Electro-Physical Interpretation of the Degradation of the Fill Factor of Silicon Heterojunction Solar Cells Due to Incomplete Hole Collection at the a-Si : H / cSi Thermionic Emission Barrier

An electro-physical interpretation for the degradation of the Fill Factor in p+/n silicon heterojunction solar cells (SHJ) due to incomplete hole collection at the thermionic emission barrier at the amorphous/crystalline silicon (a-Si:H/c-Si) hetero-interface is proposed supported by results of AFORS-HET device simulations. Under illumination, reflected holes at the thermionic barrier pile up at the hetero-interface which strengthens the dipole with the negative dopant ions in the doped a-Si:H(p+) layer and enhances the electric field passing through the a-Si:H layer. Such an enhanced electric field sweeps back the free holes spilling over in the intrinsic a-Si:H(i) layer from the a-Si:H(p+) layer considerably depleting the double a-Si:H layer and enhancing its resistance and the overall cell series resistance. Therefore, the degradation due to incomplete hole collection at the thermionic emission barrier under illumination can be assimilated to the effect of a series resistance does not affect the cell open circuit voltage but degrades only its fill factor. The resistance enhancement is found to be bias-dependent and to increase with decreasing the doping level in a-Si:H(p+). Predictions of the proposed model for different hole reflection probability at the barrier and for different thicknesses of the intrinsic a-Si:H(i) layer agree perfectly with the results of simulations.


