Articles de revista
http://hdl.handle.net/2117/2413
2019-12-06T06:26:11Z
2019-12-06T06:26:11Z
A self-branched lamination of hierarchical patronite nanoarchitectures on carbon fiber cloth as novel electrode for ionic liquid electrolyte-based high energy density supercapacitors
Manikandan, Ramu
Justin Raj, Chellan
Nagaraju, Goli
Puigdollers i González, Joaquim
Voz Sánchez, Cristóbal
http://hdl.handle.net/2117/172952
2019-11-23T06:35:59Z
2019-11-22T14:19:03Z
A self-branched lamination of hierarchical patronite nanoarchitectures on carbon fiber cloth as novel electrode for ionic liquid electrolyte-based high energy density supercapacitors
Manikandan, Ramu; Justin Raj, Chellan; Nagaraju, Goli; Puigdollers i González, Joaquim; Voz Sánchez, Cristóbal
The developments of rationally designed binder-free metal chalcogenides decorated flexible electrodes are of paramount importance for advanced energy storage devices. Herein, binder-free patronite (VS4) flower-like nanostructures are facilely fabricated on a carbon cloth (CC) using a facile hydrothermal method for high-performance supercapacitors. The growth density and morphology of VS4 nanostructures on CC are also controlled by varying the concentrations of vanadium and sulfur sources along with the complexing agent in the growth solution. The optimal electrode with an appropriate growth concentration (VS4-CC@VS-3) demonstrates a considerable pseudocapacitance performance in the ionic liquid (IL) electrolyte (1-ethyl-3-methylimidazolium trifluoromethanesulfonate), with a high operating potential of 2 V. Utilizing VS4-CC@VS-3 as both positive and negative electrodes, the IL-based symmetric supercapacitor is assembled, which demonstrates a high areal capacitance of 536 mF cm-2 (206 F g-1) and excellent cycling durability (93%) with superior energy and power densities of 74.4 µWh cm-2 (28.6 Wh kg-1) and 10154 µW cm-2 (9340 W kg-1), respectively. As for the high energy storage performance, the device stably energizes various portable electronic applications for a long time, which make the fabricated composite material open up news for the fabrication of fabrics supported binder-free chalcogenides for high-performance energy storage devices.
This is the peer reviewed version of the following article: Ramu, M., Chellan, J. R., Goli, N., Joaquim, P., Cristobal, V., Kim, B. C., A Self‐Branched Lamination of Hierarchical Patronite Nanoarchitectures on Carbon Fiber Cloth as Novel Electrode for Ionic Liquid Electrolyte‐Based High Energy Density Supercapacitors. Adv. Funct. Mater. 2019, 1906586. https://doi.org/10.1002/adfm.201906586, which has been published in final form at https://doi.org/10.1002/adfm.201906586. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.
2019-11-22T14:19:03Z
Manikandan, Ramu
Justin Raj, Chellan
Nagaraju, Goli
Puigdollers i González, Joaquim
Voz Sánchez, Cristóbal
The developments of rationally designed binder-free metal chalcogenides decorated flexible electrodes are of paramount importance for advanced energy storage devices. Herein, binder-free patronite (VS4) flower-like nanostructures are facilely fabricated on a carbon cloth (CC) using a facile hydrothermal method for high-performance supercapacitors. The growth density and morphology of VS4 nanostructures on CC are also controlled by varying the concentrations of vanadium and sulfur sources along with the complexing agent in the growth solution. The optimal electrode with an appropriate growth concentration (VS4-CC@VS-3) demonstrates a considerable pseudocapacitance performance in the ionic liquid (IL) electrolyte (1-ethyl-3-methylimidazolium trifluoromethanesulfonate), with a high operating potential of 2 V. Utilizing VS4-CC@VS-3 as both positive and negative electrodes, the IL-based symmetric supercapacitor is assembled, which demonstrates a high areal capacitance of 536 mF cm-2 (206 F g-1) and excellent cycling durability (93%) with superior energy and power densities of 74.4 µWh cm-2 (28.6 Wh kg-1) and 10154 µW cm-2 (9340 W kg-1), respectively. As for the high energy storage performance, the device stably energizes various portable electronic applications for a long time, which make the fabricated composite material open up news for the fabrication of fabrics supported binder-free chalcogenides for high-performance energy storage devices.
