Let us consider the basic physical processes taking place in a semiconductor solar cell with a heterojunction (Fig. 1). The device is composed by two semiconductors with different band gap values [1, 6]. The widerband material forms socalled window layer (for which the corresponding characteristics in Fig. 1 have the subscript "W") and is used to process highenergy photons, allowing lowenergy photons to pass through. These became absorbed in the narrowerband material forming the absorber layer (hence the subscript "A" in Fig. 1). The thickness of the corresponding layers will be referred to as DW and DA, correspondingly, making the total device thickness equal to DWA = DW + DA. The presence of window and absorber layers allows to optimize solar spectrum use and reduce device heating that is more prominent in pn junctions, where more absorbed photons have the energies exceeding band gap of the material. The other benefit consists in increase of material choice for creating the junction, because not all semiconductors can be obtained in both modifications with p – and n – conductivity. On the negative side, the mismatch of lattice parameters of junction components create undesirable defects, and difference in thermal expansion coefficient may be critical for stability of the device if used under the elevated temperatures.
The contact of two materials with different conductivity type leads to the formation of space charge region [7] associated with diffusion potential difference Ud. Within this so
called depletion region of width wn + wp (Fig. 1) energy band bending occurs. In general case, due to the difference of band gap values of window and absorber layers (EgW and EgA, respectively) there will be band discontinuities AEV and AEC, introducing additional energy barriers for the carriers and paving the way for different types of tunneling effects. The band diagram of a heterojunction can be constructed using electron affinities for both materials – xW and xA, respectively, which allows to construct band diagram of the structure.
Under illumination, the electrons obtain the energy sufficient for moving into conduction band, creating holes in the valence band. The resulting nonequilibrium electronhole pair can disappear due to recombination. However, if it is generated in the vicinity of the junction, the embedded electric field of the space charge region will exert different forces on the carriers in accordance with their charges, moving them towards the contacts where they contribute to the photocurrent of the external circuit.
The transport of the carriers, in addition to the action of the embedded electric field, is also governed by the diffusion caused by the difference in concentrations of electrons and holes in the corresponding parts of the device. This mechanism can be described as [8]
Jn = en (XKE( x)+ eDn^
where n, p are concentration of electrons and holes, pn and pp are their mobilities and Dn, Dp are diffusion coefficients. The action of the electric field is limited to the space charge region that is characterized by the thickness
with dielectric constants of window and absorber materials ew and eA, respectively. The values of Nd and Na correspond to the concentration of donors and acceptors that define conductivity type of the materials forming heterojunction. The height of the energy barrier, Ud, can be manipulated by application of a voltage U, which is also included into (3). Calculating the value of wn + wp it can be shown that the space charge region usually is of negligible thickness in comparison with that of the entire device. Therefore, one can simplify the equations (1) and (2) by keeping only diffusion terms:
The resulting expressions can be rewritten relating current variation to the difference of recombination and generation rates. For simplicity, we will present here only equations describing window layer, for which the minority carriers are holes:
Recombination rate rp = (p – pn 0) / zpW depends on the difference between nonequilibrium
and equilibrium concentrations of holes given in numerator, and a characteristic time TpW defined by recombination processes taking place in the system. Generation rate gW = aWФ0exp(aWx) includes absorption coefficient of the material aW, spectral power of incident light flux Ф0 and distance from the surface x. Substituting these expressions into formula (5) one will obtain
d2p _ p – Vn0 – ®0aW *wx,,,
dx2~ Lp ~ Dp Є (6)
where L p = "ijDpzp is the diffusion length for the holes. Equation (6) should be solved taking into account the boundary conditions. On the front of the cell, the variation of hole concentration is connected with surface recombination rate sp:





At the boundary with space charge region, the concentration of holes is equal to:
p(DW – Wn) = Pn0eXP (eU / kBT) (8)
where kBT is a product of the Boltzmann constant and the temperature. The solution of the equation (6) is usually written in the form


with CrP = D °2 L 2) and coefficients Ap and Bp that can be found from boundary condi
Dp(1 aWL p ) F F
tions. The similar equations can be obtained for electrons as minority carrier in absorber part of the device, yielding the general solution for carrier concentration as



where DW appearing in the first exponent denotes the decrease of light flux upon passing through the window layer. Now, the total current through passing to the external circuit can be obtained as the sum of electron and hole currents at the contacts together with total photogeneration current reduced by integral describing recombination losses:
Dwa Dwa
J = JP (0) + Jn (Dwa ) + e J g(x)dx – e J r(x)dx (11)
00
Two first terms in (11) can be easily found using the expressions for carrier concentrations
(9) , (10) and their relation to the corresponding current components (4):