Introduction
The carrier selection property of the amorphous/crystalline silicon (a-Si:H/c-Si) hetero-junction has been demonstrated for the first time more than three decades ago on npn bipolar junction transistors, which showed enhanced a-Si:H emitter efficiency and current gain [1,2].Such a property has then been exploited later to reduce the dark current and increase the open circuit voltage (V OC ) of the a-Si:H/c-Si heterojunction solar (SHJ) cell [3,4].Extensive research and development have been devoted to the design and technology of such a cell, especially p + doped a-Si:H(p + ) on n-type c-Si substrate, which led to a continuous improvement in the cell efficiency [5,6] to presently exceed 26% [7].
The superior performance of the SHJ cell is mainly the result of the reduced saturation current of the p-type inversion layer emitter induced in the n-type c-Si side of the a-Si:H/c-Si hetero-interface [8,9].This results in a strong boost in V OC to reach values close to 760 mV, which cannot be achieved in any other type of high efficiency c-Si cell.Several factors contribute to minimizing the emitter dark saturation current and hence to the V OC boost in the SHJ cell.Minimum interface recombination is ensured by inserting an ultrathin intrinsic a-Si:H(i) buffer layer between the a-Si:H(p + ) layer and the c-Si substrate [3,4].In addition, band-gap narrowing in the heavily populated inversion layer is expected to be less significant than in doped layers.Band-gap widening has even been reported in strong inversion layers due to carrier quantum confinement [10,11], which results in a much smaller intrinsic carrier concentration in the p + inversion layer compared to heavily doped emitters.Finally, bulk recombination in the very thin inversion layer emitter is expected to be negligible.All these factors are usually lumped in the statement: "excellent passivation property of the a-Si:H contact" commonly used to justify the very small emitter saturation current of SHJ cells.It has been shown that the stronger the inversion, the better the cell performance [8], which means that the equilibrium band bending at the hetero-interface and consequently the difference between the work function of a-Si:H and that of c-Si should be large.Therefore, maximum free hole concentration in the p-type doped a-Si:H(p + ) layer should be ensured.On the other hand, low a-Si:H doping levels or significant depletion of the layer results in a weak inversion layer with degraded cell performance [8,12].It has also been shown that using non-appropriate material with a relatively small work function for the Transparent Conducting Oxide (TCO) used at the front contact results in a front Schottky rather than ohmic contact [12][13][14][15][16][17][18][19], which leads to further depletion of the Si:H(p + ) layer and hence to weak inversion which substantially degrades the cell V OC and FF, as detailed in [12].Therefore, extensive research is being carried out exploring novel TCO materials that are transparent, conducting, and having an adequately large work function, e.g., [20][21][22][23].
In addition to a high V OC , and apart from slight optical absorption loss in the very thin TCO and a-Si:H layers, there are no fundamental constraints that limit the short circuit current, I SC , in SHJ cells to values smaller than that achieved in high efficiency c-Si cells.Such values could be achieved by optimizing the cell technology, design, and using proper materials.Optical absorption in SHJ cells is minimized by making the a-Si:H layer ultrathin and, as mentioned earlier, by selecting an appropriate TCO material.In general, values of I SC comparable to those of high efficiency c-Si cells are steadily reached in present SHJ cells.
On the contrary to the superior V OC and equally high I SC , the FF in SHJ cells rarely exceeds the range 75-79% compared to over 84% achieved in high efficiency c-Si cells.One of the main reasons for such a deficiency relates to inefficient hole transport across the energy barrier at the hetero-interface, resulting from the difference between the energy band-gap of a-Si:H and that of c-Si.In principle, thermionic emission is the fundamental mechanism that controls such a transport.However, several tunneling mechanisms through this barrier have been evoked for a relatively long time.These mechanisms remain in dispute with no general agreement.Direct tunneling, capture-emission tunneling, multistep tunneling, tunnel by hopping via tail states, as well as mixed bias-dependent tunneling/diffusion modes have been suggested but yet not confirmed with full certainty [24][25][26][27][28][29][30][31][32].Some studies have shown that tunneling does not prevail in p + /n SHJ cells e.g., [28] or may not be needed to describe the performance of the cell with energy band-gap up to 1.85 eV [32], while others have concluded that tunneling is critical and needed [26].It is believed that selecting the optimum a-Si:H band-gap and thickness in conjunction with effective tunneling is necessary to achieve a high FF close to 84%, as is reported for the record efficiency SHJ cell [7] but is not always possible to ensure.Thermionic emission transport over the energy barrier, however, remains the fundamental mechanism controlling and, in many cases, hindering hole transport across the a-Si:H/c-Si hetero-interface.Therefore, a good understanding of the constraints imposed by such a transport mechanism and its effect on the device performance is necessary in order to optimize the cell design and technology.
Using AFORS-HET device simulations [33,34], it has been recently shown that thermionic emission hole transport across the hetero-interface in SHJ cells, with flat band perfectly ohmic contacts, results in distortion in the I-V characteristics under illumination that becomes more serious as the active doping level in a-Si:H decreases [35].It was suggested that incomplete hole collection (hole reflection) at the thermionic barrier results in a hole forward current through the pn junction, establishing a forward bias V SC , even at zero terminal voltage.The study concentrated on V SC , which was found to depend on the applied bias and on the doping level in a-Si:H(p + ) [35].The link between V SC and the thermionic emission transport mechanism has been confirmed by studying the dependence of V SC on the thermionic emission parameters.Although all the results presented in [35] are consistent and seem to indicate the role of hole reflection at the thermionic barrier in the degradation, the work lacks answers to several questions.For instance, it is not shown how the forward voltage V SC established at the pn junction is compensated in the cell closed loop (at short circuit as well as under bias).Without such a compensation, the loop voltage equation is offended and the presence of V SC becomes questionable.In addition, the bias dependence of V SC was not justified, and its dependence as well as that of the distortion of the I-V characteristics on the doping level in a-Si:H(p + ) were not discussed nor interpreted in [35].
Therefore, in an attempt to find answers to these questions, the present analysis elaborates on the hole reflection model proposed in [35] by carrying out extensive AFORS-HET simulations of SHJ cells and investigating carefully all possible details offered by the results of these simulations.The study aims at deriving a higher level model that can give more measurable and quantitative indications that answer the questions raised above.With such a model, the responsibility of incomplete collection (hole reflection) at the thermionic barrier in the degradation of the FF of the SHJ cell would be clearly established so that the cell could be optimized accordingly.In general, this study contributes to a better understanding of the SHJ principle of operation and limitations and enriches the physics of the device, without denying or favoring any other transport mode.