Capacitance study of a polystyrene nanoparticle capacitor using impedance spectroscopy
Véliz Noboa, Bremnen Marino
Orpella García, Alberto
Bermejo Broto, Sandra
http://hdl.handle.net/2117/172001
2019-11-20T04:55:13Z
2019-11-08T16:52:14Z
Capacitance study of a polystyrene nanoparticle capacitor using impedance spectroscopy
Véliz Noboa, Bremnen Marino; Orpella García, Alberto; Bermejo Broto, Sandra
In this study, a metal-insulator-metal capacitor structure is fabricated using polystyrene nanoparticles. Impedance spectroscopy is used to evaluate the performance of this capacitor in which we found a significant magnitude increment in capacitance and loss tangent compared with an equivalent ideal capacitor with continuous polystyrene layer and same geometry. Capacitance values up to 11.7 and loss tangent values up to 387 (at 0.1 Hz) larger than the expected for a continuous polystyrene MIM capacitor are achieved. The capacitor shows a good stable capacitive behaviour in the frequency range from 0.1 Hz to 100 kHz at room temperature, 30 °C, 40 °C and 50 °C without an effective relaxation process. Nyquist, capacitance, loss tangent and normalized powers curves are analysed by modified Randles model. Also, a slight decrease in the capacitance value at 50 °C is observed, which that may be attributed to space charge localized at the nanoparticles interface and that are affected by the temperature changes.
2019-11-08T16:52:14Z
Véliz Noboa, Bremnen Marino
Orpella García, Alberto
Bermejo Broto, Sandra
In this study, a metal-insulator-metal capacitor structure is fabricated using polystyrene nanoparticles. Impedance spectroscopy is used to evaluate the performance of this capacitor in which we found a significant magnitude increment in capacitance and loss tangent compared with an equivalent ideal capacitor with continuous polystyrene layer and same geometry. Capacitance values up to 11.7 and loss tangent values up to 387 (at 0.1 Hz) larger than the expected for a continuous polystyrene MIM capacitor are achieved. The capacitor shows a good stable capacitive behaviour in the frequency range from 0.1 Hz to 100 kHz at room temperature, 30 °C, 40 °C and 50 °C without an effective relaxation process. Nyquist, capacitance, loss tangent and normalized powers curves are analysed by modified Randles model. Also, a slight decrease in the capacitance value at 50 °C is observed, which that may be attributed to space charge localized at the nanoparticles interface and that are affected by the temperature changes.
Near 5% DMSO is the best: a structural investigation of PEDOT: PSS thin films with strong emphasis on surface and interface for hybrid solar cell
Mahato, Somnath
Puigdollers i González, Joaquim
Voz Sánchez, Cristóbal
Mukhopadhyay, Mala
Mukherjee, Manabendra
Hazra, Satyajit
http://hdl.handle.net/2117/169537
2019-11-05T19:26:30Z
2019-10-09T12:57:26Z
Near 5% DMSO is the best: a structural investigation of PEDOT: PSS thin films with strong emphasis on surface and interface for hybrid solar cell
Mahato, Somnath; Puigdollers i González, Joaquim; Voz Sánchez, Cristóbal; Mukhopadhyay, Mala; Mukherjee, Manabendra; Hazra, Satyajit
Effect of dimethyl sulfoxide (DMSO) doping on poly(3,4-ethylenedioxythiophene)-poly(styrenesulfonate) (PEDOT:PSS) thin films have been optimized for obtaining better hole transport layer in hybrid solar cell. The correlation between morphology and conductivity is established through atomic force microscopy and transmission length method measurements. On the other hand, change in the shape of the building blocks (from spheroidal-like to ellipsoidal-like) in the PEDOT:PSS films with DMSO concentration is apparent from their electron density profiles and topographies, suggesting possible conformational change (from coil-like to rod-like) in film by X-ray reflectivity. Such change is further evident from their compositional profiles, work functions and electronic band structures estimated from X-ray and ultraviolet photoelectron spectroscopies. In fact, complementary information suggest that near 5% DMSO doped PEDOT:PSS film is governed through maximum in-plane extended ellipsoidal-like blocks as well as well-organized in out-of-plane ordering which is likely to be the optimum structure for increased the highest electrical conductivity up to 1230¿S/cm. Finally, maximum power conversion efficiency of 11% with open-circuit voltages around 600¿mV, a short-circuit current density higher than 30¿mA/cm2 and a fill factor of 59.4% is achieved for the 5% DMSO doped PEDOT:PSS/n-Si hybrid solar cell, which is perfectly correlated with their structure.
