3.2.1 Finite Difference Time Domain Model
For optical simulation, three-dimensional finite difference time domain (FDTD) method was used. The solar cells were modeled using complex refractive index ᵰ( ᵰ ) = ᵊ(ᵰ) + ᵆᵇ(ᵰ) of each layer, where ᵰ, ᵊ, ᵝᵊᵠ ᵇ are the dielectric permittivity, refractive index, and extinction coefficient, respectively. AM 1.5G standard solar spectrum was used to model the perpendicularly incident solar irradiance. The computational domain and its boundary conditions are shown in Figure 3.1(e) for the NW SC. In the case of normal incidence, periodic boundary conditions were used along the horizontal dimensions due to the symmetry of the cell, whereas perfectly matched layers (PML) were placed at the top and bottom to avoid any reflection. For oblique irradiance, Bloch boundary conditions were employed along the dimension of solar angle variation. Similar boundary conditions were employed for the PSCs with and without forward scatterers. The FDTD simulator (implemented using ANSYS Lumerical in this work) is based on the simultaneous time-dependent solution of Maxwell's third and fourth equations, which are given by
42 ᵱᵆ
ᵱᵐ = ∇ × ᵊ (3.1) ᵱᵊ
ᵱᵐ = − 1
ᵱ ∇ × ᵇ (3.2) Where, ᵆ(ᵳ) = ᵰ ᵰ∗ᵇ (ᵳ)
Here, ᵇ is the optical electric field inside the structure, ᵆ is the displacement vector, ᵊ is the magnetic field intensity, ᵳ is the angular velocity, ᵰ is the permittivity of free space and ᵰ is the relative permittivity of the material under consideration (ᵰ = ᵰ ᵰ ).
The time-dependent Maxwell’s equations are discretized using central difference approximations to space and time partial derivatives. The leapfrog method is then used to solve the finite-difference equations. To put it another way, the electric field vector components in a volume of space are solved at one moment, and the magnetic field vector components in the same spatial volume are solved at the next time. This procedure is performed numerous times. These leapfrogging time steps are executed a finite number of times depending on the type of source employed in the simulation area. Figure 3.2 depicts the process of solving Maxwell's equations.
Figure 3.2 Simulation workflow of Maxwell’s equation solution.
43
Figure 3.3 Yee grid in three-dimensional simulation domains [136].
Yee proposed the leapfrog technique of solving Maxwell's equation in a seminal work published in 1966. That work proposed solutions to a set of finite difference equations for time and space-dependent equations for loss-less materials. Yee's technique is also included in the commercial electromagnetic simulator we used in our research. This approach introduced a spatially staggered vector component of the E-field and H- field about rectangular unit cells of a Cartesian computing grid. It was set up in such a way that each E-field vector component is in the middle of two H-field vector components. This grid was later referred to as the Yee grid. The Yee grid for three- dimensional simulation regions is shown in Figure 3.3. Apart from presenting spatially staggered field vectors, Yee also developed a leapfrog technique for stepping in time, resulting in temporally staggered electric and magnetic fields. As a result, between consecutive H field updates, electric field updates are computed halfway through each time step and vice versa. This technique of solving Maxwell’s equation is utilized in the FDTD solver used in this study for optical simulation.
In optical solver, the electric field was obtained by simultaneously solving Equations (3.1) and (3.2), from which the optical power absorbed (ᵒ ) is calculated. The absorbed irradiance gives rise to electron-hole pair (EHP) generation measured by the generation rate (G),
ᵒ (ᵔ, ᵕ, ᵖ, ᵳ) = −0.5 ᵳ ᵇ ᵅᵉᵝᵣ(ᵰ) (3.3)
44
ᵉ = ∫ ℏ ᵠᵳ (3.4) From the optical simulation results, the space integrated absorbed power ᵃ(ᵰ) (λ representing wavelength) was calculated from reflection (ᵔ(ᵰ)) and transmission monitor (ᵖ (ᵰ)) data. Then, the optical performance of various cells was evaluated with respect to PSCs by using the following enhancement parameter based on the integrated quantum efficiencies of respective cells.
( ) = 1 − ( ) − ( ) (3.5)
1.5 300
300
( )
Enhancement ( 1) 100%
( )
g
g
G
REF
A I d
hc
A d
hc
(3.6)
Here, ᵃ ((λ) represents the integrated absorbed power for reference optimized planar solar cell. ᵋ is the AM 1.5G standard solar radiation and ᵰ represents the wavelength corresponding to the bandgap of absorber material, beyond which absorption in a solar cell is negligible. In order to take account of transverse magnetic and transverse electric mode of polarization in the case of nanowire structure, the enhancements obtained from two simulations using these modes were averaged to.
3.2.2 Boundary Condition
The periodic arrays of nanowires have been analyzed by simulating a unit cell containing a single nanowire; hence appropriate boundary conditions are necessary.
One of the primary sources of error in FDTD simulators when simulating periodic structures is the reflection from simulation borders. In a typical electromagnetic simulation, at least one or more of the boundaries must be infinitely expanded. It is not computationally efficient to make the simulation domain large enough to allow the scattered wave to fade. As a result, absorbing boundaries are incorporated into
45
the simulation space to represent infinitely expanded space. In order to tackle this problem, many forms of absorbing boundary conditions (ABC) might be used [137], [138]. Berenger developed the perfectly matched layer, which is one of the most effective ABCs [139]. The perfectly matched layer (PML) technique's basic idea is to add an additional lossy layer around the simulation region with intrinsic impedance matched with the media at the outermost simulation region, ensuring zero reflection from the interfaces and attenuating the field as it propagates through the media. Before the field intensity reaches the simulation border, it is ensured that it will attenuate to zero. In our simulation, the PML boundary condition is employed at the boundaries perpendicular to the source. The whole response of a periodic structure can be derived from the simulation results of just one unit cell in FDTD simulations.
Figure 3.4 Device structure and boundary conditions for oblique incidence of solar light.
The periodic boundary condition is applied along the axis of symmetry to achieve this. A periodic boundary condition can be applied to one or more simulation boundaries. However, for oblique incidence, the structure loses its symmetry along
46
the direction of variation of incident angle. Hence, Bloch boundary condition is needed. When the structure is periodic, but the EM field has a phase change between each period, this boundary condition is required. A situation like this emerges when the incident light is obliquely incident. Both normal and oblique incident sunlight have been explored in our simulation; hence Periodic and Bloch boundary conditions were applied in respective cases.