3.3 Dielectric Response

We compare our TB results with first principles calculations introduced in Sec. 3.1.1. The optical polarization vector is assumed to be along the transport direction similar to the assumption made in the TB calculation.

Figure 3.4(a) exhibits ε_i(ω) and the JDOS of AGNR₂₀BN₄₀. In the energy range 0 < ℏω < 2 eV the JDOS has maxima at ℏω = 0.3019, 0.62214, 0.92657, 0.91883, 1.2289,1.5256, and 1.7349 eV. However, only four of these maxima (ℏω = 0.3019, 0.91883,0.92657, and 1.5256 eV) appear in ε_i(ω). From the electronic band-structure in Fig. 3.4(b) it can be shown that the peaks in ε_i(ω) are related to transitions from n_v = 1 to n_c = 1 (A), n_v = 1 to n_c = 3 (B), n_v = 2 to n_c = 2 (C), and n_v = 2 to n_c = 4 (D). Disappeared peaks in ε_i(ω) are due to zero momentum matrix elements in Eq. 3.4. This transition rule confirms previous results which are explained by the symmetry properties of the wave functions.

Figure 3.4(a) compares the dielectric functions of an AGNR₂₀BN₄₀ obtained from TB and first principle calculations. Excellent agreement between these results confirms the transition rules obtained from TB calculations. The energy of the first peak matches well, however, the discrepancies increase for higher peaks. This behavior is related to the differences between the predicted energy-gaps obtained from SIESTA and TB at higher energies, see Fig. 3.4(b).

In order to investigate AGNRs/BN for photodetection application, we study the quantum efficiency defined as α = ∕, where I_ph is the photo current and P_op is the incident optical power. We assumed that all absorbed photons contribute to the photo current, such that the quantum efficiency can be calculated from the dielectric function (Eq. 3.4). A quantum efficiency of 6 - 16% for graphene is reported in Ref.  [82] and a maximum quantum efficiency ranging from 9% to 11% is reported for H-AGNRs in Ref.  [171]. Figure 3.5 shows the calculated quantum efficiency as a function of the incident photon energy at various GNR widths. The efficiency is maximized when the photon energy matches the bandgap of the nanoribbon (the first peak for each structure). Our results indicate a peak of quantum efficiency in the range of 14 - 15% for AGNRs/BN. The quantum efficiencies of photodetectors based on AGNRs/BN and H-AGNRs are compared in Fig. 3.5. Due to the presence of more allowed transitions a wider absorption spectrum is achieved in AGNRs/BN compared to H-AGNRs. As a H-AGNR with index 8 is metallic, the first peak is related to the second energy-gap and appears at 2.88 eV whereas the AGNR₈BN₃₀ shows three peaks below that energy due to energy-gap opening, see Fig. 3.5(a). In Fig. 3.5, the quantum efficiency decreases for the first energies, but increases at higher energies, see for example the sixth peak for AGNR₃₃BN₃₀ (Fig. 3.5(c)). This is due to different effective masses of different subbands which affect the JDOS. According to Eq. 3.4, a larger JDOS leads to a larger absorption of photons and a higher quantum efficiency.

We also investigate the photoresponsivity given by . Our calculations give an upper limit for the photoresponsivities of 0.336 A/W, 0.239 A/W, and 0.202 A/W for photon energies near the bandgaps of AGNR₈BN₃₀, AGNR₁₆BN₃₀, and AGNR₃₃BN₃₀ respectively. Due to the higher quantum efficiency of AGNRs/BN compared to AGNRs, a higher photoresponsivity is obtained for the same input optical power of 10⁷ W/m².