© 2019 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
2019-10-09T12:57:26Z
Mahato, Somnath
Puigdollers i González, Joaquim
Voz Sánchez, Cristóbal
Mukhopadhyay, Mala
Mukherjee, Manabendra
Hazra, Satyajit
Effect of dimethyl sulfoxide (DMSO) doping on poly(3,4-ethylenedioxythiophene)-poly(styrenesulfonate) (PEDOT:PSS) thin films have been optimized for obtaining better hole transport layer in hybrid solar cell. The correlation between morphology and conductivity is established through atomic force microscopy and transmission length method measurements. On the other hand, change in the shape of the building blocks (from spheroidal-like to ellipsoidal-like) in the PEDOT:PSS films with DMSO concentration is apparent from their electron density profiles and topographies, suggesting possible conformational change (from coil-like to rod-like) in film by X-ray reflectivity. Such change is further evident from their compositional profiles, work functions and electronic band structures estimated from X-ray and ultraviolet photoelectron spectroscopies. In fact, complementary information suggest that near 5% DMSO doped PEDOT:PSS film is governed through maximum in-plane extended ellipsoidal-like blocks as well as well-organized in out-of-plane ordering which is likely to be the optimum structure for increased the highest electrical conductivity up to 1230¿S/cm. Finally, maximum power conversion efficiency of 11% with open-circuit voltages around 600¿mV, a short-circuit current density higher than 30¿mA/cm2 and a fill factor of 59.4% is achieved for the 5% DMSO doped PEDOT:PSS/n-Si hybrid solar cell, which is perfectly correlated with their structure.
Acceleration of the measurement time of thermopiles using sigma-delta control
Domínguez Pumar, Manuel
perez, eduard
Ramon, Marina
Jiménez Serres, Vicente
Bermejo Broto, Sandra
Pons Nin, Joan
http://hdl.handle.net/2117/168400
2019-09-19T05:28:05Z
2019-09-18T18:46:24Z
Acceleration of the measurement time of thermopiles using sigma-delta control
Domínguez Pumar, Manuel; perez, eduard; Ramon, Marina; Jiménez Serres, Vicente; Bermejo Broto, Sandra; Pons Nin, Joan
This work presents a double sliding mode control designed for accelerating the measurement of heat fluxes using thermopiles. The slow transient response generated in the thermopile, when it is placed in contact with the surface to be measured, is due to the changes in the temperature distributions that this operation triggers. It is shown that under some conditions the proposed controls keep the temperature distribution of the whole system constant and that changes in the heat flux at the thermopile are almost instantaneously compensated by the controls. One-dimensional simulations and experimental results using a commercial thermopile, showing the goodness of the proposed approach, are presented. A first rigorous analysis of the control using the Sliding Mode Control and Diffusive Representation theories is also made.
2019-09-18T18:46:24Z
Domínguez Pumar, Manuel
perez, eduard
Ramon, Marina
Jiménez Serres, Vicente
Bermejo Broto, Sandra
Pons Nin, Joan
This work presents a double sliding mode control designed for accelerating the measurement of heat fluxes using thermopiles. The slow transient response generated in the thermopile, when it is placed in contact with the surface to be measured, is due to the changes in the temperature distributions that this operation triggers. It is shown that under some conditions the proposed controls keep the temperature distribution of the whole system constant and that changes in the heat flux at the thermopile are almost instantaneously compensated by the controls. One-dimensional simulations and experimental results using a commercial thermopile, showing the goodness of the proposed approach, are presented. A first rigorous analysis of the control using the Sliding Mode Control and Diffusive Representation theories is also made.
Transport mechanisms in silicon heterojunction solar cells with molybdenum oxide as a hole transport layer
García Hernansanz, Rodrigo
Garcia Hemme, E.
Montero, D.
