Magdalini Titirla
email: magdalini.titirla@lecnam.net

Georgios Aretoulis

Comparison of linear regression and neural network models to estimate the actual duration of Greek highway projects

Keywords: Highway Construction, Linear Regression, Neural Networks, Predicting Models, Project Actual Duration

The aim of this study is to highlight and evaluate parameters that correlate with the actual project duration and compare performance of linear regression with neural network models to predict a reliable final duration for highway construction projects based on data available at the bidding stage. The thirty-seven highway projects that are examined, constructed in Greece, are similar in available data like the extent, the type of work packages and the significance. Considering each project's characteristics and the actual construction duration, correlation analysis is implemented, with the aid of SPSS 25. Correlation analysis identified the most significant project variables. These include archeological findings, type of terrain, land expropriation, the existence of bridge, tunnel and embankment. Furthermore, the WEKA application, through its attribute selection function, highlighted the most efficient subset of variables, both within the initial available attributes but also within the set of attributes identified through correlation analysis. These selected variables through correlation analysis and / or WEKA and appropriate combinations of these are used as input data for linear regression models and neural network models. The corresponding models (linear regression and neural networks) are presented along with their performance. Finally, this paper presents the comparison of linear regression and neural network models to estimate the actual duration of Greek highway projects. Results' discussion and conclusions along with limitations and further research are appropriately analyzed.

INTRODUCTION AND BACKGROUND

 [START_REF] Kim | Relationship between the financial crisis of Korean construction firms and macroeconomic fluctuations[END_REF], cited in [START_REF] Tsaousoglou | Forecasting the trends of financial ratios of Greek construction companies[END_REF] suggest that construction industry plays an important role in leading the national economy and macroeconomic fluctuations substantially influence the construction business. It is therefore essential for each country to promote the implementation of public infrastructure construction projects. During project implementation it is common that variations occur among the initial planning and the actual ongoing project. Variation in project costs usually results in exceeding the initial estimated cost [START_REF] Aretoulis | A prototype system for the prediction of final cost in construction projects[END_REF]. At the same time projects suffer from increase in their duration. In this context, it is useful to keep in mind that each highway project is different to another one as there are a lot of parameters which play a role in the differentiation of projects. Research suggests that project location is one of the major differences in construction of highway projects. The unique legal framework in each country influences project implementation, and project procurement systems and contract types [START_REF] Antoniou | Choosing the Most Appropriate Contract Type for Compensating Major Highway Project Contractors[END_REF]. Furthermore, construction projects are influenced by many parameters which alter the cost and schedule planning, both internal and external. Furthermore, [START_REF] Glymis | Predicting highway projects' actual duration using neural networks[END_REF] based on FANN Tool proposed three, selected, neural network models for the prediction of actual project duration for highways, based on tender budget, length of highway project, number of lanes, number of technical projects, number of bridges, tunnels and road total length. A further research approach focused on highway project duration. The latter is estimated on the basis of variables which were known at the planning phase, and included planned cost, project type and contract type [START_REF] Irfan | Planning-stage estimation of highway project duration on the basis of anticipated project cost, project type, and contract type[END_REF]. Moreover, [START_REF] Aziz | Performance Analysis and Forecasting for WSDOT Highway Projects[END_REF] produced an approach, based on statistical regression analysis. The proposed approach predicts the cost and time of a project and, also, facilitates evaluation of the project's performance during construction. The current study includes, three methods of analysis: ridge regression analysis, general regression analysis and non-linear partial leastsquare regression analysis. More specifically, data collected from completed projects included the type of pavement, contract value, duration and project miles. [START_REF] Jiang | A method for highway agency to estimate highway construction durations and set contract times[END_REF], considered 1818 projects from the area of Indiana. Their research identified factors which affect directly the duration and the cost of each project. These critical factors include type of the project, production rates of the construction firm, location of the project and weather condition. appeared. [START_REF] Anastasopoulos | Empirical assessment of the likelihood and duration of highway project time delays[END_REF], used data from 1,722 projects from Indiana. His research identified critical parameters for consideration in models. The variables which were chosen as influential included: project time delay, project cost, project type, weather, level of competition, difference between the winning bid amount and the engineer's estimate and finally difference between the winning bid amount and the next lowest bid. [START_REF] Pewdum | Forecasting final budget and duration of highway construction projects[END_REF] analyzed highway construction project data. Their research factors affecting project final budget and duration before developing the forecasting models. They employed Artificial Neural Network (ANN). The forecasting results obtained from the proposed method were compared with those obtained from models based on earned value. In a similar context, Hola and Schabowicz (2010) produced a neural network based model to predict productivity for selected sets of machines and to calculate the task execution time and cost. Aretoulis (2019) examined selected public Greek highway projects in order to produce models to predict their actual construction cost based on data available at the bidding stage. Twenty highway projects, constructed in Greece, with similar type of available data were examined. Considering each project's attributes and the actual cost, correlation analysis is implemented, with the aid of SPSS. The optimum neural network model produced a mean squared error with a value of 7.68544E-05 and was based on budgeted cost, lowest awarding bid, technical work cost and electromechanical work cost. The aim of this study is to highlight and evaluate parameters that correlate with the actual project duration and compare linear regression with neural network models to predict a reliable final duration for highway construction projects based on data available at the bidding stage. The current research approach also aims at considering special characteristics of projects, realized in Greece. These include archeological findings, type of terrain and land expropriation. Considering each project's characteristic and the actual construction duration, correlation analysis is implemented, with the aid of SPSS 25. This paper presents the results of a relevant literature review on construction project duration, and the recorded corresponding findings, with emphasis on the parameters that have been identified. Furthermore, the WEKA application, through its attribute selection function, highlighted the most efficient subset of variables, both within the initial available attributes but also within the set of attributes identified through correlation analysis. These selected variables through correlation analysis and / or WEKA and appropriate combinations of these are used as input data for linear regression models and neural network models. The corresponding models (linear regression and neural networks) are presented along with their performance. Finally, this paper presents the comparison of linear regression and neural network models to estimate the actual duration of Greek highway projects. Results' discussion and conclusions along with limitations and further research are appropriately analyzed.