Methodology
The p + /n SHJ cell under study is sketched in Figure 1.The substrate of the device consists of 200 µm high quality crystalline silicon (c-Si) substrate with a Shockley-Read-Hall (SRH) carrier lifetime of 5 ms and a default n-type doping concentration N D = 1.5 × 10 16 cm −3 .A double ultrathin intrinsic a-Si:H(i)/p-type doped a-Si:H(p + ) layer (5 nm/10 nm) is deposited on the surface of the c-Si substrate.The AFORS-HET device simulator is capable of dealing with pyramidal surface texturing, which is therefore implemented in the structure under study to ensure maximum light coupling and maximum I SC .A TCO layer is normally deposited on top of the a-Si:H(p + ) for the front contact.However, as mentioned earlier, the study deals with FF degradation due to the thermionic emission barrier at the hetero-interface, and any other source of performance degradation should be turned off.Hence, the TCO/a-Si:H(p + ) interface (the front contact) is set always to a flat band condition, unless stated differently, and assumed to be perfectly ohmic with a surface recombination velocity of 10 6 cm/s, as expected at a metal/Si contact.Furthermore, the TCO is assumed to be transparent with no light absorption, so its material and thickness are irrelevant to the present study.Consequently, the TCO layer is omitted from the structure to be simulated setting the top of the cell at the a-Si:H(p + ) top surface.Default values of the a-Si:H energy band-gap (E G,a-Si ) and electron affinity (χ a-Si ) are set to 1.72 eV and 3.9 eV, respectively, as commonly used [36][37][38][39].The default physical models and properties of a-Si:H implemented in the AFORS-HET simulator are maintained in all simulations.The work function of a-Si:H(p + ) depends on the free hole concentration, hence on the chemical doping concentration in the layer and on the dangling bond defect density in the bulk and at the interfaces.These dangling bond defects are represented in the AFORS-HET simulator by Gaussian distributions with a peak density N DB0 in the bulk and D it0 at the a-Si:H/c-Si interface.It is well known that the dangling bond density in the bulk of a-Si:H increases as the chemical doping the concentration is increased [40,41].Therefore, the peak value N DB0 cm −3 eV −1 in the bulk is set to the value of the doping concentration in the a-Si:H(p + ) layer, and to a minimum value equal to 1.385 × 10 16 cm −3 eV −1 in the intrinsic a-Si:H(i) layer.The peak dangling bond defect density at the a-Si:H(i)/c-Si hetero-interface, D it0 , is set to 10 10 cm −2 eV −1 , which ensures excellent passivation and negligible recombination at the interface.Shockley-Read-Hall recombination and Auger recombination are considered in the simulations, while band-gap narrowing (BGN) is ignored in the heavily doped a-Si:H due to lack of data, and in the heavily populated inversion layer where it is expected to be negligible, as mentioned earlier.
All simulations are carried out at room temperature (T = 300 K).
Since the performance of the SHJ cell is fundamentally determined by the difference between the work function of a-Si:H and that of c-Si, and the former depends on the hole activation energy (E F -E V ) and the associated p-type doping level N a-Si in a-Si:H(p + ), two extreme doping levels will be considered: (1) the lowest doping level N a-Si = 5 × 10 18 cm −3 in cell SHJ 1 , and ( 2) the highest doping level N a-Si = 1 × 10 20 cm −3 in cell SHJ 4 .Results for cells with intermediate doping levels taken from [35] may be used whenever necessary.

Cell Characterization at Equilibrium: Energy Band Diagrams
The simulated band diagram at equilibrium reproduced from [35] for cell SHJ 1 and obtained in the present work for cell SHJ 4 are displayed in Figure 2a,b, respectively.Conduction and valence band offsets (discontinuities) ∆E C and ∆E V occur at the a-Si:H(i)/c-Si hetero-interface due to different values of electron affinity and energy band-gap in a-Si:H and in c-Si.As predicted by band line-up and the electron affinity rule [42,43], these offsets are independent of the doping level.Hence, for both cells where χ c-Si and E G,c-Si represent the electron affinity and the energy band-gap of the c-Si substrate.
At the hetero-interface, the valence band bends and nears the Fermi energy, which induces p-type inversion in the c-Si side of the interface.Such an inversion layer, confirmed experimentally [44][45][46], plays a major role in boosting V OC of the cell and in controlling the overall SHJ cell performance, as previously detailed [8,9,12].The total band bending or built-in potential V bi,total in the structure is given by the difference between the work functions of the doped a-Si:H(p + ) layer and of the n-type c-Si substrate.Such a potential is distributed between the two regions: (1) V bi,a-Si in a-Si:H, and (2) V bi,c-Si in the c-Si substrate, such that The values of V bi,total , V bi,a-Si , and V bi,c-Si at equilibrium extracted from the band diagrams of Figure 2a,b are listed for both cells in Table 1.Due to the larger band bending in c-Si in cell SHJ 4 than in cell SHJ 1 , the induced inversion layer in the former is much stronger, and its peak hole concentration at the hetero-interface is much higher and amounts to 4.8 × 10 19 cm −3 compared to 8.55 × 10 18 cm −3 in cell SHJ 1 [35], as depicted in Figure 3a.Additionally, the free hole concentration in the p-type a-Si:H(p + ) of cell SHJ 4 is much larger and amounts to 2.1 × 10 19 cm −3 , compared to 3.5 × 10 16 cm −3 in cell SHJ 1 [35].The electric field distributions depicted in Figure 3b show that, due to the larger band bending and larger inversion hole population in cell SHJ 4 , the electric field at the hetero-interface is higher and mainly confined in the depleted a-Si:H(i) layer due to the availability of a large concentration of negative dopant ions in the a-Si:H(p + ) layer.On the other hand, the electric field at the hetero-interface in SHJ 1 is much smaller, and extends deep in the a-Si:H(p + ) layer with fewer less negative ions capable of terminating the field.In addition, the much larger concentration of free holes spilling-over in the intrinsic a-Si:H(i) layer of cell SHJ 4 compared to cell SHJ 1 reduces the layer effective thickness in the former, as depicted in Figure 3a.This explains the gradual increase in the electric field in the a-Si:H(i) layer of SHJ 4 and the peak shown close to the edge of the layer.The flat electric field distribution over the whole a-Si:H(i) layer of cell SHJ 1 is due to much less hole spill-over, as explained above.While the heavy hole population in the p + inversion layer is confined in the 5nm high field region at the hetero-interface, p-type inversion extends deeper in the c-Si substrate, which shifts the pn junction approximately 85 nm away from the hetero-interface.

