Formulation of a Self-Consistent Model for Quantum Well pin Solar Cells : Dark Behavior

A self-consistent numerical simulation model for a pin single-cell solar cell is formulated. The solar cell device consists of a p–AlGaAs region, an intrinsic i–AlGaAs/GaAs region with several quantum wells, and a n–AlGaAs region. Our simulator solves a field-dependent Schrodinger equation self-consistently with Poisson and drift-diffusion equations. The field-dependent Schrodinger equation is solved using the transfer matrix method. The eigenfunctions and eigenenergies obtained are used to calculate the escape rate of carriers from the quantum wells, the capture rates of carriers by the wells, the absorption spectra in the wells, and the non-radiative recombination rates of carriers in the quantum wells. These rates are then used in a self-consistent finite-difference numerical Poisson-drift-diffusion solver. We believe this is the first such comprehensive model ever reported.


INTRODUCTION
The conversion efficiency of a single cell pin solar cell can be enhanced by incorporating quantum wells in the intrinsic region of the device.[1] The incorporation of the quantum wells has two counteracting effects: the short-circuit current is increased because of the additional absorption of the low-energy photons in the lower bandgap quantum well and the open-circuit voltage is decreased because of the increase in the recombi- nation of the photoexcited carriers trapped in the quantum well.Experimental results have shown, nevertheless, that the additional photocurrent resulting from the extension of the absorption spectrum to lower energies can outweigh the accompanying drop in the open-circuit voltage [2][3].
Along with these experimental studies, a number of theoretical investigations have been performed.Corkish and Green [4] studied the effects of recombination of carriers in the quantum well and concluded that although the increased recom- bination reduces the open-circuit voltage, limited Corresponding author.enhancement in the conversion efficiency can be obtained with incorporation of the quantum well, albeit not as much as previously reported by Barnham and Duggan [1].Araujo et al. [5] used detailed balance theory and predicted that the conversion efficiency of the quantum well cell would not exceed that of the base-line device.The results of photoresponse calculations by Renaud et al. [6] revealed that introducing the quantum wells can lead to improved photocurrent without much degradation of the open-circuit voltage.Most recently, Anderson  [7] presented an ideal model for the quantum well solar cell device, incorporating the recombination and generation in the quantum wells.Anderson concluded that the improvement in efficiency is achieved only when the depth of the quantum well is less than about 200 meV.
The need for a comprehensive model is rather obvious, now that there seems to be an unsolved debate as to the ultimate advantage of incorporat- ing quantum wells in the intrinsic region of a pin solar cell.In this paper we present formulation of one such model in which we self-consistently include the effects of: 1) Capture of electrons by the wells, 2) Escape of electrons from the wells, 3) Absorption of light in the wells, and 4) Recombination of carriers in the wells.
The standard drift-diffusion equations are modified to account for generation and recombination in the quantum wells and the transfer of electrons and holes between the bulk and quantum well systems.

SELF-CONSISTENT MODEL
The steady-state transport of carriers in the pin structure is described by current continuity equa- tion written for the bulk regions as: Onb nw nb dJn =o, Ot "re nq dx where nb and nw are electron densities in the bulk and quantum wells, respectively.Jn is electron current density.The two additional terms to the standard bulk current continuity equation incor- porate the effect of carrier transfer into and out of the quantum wells.The terms nw/% n and nb/7 are the electrons escape and capture rates, respec- tively.These rates are also used in the continuity equations for the quantum wells, as given by: Onw nb nw at 7-f 7-e n f-Gw Uw O. (2) In the above equations -e n and %n are the electrons escape and capture times, respectively.The escape times of carriers are calculated using the model reported by Moss et al. [8] and capture times are extrapolated from theoretical and experi- mental data reported by Blom et al. [9].Similar equations are written for holes.The boundary conditions for continuity equations are derived from surface recombination velocity model.
In the above continuity equations, the recombi- nation in the bulk is modeled with radiative and non-radiative mechanisms.The term Gb is bulk generation rate and is given by: Ac ab Nigh" exp ( foX a()dx)] dA (3 where Ac is set to correspond to the bandgap of the material.The recombination in the quantum well Uw is a modified Shockley-Read-Hall recombina- tion rate given by: Uw CrnCrpvthNt[pn pn] (4) Crn[n + n,] + Crp[p 4-p,]   where the trap density, Nt, is derived from the density of the interface states.The generation term Gw is calculated from Eq. ( 3) with the bulk absorption coefficient replaced with that of the quantum well.The absorption coefficients of the quantum wells are calculated by a model reported by Stevens et al. [10].
Calculation of the absorption spectra in the quantum wells as well as the escape and capture times of carriers require the eigenfunctions and eigenenergies of the carriers in the quantum wells, which are obtained from a field-dependent Schr6dinger equation given by:

RESULTS
The energy band diagram of the pin solar cell device with four quantum wells in the intrinsic region is shown in Figure 1.We simulated five pin devices all with a acceptor doping level of 108/cc where (x) is the envelope function, Ei are the eigenenergies and V(x) is the potential profile.
Non-constant effective mass m*(x) is assumed.
The Schr6dinger equation is solved using the transfer matrix method [11].The above equations, together with Poisson and drift-diffusion equa- tions are solved using a finite difference scheme.and donor doping level of 4 x 1017/cc in the 0.2 lam p-region and 0.3 lam n-region, respectively.The intrinsic region of all five devices is 0.5 lam.The first device is a base-line GaAs, the second device is a base-line A1GaAs device and the remaining three devices are A1GaAs devices with 1, 10 and 20 quantum wells, respectively.The mole fraction of A1 is 0.3 in all A1GaAs devices and all quantum wells are 100 wide.
We simulated the dark characteristics of all five cells and the results are shown in Figure 2. Because the model incorporates both radiative and non- radiative recombination for the bulk material, the slope of the dark current is qv/nkT, with the ideality factor n ranging from one for exclusively radiative recombination to two for exclusively non-radiative.In the pin devices without quantum wells, n approaches one at high forward bias where radiative recombination becomes more important.
At low biases the non-radiative recombination dominates and n is 1.9 for the A1GaAs device at 0.1 volts applied.
The recombination in the quantum wells is modeled as non-radiative interface recombination using a modified Shockley-Read-Hall expression.With the introduction of quantum wells into the pin device, this non-radiative recombination dom- inates at all biases and the ideality factor equals about 1.7 for all the quantum well devices.The observed crossover of the i-v curves of the quantum well devices and the GaAs device has also been reported by Ragay et al. [3].

FIGURE
FIGUREEnergy band diagram of the solar cell with four quantum wells.

FIGURE 2
FIGURE 2 Dark characteristics of MQW pin solar cells.