Olea Ariza, Javier
Prado Millán, Álvaro del
Martil, Ignacio
Voz Sánchez, Cristóbal
Gerling Sarabia, Luis Guillermo
Puigdollers i González, Joaquim
Alcubilla González, Ramón
http://hdl.handle.net/2117/133721
2019-09-11T03:31:17Z
2019-05-30T12:41:05Z
Transport mechanisms in silicon heterojunction solar cells with molybdenum oxide as a hole transport layer
García Hernansanz, Rodrigo; Garcia Hemme, E.; Montero, D.; Olea Ariza, Javier; Prado Millán, Álvaro del; Martil, Ignacio; Voz Sánchez, Cristóbal; Gerling Sarabia, Luis Guillermo; Puigdollers i González, Joaquim; Alcubilla González, Ramón
Heterojunction solar cells based on molybdenum sub-oxide (MoOx) deposited on n-type crystalline silicon have been fabricated. The hole selective character of MoOx is explained by its high workfunction, which causes a strong band bending in the Si substrate. This bending pushes the surface into inversion. In addition, the sub-stoichiometry of the evaporated MoOx layers leads to a high density of states within the bandgap. This is crucial for charge transport. The J-V electrical characteristics at several temperatures were analysed to elucidate the dominant charge transport mechanisms of this heterojunction structure. We have identified two different transport mechanisms. At low bias voltage, transport is dominated by hole tunnelling through the MoOx gap states. At higher voltage the behaviour is similar to a Schottky junction with a high barrier value, due to the high MoOx work function. These results provide a better understanding of the hole selective character of MoOx/n-type silicon heterocontacts, which is key to further improve this new kind of solar cells.
© <2018>. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
2019-05-30T12:41:05Z
García Hernansanz, Rodrigo
Garcia Hemme, E.
Montero, D.
Olea Ariza, Javier
Prado Millán, Álvaro del
Martil, Ignacio
Voz Sánchez, Cristóbal
Gerling Sarabia, Luis Guillermo
Puigdollers i González, Joaquim
Alcubilla González, Ramón
Heterojunction solar cells based on molybdenum sub-oxide (MoOx) deposited on n-type crystalline silicon have been fabricated. The hole selective character of MoOx is explained by its high workfunction, which causes a strong band bending in the Si substrate. This bending pushes the surface into inversion. In addition, the sub-stoichiometry of the evaporated MoOx layers leads to a high density of states within the bandgap. This is crucial for charge transport. The J-V electrical characteristics at several temperatures were analysed to elucidate the dominant charge transport mechanisms of this heterojunction structure. We have identified two different transport mechanisms. At low bias voltage, transport is dominated by hole tunnelling through the MoOx gap states. At higher voltage the behaviour is similar to a Schottky junction with a high barrier value, due to the high MoOx work function. These results provide a better understanding of the hole selective character of MoOx/n-type silicon heterocontacts, which is key to further improve this new kind of solar cells.
3D simulations of interdigitated back-contacted crystalline silicon solar cells on thin substrates
Jin, Chen
Martín García, Isidro
Ortega Villasclaras, Pablo Rafael
Calle Martín, Eric
Alcubilla González, Ramón
http://hdl.handle.net/2117/132892
2019-11-20T04:56:05Z
2019-05-10T19:04:07Z
3D simulations of interdigitated back-contacted crystalline silicon solar cells on thin substrates
Jin, Chen; Martín García, Isidro; Ortega Villasclaras, Pablo Rafael; Calle Martín, Eric; Alcubilla González, Ramón
Interdigitated back contact technology is a promising candidate to be applied to thin crystalline silicon solar cells because of its simpler one-side interconnection while allowing a more flexible front surface treatment. This work explores the efficiency potential of IBC c-Si solar cells applied to thin c-Si substrates through 3D device simulations. In particular, we explore the impact of substrate thickness and front surface recombination velocity on cell performance with special attention to the different behavior in carrier collection of two different rear-surface doping structures. Firstly, the model is validated by comparing simulation results to a fabricated device on 280¿µm-thick substrates with stripe-like p+ and n+ diffusions. It is revealed that efficiencies of 16–17% are reachable for substrates on the 10–15¿µm range without changing the technology developed for thick ones. Next, the rear doping structure is modified leading to doped regions just under the metal contacts. This type of structure is expected in solar cells where high-temperature diffusions are replaced by point-like laser doped contacts, which is a feasible alternative to be applied to thin substrates. Simulation results show that diffusion length requirements for those locally-doped structures are more demanding due to the reduction of emitter regions. As a result, very well passivated front and rear surfaces are required to maintain short-circuit current densities to reasonable values. Finally, for both structures open-circuit voltage is kept almost constant with reduced thickness, despite the strong reduction in short-circuit current. Simulations show a reduction of dark saturation current density with substrate thinning due to the redistribution of dark current densities that flow parallel to the device surface.