SHJ Cell Characterization under Illumination
AFORS-HET simulations are carried out under AM1.5G illumination assuming that carrier transport across the a-Si:H/c-Si hetero-interface occurs by thermionic emission [47][48][49][50][51][52][53][54][55][56][57][58][59][60][61], which fundamentally occurs due to the inevitable presence of the valence band discontinuity ∆E V at the a-Si:H/c-Si hetero-interface.In such a case, the hetero-interface is considered to be a boundary with the thermionic current setting the boundary condition.Hence, it is expected that the simultaneous solution of the continuity, current and Poisson's equation carried out by the simulator would depend on the thermionic emission parameters, namely, the Richardson coefficient, the temperature, and the thermionic barrier height, which has been confirmed in [35].In some situations, hole transport across the hetero-interface is assumed to occur by drift-diffusion.In that case, the thermionic emission barrier is ignored such that holes are fully transported across the hetero-interface, and the losses attributed to the barrier are avoided.Therefore, the I-V characteristics simulated assuming drift-diffusion transport will only be used as a reference for comparison.The front contact is assumed always to be at flat band unless stated differently.Like in [35], the default value for the Richardson constant A * = 9.56 A K −2 cm −2 implemented in AFORS-HET is maintained.Such a value is considered to be reasonable taking into account the uncertainties in the value of the hole effective mass [62-65] and in the probability of hole reflection at the hetero-interface [50,51,[59][60][61].

Current-Voltage (I-V) Characteristics
Figure 4a displays the I-V characteristics simulated under illumination for the cells SHJ 1 and SHJ 4 to be investigated here, together with those of two other cells SHJ 2 and SHJ 3 with intermediate doping concentrations in a-Si:H(p + ) N a-Si = 1 × 10 19 cm −3 and 5 × 10 19 cm −3 , respectively [35].While V OC and I SC maintain their proper values, the FF in the cells degrade considerably to reach 40% in cell SHJ 1 with N a-Si = 5 × 10 18 cm −3 and 75.5% in cell SHJ 4 with N a-Si = 10 20 cm −3 , which is well below what is expected from a high efficiency c-Si cell.It has been suggested, in a merely descriptive way, that such distortion relates to hole reflection at the a-Si:H/c-Si thermionic emission barrier [35].In order to validate such a model, however, a link to a measurable quantity that satisfies the model predictions must be found.For instance, the shape of the distorted I-V characteristics suggests the presence of a high cell series resistance since V oc is practically unaffected by the distortion.When drift-diffusion rather than thermionic emission transport is assumed, however, the severe distortion disappears from the I-V characteristics of cell SHJ 1 as depicted in Curve 4 of Figure 4b, which confirms that the distortion is related to thermionic emission transport but does not systematically link it to a high series resistance.Different I-V characteristics for SHJ 1 simulated under different conditions with and without TCO at the front contact are compared in Figure 4b.In Curves 1, 2, and 3, the distortion caused by reflection at the thermionic emission barrier is clearly recognized.In Curve 1, however, Voc also degrades considerably due to weak inversion caused by the front Schottky contact when the TCO work function is low.In Curve 2, the TCO work function is high enough to avoid such a V OC degradation.Curve 3 is identical to Curve 1 in Figure 4a with flat band front contact.Finally, Curve 4 displays I-V characteristics of the reference simulation assuming drift-diffusion transport and flat band front contact.I-V characteristics similar to those of Curves 1 and 2 were reported previously [26,32], with no clear interpretation for the shape of the distortion.