METHODOLOGICAL APPROACH

The present study is aimed at the production of reliable and efficient models (linear regression and neural networks) for the prediction of actual construction duration of highway projects in Greece. The sample projects under examination include 37 highway projects of similar extent and content [START_REF] Titirla | Neural network models for actual duration of Greek highway projects[END_REF]. For these specific projects it became possible to record detailed amount of the same type of data, both quantitative and qualitative. The methodology steps included the following:

Step 1: Consideration of 37 selected highway projects and collection of corresponding data.

Step 2: Construction of an appropriate SPSS database, including both quantitative and qualitative variables.

Step 3: Descriptive statistics of the sample projects' variables.

Step 4: Correlation statistical analysis among the available variables and actual project duration. Analysis was conducted one time for quantitative variables and a second time for mixed variables, both quantitative and qualitative variables.

Step 5: Creation of a ranked list of variables, based on a decreasing degree of correlation coefficients.

Step 6: Proposal of Linear Regression (LR) and Neural Network (NN) models. The models were created starting with the highest correlating variable and then adding one more variable each time from the ordered list based on correlation degree (correlation coefficient) (Step 5).

Step 7: Comparison of the produced LR and NN models.

Database description

The sample projects under examination include 37 highway projects of similar extent and content. The database included data from 36 Egnatia Odos' sub-projects and 1 smaller project in terms of acreage, cost, and time of completion. Egnatia Odos is one of the most modern highways in the Southeast Europe and part of the TEN-T corridors. The projects were selected on the basis of data uniformity and availability. The available variables are the followings: Project length, Initial cost, Land requirement, Landfill, Tender offer, Embankment, Initial duration, Geotechnical project, Number of lanes, Existence of Tunnel, Number of technical projects, Existence of Bridge. These variables are characterized as quantitative and qualitative and are presented in Table 1. The quantitative variables take on numerical values, and the qualitative variables take on binary (Yes/No) or ordinal values.

Correlation analysis

An SPSS database was created to record all the available variables. The database consists of 37 cases (number of projects) and 12 variables, which cover common available data across all the projects and according to expert opinion greatly affect the project duration. These variables were recorded as quantitative and also transformed, where possible, into qualitative ones to facilitate additional statistical analyses. Six variables remained as quantitative, while the other six were transformed into qualitative. The values of the IBM SPSS Statistics database were used to identify the correlations among the independent quantitative variables and actual project duration. Furthermore, based on the findings of the correlation analysis, a number of linear regression and neural networks models were designed and implemented, to predict the actual duration of the projects. A brief look on the correlation analyses' results reveals that actual project duration is associated in descending correlation degree with the following variables:

 initial cost;  tender offer;  initial duration;  embankment;  number of lanes;  existence of geotechnical projects;  length;  existence of bridges;  and existence of tunnels.