© <2018>. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
2019-05-10T19:04:07Z
Jin, Chen
Martín García, Isidro
Ortega Villasclaras, Pablo Rafael
Calle Martín, Eric
Alcubilla González, Ramón
Interdigitated back contact technology is a promising candidate to be applied to thin crystalline silicon solar cells because of its simpler one-side interconnection while allowing a more flexible front surface treatment. This work explores the efficiency potential of IBC c-Si solar cells applied to thin c-Si substrates through 3D device simulations. In particular, we explore the impact of substrate thickness and front surface recombination velocity on cell performance with special attention to the different behavior in carrier collection of two different rear-surface doping structures. Firstly, the model is validated by comparing simulation results to a fabricated device on 280¿µm-thick substrates with stripe-like p+ and n+ diffusions. It is revealed that efficiencies of 16–17% are reachable for substrates on the 10–15¿µm range without changing the technology developed for thick ones. Next, the rear doping structure is modified leading to doped regions just under the metal contacts. This type of structure is expected in solar cells where high-temperature diffusions are replaced by point-like laser doped contacts, which is a feasible alternative to be applied to thin substrates. Simulation results show that diffusion length requirements for those locally-doped structures are more demanding due to the reduction of emitter regions. As a result, very well passivated front and rear surfaces are required to maintain short-circuit current densities to reasonable values. Finally, for both structures open-circuit voltage is kept almost constant with reduced thickness, despite the strong reduction in short-circuit current. Simulations show a reduction of dark saturation current density with substrate thinning due to the redistribution of dark current densities that flow parallel to the device surface.
Bipolar transistor vertical vertical scaling framework
Castañer Muñoz, Luis María
Alcubilla González, Ramón
Benavent, A
http://hdl.handle.net/2117/131231
2019-09-11T03:06:09Z
2019-04-03T16:44:26Z
Bipolar transistor vertical vertical scaling framework
Castañer Muñoz, Luis María; Alcubilla González, Ramón; Benavent, A
Scaling factors for current and transit time are derived for polysilicon emitter, silicon based heterojunction bipolar transistors. It is shown that a simple set of analytical equations in integral form can be used to analyse the above scaling properties and by introducing two “heterojunction factors” can be extended to the scaling analysis of heterojunction bipolar transistors.
2019-04-03T16:44:26Z
Castañer Muñoz, Luis María
Alcubilla González, Ramón
Benavent, A
Scaling factors for current and transit time are derived for polysilicon emitter, silicon based heterojunction bipolar transistors. It is shown that a simple set of analytical equations in integral form can be used to analyse the above scaling properties and by introducing two “heterojunction factors” can be extended to the scaling analysis of heterojunction bipolar transistors.
Analysis of the conductance transient in thick film tin oxide gas sensors
Vilanova, X
Llobet Valero, Eduard
Alcubilla González, Ramón
Sueiras, Jl
Correig, X
http://hdl.handle.net/2117/131230
2019-09-11T03:06:09Z
2019-04-03T16:27:48Z
Analysis of the conductance transient in thick film tin oxide gas sensors
Vilanova, X; Llobet Valero, Eduard; Alcubilla González, Ramón; Sueiras, Jl; Correig, X
In this paper, we analyse the conductance transient of a Taguchi TGS-822 sensor under a step change in the vapour concentration. A diffusion-limited range in the conductance transient is observed. Adjustments between the theoretical calculations based on a non-linear diffusion-reaction model and the experimental results alloy a constant, t', to be estimated, which is independent of the final conductance value and depends, among other parameters, on the effective diffusion coefficients of vapours in the porous tin oxide sensor. From transient measurements of organic solvents (benzene and o-xylene) we have obtained t' values that are independent of concentration and characteristic for each vapour. This new parameter can give useful information for gas/vapour recognition.