Energy Band Diagrams under Illumination
The energy band diagram simulated for cell SHJ 1 and displayed in Figure 5a at non-equilibrium under AM1.5G illumination and short circuit conditions (illumination-SC), and assuming drift diffusion hole transport indicates the presence of a quasi-Fermi energy split δ, or equivalently a forward bias, amounting to 80 mV on the pn junction, despite the short circuit condition.Such a bias is established in order to compensate the voltage drop on the cell dark series resistance R s and hence to satisfy the cell closed loop equation with short circuit terminals.Therefore, δ represents the minimum junction bias at short circuit condition irrespectively of the transport mechanism and of the a-Si:H doping level.The value of R s is estimated from Ohm's law R s =δ/I SC (3) to 1.9 Ω cm 2 , which accounts for the resistance of the bulk and of the back contact.Although such a resistance is relatively high and can be reduced to less than half this value if the back contact is optimized, it will be maintained throughout the present work.On the other hand, a much larger quasi-Fermi energy split, designated by V SC in the band diagram of Figure 5b, has been reported when thermionic emission hole transport is considered, and qualitatively linked to hole reflection at the hetero-interface [35].A clear interpretation, however, is still needed for the exact process leading to the creation of Vsc and for its dependence on the a-Si:H(p + ) doping concentration and on the applied bias.In addition, as can be deduced from Figure 5b and as listed in Table 2, the built-in potentials V bi,c-Si in c-Si and V bi,a-Si in a-Si:H at short circuit condition are quite different from their values at equilibrium depicted in Figure 2a and listed Table 1 when thermionic emission transport is assumed, while they remain practically unchanged under drift-diffusion transport, as shown in Figure 5a.Such changes in the built-in potentials may significantly affect the hole profiles, which is relevant to the investigation under study, and hence need to interpreted.Finally, it is useful to reiterate that such important differences between the band diagrams in Figure 5a resulting from simulations assuming drift-diffusion transport and in Figure 5b assuming thermionic emission transport result from the hetero-interface being considered as a boundary in the latter case, as explained in Section 3.2.Table 2. Built-in potential in cells SHJ 1 and SHJ 4 under illumination-SC.

Enhanced Series Resistance Model
As mentioned earlier, hole reflection at the hetero-interface has been suggested to be the origin of the forward bias established at the pn junction under illumination and was taken as a basis for a model that interprets the FF degradation due to the thermionic emission barrier at the a-Si:H/c-Si hetero-interface [35].So far, however, such a model lacks answers to the several questions raised in the introduction.An attempt is made in the following analysis to develop a higher level model that gives acceptable and justified answers to these questions.

