AGNR were set equally. (d) Phase shift vs energy for AGNR with strain ${\sigma }_x=4.7\%$. Note that the red dashed line in (a), (b), and (c) indicates the allowed $k$ values governed by hard-wall boundary $k_n^0$, and the blue lines in (c) indicate the allowed $k$ values governed by real edge barrier $k_n$.

Figure 3

Phase shift of QW states at the ribbon edge as a function of energy in $N$-AGNRs under various $x$ strains for (a) ${\pi }_1$ and (b) ${\pi }_2$ bands.

Figure 4

(a) Position of the Fermi point relative to the $\Gamma$ point in strained graphene. (b) Slopes of the Dirac cone, $d{E}_4^L/dk$ (blue dashed line), $d{E}_4^R/dk$ (blue solid line), $d{E}_5^L/dk$ (black dashed line), and $d{E}_5^R/dk$ (black solid line), as function of $x$ strain. The effects of $y$ strain induced by $x$ strain were included.

Figure 5

(right) The band gaps of $N$-AGNRs and $N$-ZCNTs. The blue solid circles and red open circles denote N-AGNRs and N-ZCNTs [1], respectively, by direct LDA calculation. The blue solid lines and red dashed lines denote the results calculated by the ZF method using $\varphi =0.75\pi$ and $\varphi$ = 0, respectively. (left) The band structure of unstrained graphene and the allowed $k$ lines of various AGNRs for $\varphi =0.75\pi$ (blue lines) and $\varphi$ = 0 (red dashed lines). The LDA data of ZCNTs are taken from [36].

Figure 6

The band gap of $N$-AGNRs and $N$-ZCNTs as a function of $x$ strain ${\sigma }_x$. The blue solid circles and red open circles denote $N$-AGNRs and $N$-ZCNTs, respectively, by direct LDA calculation. The blue lines and red dashed lines denote the results calculated by the ZF method using $\varphi =0.75\pi$ and $\varphi =0$, respectively. The LDA data of ZCNTs are taken from [14].