aAC0°e aaDa + —(Bne Dwa/Ln – AneDwA/Ln)
The generation term from (11) can be calculated analytically:
dwa dw Da
e J g(x)dx = Ф^ащв J e~a’W%dx + аАФ0е~а’А/В’^ e J e 0 0 0
= Ф0є Гі – e (awDw
The recombination term is calculated numerically, taking into account the distribution of nonequilibrium carrier concentration in window/absorber and various recombination mechanisms such as direct recombination, HallShockleyRead recombination involving impurity levels in band gap, and Auger recombination for highenergy carriers that transfer their excess energy to another particle [8].
For numerical simulations we considered the heterojunction AlxGa1xAs/GaAs [9] characterized by a small lattice mismatch of 0.127%. The window layer remains directband semiconductor for aluminium contents less than 45%. Material parameters used in our simulations are listed in Table 1 as functions of aluminium content x and temperature T.
Parameter 
Value or calculation formula 
Dielectric constant e 
12.90 – 2.84x 
Electronic affinity x, eV 
4.07 1.1 x 
DOS effective masses 
mn*= (0.067 + 0.083x) m0; mp* = (0.62 + 0.14x) m0 
Electron mobility цп, cm2/(Vs) 
( 8 – 22x + 10x 2 )103 
Hole mobility Цр, cm2/(Vs) 
( 3.7 – 9.7x + 7.4x 2 )102 
Band gap Eg, eV 
1.424 + 1.247x – (5.4104T2)/(T+204[K]) 
Table 1. Parameters of AlxGa1xAs used in calculations 
In the framework of the current chapter, we studied the dependence of window layer thickness DW on the efficiency of AlGaAs/GaAs solar cell. All calculations were done for AM1.5 illumination [10]. First, we considered the question about the thickness ratio of window/ absorber layers (Fig. 2). The resulting plot has roughly triangular shape with the grayed out area in the bottom left corner where the system is not converging to any solution. As one can see from the figure, the efficiency ц exceeding 24% is obtained for thicker cell (DWA = 300 om), which is expected because the junction should have enough material for a considerable absorption of solar light. When the cell is 3 om thick, the value of ц reaches 20% at most. It is also clear from the figure that the cell performs better with a thin window layer. For example, the efficiency over 20% is reachable for a thick cell with window layer thickness under
0. 5% of Dwa, i. e., DW < 1.5 jm. This result proves that the embedded electric field of space
charge region should be located quite close to the surface where the major photogeneration of nonequilibrium carriers takes place, ensuring efficient separation of electronhole pairs and reducing losses due to the recombination processes. If the junction is located deeper into the cell, the embedded field becomes less efficient. Also, thicker absorber layer augments the number of processed photons that increases the current flowing to the external circuit.
Figure 2. Dependence of Al029Ga071As/GaAs solar cell performance as a function of window and cell thickness 
Another important point is the adjustment of band gap difference between the heterojunction components that modifies the percentage of light processed by window and absorber layers. Having in mind that thicker cell has better overall performance, we performed calculations varying the aluminum contents x in AlxGa1xAs and thickness of the window layer. These results are presented in Fig. 3. Similarly to Fig. 2, the case of ultrathin window precludes numerical convergence and is greyed out. As one can see from the figure, the efficiency landscape has two prominent details. For comparative thick window layer with DW above a micron, it has a pronounced maximum at x = 29.5%that does not shift with variation of DW by two orders of magnitude. The maximum changes into a wide plateau with ц > 17% for x exceeding 20%, following with a quick drop of efficiency for decreasing aluminum content. Increase of x above 30% also causes abrupt drop of the efficiency. We explain this behavior by optimal adjustment of band gaps EgW and EgA ensuring good separation of solar spectra into "high" and "low"energy parts processed by window and absorber layer, respectively, for 20% < x < 30%. For lower x, the difference of band gaps is insufficient. For higher x, the difference is too big so that the energy of the photons passing the window layer is high in comparison with EgA, which will result in excess Auger recombination rate and increased cell heating.
Figure 3. Efficiency of AlxGa1xAs/GaAs solar cell as a function of window layer thickness DW and aluminum contents x 
However, when window layer becomes very thin (DW < 1am), the system starts to behave differently. Now, the generation of the carriers in direct vicinity of junction boundary provides a significant benefit by quick and efficient separation of carriers by electric field associated with space charge region, reducing recombination losses. Nevertheless, even the contour line for n = 23% shows that the cell performs slightly better when aluminum contents gravitates towards x = 30%.
Therefore, theoretical treatment of semiconductor solar cell followed by numerical simulations allowed to obtain useful information about the system, which can significantly simplify the experimental optimization of solar cell parameters by suggesting the most promising ranges of parameters that corresponds to the highest efficiency of photovoltaic conversion of solar energy.