Enhanced a-Si:H Resistance under Illumination and Its Relationship with Hole Reflection at the Thermionic Barrier
By inspecting the band diagrams of Figure 5a,b more closely, following the behavior of the hole quasi-Fermi energy and the quasi-Fermi energy split at the junction, and studying the cell short circuit closed loop equation, the forward bias V SC at the junction should be given by where V a-Si is the total voltage drop on the double a-Si:H layer, ∆V is the sharp voltage drop on the a-Si:H(i) layer, well identified in the band diagram of Figure 5b, and V a-Si(p+) is the voltage drop on the doped a-Si:H(p + ) layer.As concluded in Section 3.2.1, the presence of a naturally high dark a-Si:H resistance that justifies the voltage drop V a-Si is ruled out, since it does not show when drift-diffusion transport is assumed.Consequently, the voltage drop on the double a-Si:H layer could only be explained by the presence of an enhanced a-Si:H resistance under illumination.The origin of such enhanced resistance and its exclusive occurrence under thermionic emission transport needs to be clarified.The hole concentration profile in cell SHJ 1 displayed in Figure 6a shows that, when thermionic emission transport is assumed, the peak hole concentration p peak at the hetero-interface increases from typically 8.55 × 10 18 cm −3 at equilibrium to 3.74 × 10 19 cm −3 under illumination-SC, which represents a substantial boost of 400%.On the other hand, the value of p peak practically does not change from its equilibrium value when drift-diffusion hole transport is assumed, and is maintained around 8.7 × 10 18 cm −3 .Therefore, such a boost in p peak under thermionic emission transport can surely be attributed to incomplete hole collection, or equivalently to holes reflected at the thermionic barrier and piling-up as excess holes at the hetero-interface.These excess holes create a dipole with the negative acceptor ions in the doped a-Si:H(p + ) that enhances the electric field crossing the a-Si:H(i) layer well above its value at equilibrium, as depicted in Figure 6b.Such a high field repels the free holes spilling-over in a-Si:H(i) back towards the a-Si:H(p + ) layer, causing full depletion of the a-Si:H(i) layer, which explains the very low and flat hole concentration as well as the very high and flat field distribution in the layer depicted in Figure 6a,b, respectively.The enhanced electric field causes further depletion of the a-Si:H(p + ) layer and decays gradually as it penetrates deeper in the layer.Consequently, the resistance of the double a-Si:H layer is strongly enhanced, which justifies the voltage drop V a-Si of Equation (4a), and may finally lead to a considerable increase in the cell series resistance that explains the distorted shape of the I-V characteristics displayed in Figure 4a.Furthermore, the integration of the enhanced and extended electric field in a-Si:H explains the increase in the built-in potential in a-Si:H, V bi,a-Si , compared to its value at equilibrium, as discussed in Section 3.2.2.Such an increase is balanced by a decrease in the built-in potential in c-Si, V bi,c-Si , in order to force the total built-in potential at short circuit to be equal to its value at equilibrium, taking into account the dark resistance voltage drop δ, as confirmed in Table 2.
At short circuit condition and using Equation (4a), the enhanced a-Si: where V SC is to be extracted from the band diagram under illumination-SC, δ to be accurately determined from the band diagram under drift diffusion transport or evaluated knowing the dark series resistance, and I SC determined from the I-V characteristics under illumination.Typically, R a−Si in cell SHJ 1 is estimated to 9 Ω cm 2 .In addition, the latter is to be divided into (1) a dominant a-Si:H(i) resistance given by and ( 2) a resistance of the depleted region in the a-Si:H(p + ) layer to be determined from Hence, R a−Si(i) can be estimated with ∆V in Equation (5b) extracted from the band diagram of Figure 5b, or can be calculated knowing the low flat free hole concentration in the a-Si:H(i) layer in Figure 6a and the layer thickness.Finally, R a−Si(p+) follows from Equation (5c).Typically, R a−Si(i) and R a−Si(p+) in SHJ 1 at short circuit are estimated to be 6.25 and 2.75 Ω cm 2 , respectively.It is worth noting that the hole profile, the electric field, and the built-in potential close to the interface remain intact when drift-diffusion transport is assumed, which confirm the proposed relationship between hole reflection at the thermionic barrier and the enhanced a-Si:H series resistance under illumination.
Finally, the proposed enhanced resistance model developed here explains clearly that the forward bias V SC revealed in [35] is compensated in the cell closed loop at short circuit or at any bias by the voltage drop on the enhanced a-Si:H resistance.In addition, it also becomes clear that, since the effect of hole reflection at the thermionic emission barrier is finally assimilated to the effect of an enhanced series resistance, the fact that only the cell FF is degraded while the cell open circuit voltage is not affected is now clarified.