2019-04-03T16:27:48Z
Vilanova, X
Llobet Valero, Eduard
Alcubilla González, Ramón
Sueiras, Jl
Correig, X
In this paper, we analyse the conductance transient of a Taguchi TGS-822 sensor under a step change in the vapour concentration. A diffusion-limited range in the conductance transient is observed. Adjustments between the theoretical calculations based on a non-linear diffusion-reaction model and the experimental results alloy a constant, t', to be estimated, which is independent of the final conductance value and depends, among other parameters, on the effective diffusion coefficients of vapours in the porous tin oxide sensor. From transient measurements of organic solvents (benzene and o-xylene) we have obtained t' values that are independent of concentration and characteristic for each vapour. This new parameter can give useful information for gas/vapour recognition.
Space charge recombination in p-n junctions with a discrete and continous trap distribution
Pallarès Marzal, Josep
Marsal Garví, Lluís Francesc
Correig, Xavier
Calderer Cardona, Josep
Alcubilla González, Ramón
http://hdl.handle.net/2117/130962
2019-09-11T02:50:38Z
2019-03-27T17:57:45Z
Space charge recombination in p-n junctions with a discrete and continous trap distribution
Pallarès Marzal, Josep; Marsal Garví, Lluís Francesc; Correig, Xavier; Calderer Cardona, Josep; Alcubilla González, Ramón
Space charge Shockley-Read-Hall recombination currents in the presence of discrete or continuous distributions of recombination centres are analysed. For a single level trap, depending on its position inside the forbidden band, asymptotic values both for the ideality factor of the current-voltage characteristic and for the activation energy of the saturation current are obtained. The analysis is extended to continuous trap distributions and the current-voltage characteristics obtained are explained in terms of the simple theory developed for single level traps.
2019-03-27T17:57:45Z
Pallarès Marzal, Josep
Marsal Garví, Lluís Francesc
Correig, Xavier
Calderer Cardona, Josep
Alcubilla González, Ramón
Space charge Shockley-Read-Hall recombination currents in the presence of discrete or continuous distributions of recombination centres are analysed. For a single level trap, depending on its position inside the forbidden band, asymptotic values both for the ideality factor of the current-voltage characteristic and for the activation energy of the saturation current are obtained. The analysis is extended to continuous trap distributions and the current-voltage characteristics obtained are explained in terms of the simple theory developed for single level traps.
Analytical modelling of bjt neurtral base region under variable injection conditions
Bardés Llorensí, Daniel
Alcubilla González, Ramón
http://hdl.handle.net/2117/130753
2019-09-11T03:30:36Z
2019-03-21T19:05:33Z
Analytical modelling of bjt neurtral base region under variable injection conditions
Bardés Llorensí, Daniel; Alcubilla González, Ramón
We present a set of closed form analytical solutions of the transport equation in the base of bipolar transistors. The presented 1D solutions hold for variable injection conditions, arbitrary doping profiles and arbitrary intrinsic carrier concentrations along the base, allowing to consider bandgap narrowing effects and a variable composition base (e.g. Si1-xGex graded base). The above solutions lay on the commonly used assumption of zero majority carrier current in the neutral zone, and are valid up to the onset of the Kirk effect. Two solutions are analysed; the zero-order solution, found as a correction of the low injection case, and the first order solution, derived as a correction of the zero-order case. The perturbative procedure can naturally be iterated leading to an exact solution, which is used to assess the accuracy of the proposed analytical solutions. The first-order solution shows the best trade-off between accuracy and complexity.
2019-03-21T19:05:33Z
Bardés Llorensí, Daniel
Alcubilla González, Ramón
We present a set of closed form analytical solutions of the transport equation in the base of bipolar transistors. The presented 1D solutions hold for variable injection conditions, arbitrary doping profiles and arbitrary intrinsic carrier concentrations along the base, allowing to consider bandgap narrowing effects and a variable composition base (e.g. Si1-xGex graded base). The above solutions lay on the commonly used assumption of zero majority carrier current in the neutral zone, and are valid up to the onset of the Kirk effect. Two solutions are analysed; the zero-order solution, found as a correction of the low injection case, and the first order solution, derived as a correction of the zero-order case. The perturbative procedure can naturally be iterated leading to an exact solution, which is used to assess the accuracy of the proposed analytical solutions. The first-order solution shows the best trade-off between accuracy and complexity.