Impact of the Probability of Hole Reflection
The point here is just to confirm that the proposed model operates systematically and consistently.By inspecting Table 3 and the inserts in Figure 6a and in Figure 7a, it appears that under illumination and thermionic emission transport p peak in SHJ 1 decreases from 3.74 × 10 19 to 1.86 × 10 19 cm −3 when A * increases from 9.56 to 20 A K −2 cm −2 due to reduced reflection.The weaker dipole and reduced electric field enhancement confirmed in Figure 7b and in Table 3 lead to less depletion of the double a-Si:H layer, as depicted in Figure 7a compared to Figure 6a, and hence to a smaller enhancement in the resistance of the double a-Si:H layer, as listed in Table 3, which leads to an improvement in the I-V characteristics depicted in Figure 8.As expected, only at very large values of A * , i.e., when the reflection probability is very small, the cell series resistance is determined only by its value in the dark and the I-V characteristics tend to converge with those obtained assuming drift-diffusion transport.The results of the simulations of cell SHJ 1 show that reducing the thickness of the a-Si:H(i) layer (t i ) while fixing the probability of hole reflection at the hetero-interface results in larger values of p peak and of the electric field at equilibrium, as shown in Table 4.Under illumination-SC, however, the total value of p peak including the reflected holes as well as the enhanced electric field at the hetero-interface are practically independent of t i " as seen in Table 4. Therefore, the relative boost in the electric field decreases from 100% when t i = 5 nm to 61% when t i = 2 nm and to 37% when the a-Si:H(i) layer is totally omitted.The field enhancement, however, is still large enough to fully deplete the a-Si:H(i) layer and hence to enhance its resistance.As the layer thickness is reduced, however, its resistance decreases proportionally which forces V SC to decrease as confirmed in Table 4.The total a-Si:H enhanced resistance R a−Si estimated using Equation (5a), its components R a−Si(i) in a-Si:H(i) estimated using Equation (5b), and R a−Si(p+) in a-Si:H(p + ) estimated using Equation (5c) are listed, respectively, in Columns A, B, and C of Table 4.The value of R a−Si(p+) is practically independent of t i and falls in the range 2.7 to 2.85 Ω cm 2 , which is expected since the a-Si:H(i) layer is fully depleted and hence its thickness does not influence the electric field passing through.When the a-Si:H(i) layer is totally omitted, the value of R a−Si(p+) is estimated to be 2.55 Ω cm 2 , which is slightly smaller since all free holes in a-Si:H are confined now inside the doped layer.Note that the voltage drop δ attributed to dark cell series resistance at short circuit condition remains unaffected by the a-Si:H(i) thickness, which confirms that the dark resistance of the double a-Si:H layer is relatively small in the present case.The I-V characteristics simulated under illumination for different values of a-Si:H(i) thickness displayed in Figure 9 show a continuous increase in the FF from 40% when t i = 5 nm to 64% when the intrinsic layer is totally omitted.The latter value, however, is still small, which indicates that the excess hole depletion in the doped a-Si:H (p + ) under illumination and the associated enhanced resistance still play an important role in the FF degradation even if the intrinsic a-Si:H(i) layer is removed.

Bias Dependence of the Cell Series Resistance
Under illumination, the total inversion layer peak hole concentration p peak starts to decrease when the forward bias applied to the cell exceeds 0.2 V, as depicted in Figure 10a and listed in Table 5, which leads to a weaker excess dipole and reduced enhancement in the electric field at the hetero-interface, as depicted in Figure 10b and also shown in Table 5. Accordingly, the proposed model predicts less depletion of the double a-Si:H layer and smaller enhanced a-Si:H resistance with increasing the cell forward bias, which are both confirmed in Figure 10a and Table 5, respectively.The enhanced double a-Si:H layer resistance R a−Si (V) is estimated as a function of the cell forward bias from where V is the cell terminal voltage, V j (V) is the total bias at the junction to be extracted from the band diagram, and I(V) is the cell current to be extracted from the I-V characteristics.For instance, at short circuit condition V = 0, V j (0) = V SC , I(0) = I SC , I(0) × R s = δ such that Equation ( 6) reduces to Equation (5a).The resulting values of R a-Si (V) listed in Table 5 for SHJ 1 confirm that the enhanced a-Si:H resistance is bias-dependent, decreasing with increasing the forward bias, which explains the bias dependence of V SC .
The accuracy of the enhanced resistance model is checked by comparing the I-V characteristics of cell SHJ 1 regularly simulated assuming thermionic emission transport, with the I-V characteristics of the same cell simulated assuming drift-diffusion transport with a bias dependent resistance, having the values listed in Table 5, connected in series with the cell.A perfect matching is obtained, as depicted in Figure 11, which confirms the validity of the approach and the accuracy of the model.The reference I-V characteristics of cell SHJ 1 under drift-diffusion transport are also displayed in Figure 11 for comparison.5 connected in series.
Table 5. Dependence of R a-Si in cell SHJ 1 on forward bias.As discussed in Section 3.1 and depicted in Figure 3a,b, the equilibrium value of the peak inversion layer hole concentration p peak at the hetero-interface and of the electric field are much larger in cell SHJ 4 than in cell SHJ 1 .Since hole reflection is not expected to be very different in either cell at identical values of A * , of ∆E V , and of T, the relative increase in the hole concentration at the hetero-interface and the associated enhancement in the electric field upon hole reflection under illumination are expected to be much smaller in cell SHJ 4 than in cell SHJ 1 .With such a smaller enhancement in the electric field, and with an initially much larger concentration of free holes spilling-over in a-Si:H(i), the depletion of the latter and of the a-Si:H(p + ) layer is expected to be much less significant in SHJ 4 ; consequently, the enhanced a-Si:H resistance is expected to be much smaller than in SHJ 1 .Moreover, at similar current, the voltage drop on R a−Si would be much smaller in SHJ 4 , which will be reflected on the value of V SC .By interpolation, it can be concluded that the enhanced series resistance under illumination decreases continuously as the doping level in a-Si:H(p + ) increases, which perfectly explains the doping dependence of V SC recorded and not interpreted in [35].
By inspecting the results of the simulations carried out for cell SHJ 4 under illumination-SC, it appears that the value of p peak is enhanced by 17% reaching 5.6 × 10 19 cm −3 compared to 4.8 × 10 19 cm −3 at equilibrium and that the electric field in a-Si:H(i) is enhanced by only 4% on the average, as depicted in Figure 12a,b, respectively.The value of V SC extracted from the simulated band diagram for cell SHJ 4 under illumination-SC amounts to 104 mV including the 80 mV dark cell series resistance voltage drop δ, which results in a total enhanced a-Si:H resistance of 0.64 Ω cm 2 , mainly due to the partial depletion of the hole spill-over region in a-Si:H(i) layer, compared to 9 Ω cm 2 in cell SHJ 1 .Finally, apart from the effect of δ, the expanded plot of the valence band diagram displayed in Figure 13 shows only a slight change in the built-in potential in a-Si:H and in c-Si under illumination-SC compared to the situation at equilibrium, which is consistent with the relatively small degradation of the FF in cell SHJ 4 from 78.4% with drift-diffusion transport to 75.5% with thermionic emission transport.Such findings are consistent, agree perfectly with the model predictions and give a clear explanation to the doping dependence of the I-V characteristics depicted in Figure 4a.

Summary and Conclusions
A model is developed that accurately interprets the distortion of the I-V characteristics of SHJ cells under illumination linked to hole reflection at the thermionic emission barrier.In the proposed model, the reflected holes at the thermionic emission energy barrier pile up at the hetero-interface and create an excess dipole with negative ions in the a-Si:H(p + ) layer.The enhanced electric field crossing the a-Si:H(i) layer sweeps the free holes spilling-over from the a-Si:H(p + ) layer back, strongly depleting the intrinsic layer and widening the depletion of the doped layer, which enhance the total cell series resistance.With such a model, the large forward bias V SC established at the pn junction even at short circuit terminals compensates the voltage drop on the enhanced series resistance under illumination.Also, since the model explains the bias dependence of V SC, which follows that of the enhanced a-Si:H resistance linked to the bias dependence of the excess electric field at the hetero-interface.In addition, the model predicts that cells with heavily doped a-Si:H(p + ) would suffer less from hole reflection at the thermionic emission under illumination because the relative excess hole concentration piling-up at the hetero-interface is smaller.Therefore, the enhanced field, a-Si:H depletion and resistance are undermined, which results in smaller values of V SC and to improved I-V characteristics and cell FF.
Moreover, since the proposed model assimilates the impact of hole reflection at the thermionic barrier to a series resistance effect, it gives an answer to why only the cell FF is degraded, while the cell open circuit is unaffected.Accordingly, the degradation associated with the thermionic barrier can be clearly differentiated from that related to the front Schottky contact in the presence of an inadequate TCO usually characterized by a degradation in both the Voc and the FF.

Figure 3 .
Figure 3. (a) Hole concentration profile, and (b) electric field distribution, simulated for cells SHJ 1 and SHJ 4 at equilibrium.

Figure 8 .
Figure 8. Simulated I-V characteristics under illumination for cell SHJ 1 with thermionic emission transport for different values of the Richardson constant A * .The I-V characteristics assuming drift-diffusion (Curve 5) is also plotted for comparison.

Table 4 .
Enhanced cell resistance in cell SHJ 1 versus a-Si:H(i) thickness.

Figure 9 .
Figure 9. Simulated I-V characteristics under illumination for cell SHJ 1 with thermionic emission hole transport for different values of a-Si:H(i) layer thickness.

Figure 10 .
Figure 10.Simulated (a) hole concentration profile and (b) electric field distribution for cell SHJ 1 under illumination for different cell forward bias.

Figure 12 .
Figure 12.Simulated (a) free hole concentration profile and (b) electric Field distribution at equilibrium (Cont.line) and under illumination-SC (Dash) for cell SHJ 4 assuming thermionic emission hole transport at the hetero-interface with a default value of A * .

Figure 13 .
Figure 13.Simulated valence band diagram of cell SHJ 4 at equilibrium and under illumination-SC assuming thermionic emission transport across the hetero-interface with the default A * value.

Table 1 .
Built-in potential and its components in cells SHJ 1 and SHJ 4 at equilibrium.

Table 3 .
Dependence of enhanced a-Si:H series resistance in SHJ 1 on hole thermionic emission probability (A * ).