MODELS

Linear regression

Linear regression is a linear approach to modeling the relationship between a scalar response (or dependent variable) and one or more explanatory variables (or independent variables). In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Such models are called linear models. Most commonly, the conditional mean of the response given the values of the explanatory variables (or predictors) is assumed to be an affine function of those values; less commonly, the conditional median or some other quantile is used. Like all forms of regression analysis, linear regression focuses on the conditional probability distribution of the response given the values of the predictors, rather than on the joint probability distribution of all of these variables, which is the domain of multivariate analysis. Having identified the potential correlations among the selected common variables, the current paper applies the linear regression analysis. The selected method is the stepwise. The aim is to predict the final duration.

After the analysis of the 37 data cases the following linear regression equations are proposed:

𝐴𝑐𝑡𝑢𝑎𝑙 𝑝𝑟𝑜𝑗𝑒𝑐𝑡 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 841,539 + 4,632 • 10 -6 • 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑜𝑠𝑡 (1) 𝐴𝑐𝑡𝑢𝑎𝑙 𝑝𝑟𝑜𝑗𝑒𝑐𝑡 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 424,729 + 2,818 10 -6 • 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑜𝑠𝑡 + 0,843 • initial duration

(2)

The first equation includes only one variable, the initial cost and predicts the actual project duration with a predictability of 67% , while the second one includes two variables, the initial cost and the initial duration and it is able to predict the actual project duration with higher predictability equal to 72%. The above-mentioned data are presented in the following tables, namely Table 2 andTable 3.

2. Neural networks

Neural networks produced several models. The dependent variable was "Actual Project Duration" in days. Twenty-one projects were used for training the neural network and sixteen projects were used for testing the produced neural network. The methodology involving the application of neural networks for the prediction of actual duration is based on multiple neural networks designed and applied specifically for the problem at hand. The current research is contributing and focusing on identifying the best possible combination of input variables for the optimum prediction.

CONCLUSION AND FURTHER RESEARCH

This paper presents the important variables that correlate with the actual project duration and compare linear regression with neural network models to predict a reliable final duration for highway construction projects based on data available at the bidding stage. In total, 37 highway projects constructed in Greece with similar extent and content, were investigated. The corresponding models (linear regression and neural networks) are presented along with their performance. Finally, this paper presents the comparison of linear regression and neural network models to estimate the actual duration of Greek highway projects.

The proposed model of linear regression includes only 2 variables, the initial cost and the initial duration, while for the neural networks the corresponding model includes the top 12 variables, namely, initial cost, initial duration, length, lanes, technical projects, bridges, tunnels, geotechnical projects, embankment, landfill, land requirement and tender offer.

The linear regression model is able to predict the actual project duration with a predictability of 72% while the neural network model produces a MSE equal to 1.53E-06.

A further research in this area could aim at the development and the comparison of the results with other methods and tools. In addition, a wider number of projects should be included, so that correlations and trends that are not observed by this research could be identified in a more extensive research and at the same time increase models' reliability and accuracy.


Marin Rusu

Tim Kodalle

Leo Choubrac

Nicolas Barreau

Christian A Kaufmann

Rutger Schlatmann

Thomas Unold

Electronic Structure of the CdS/Cu(In

Keywords: chalcopyrites, thin-film solar cells, alkali fluoride postdeposition treatments, interfaces, band diagram, Kelvin probe, photoemission spectroscopy 2

teaching and research institutions in France or abroad, or from public or private research centers.

INTRODUCTION

Recently, the light-to-power conversion efficiency of polycrystalline thin-film solar cells based on Cu(In,Ga)(S,Se) 2 (CIGSSe) absorber layers has been boosted to 23.35%. [START_REF] Nakamura | Cd-Free Cu(In,Ga)(Se,S) 2 Thin-Film Solar Cell with Record Efficiency of 23.35%[END_REF] This remarkable result was achieved after the introduction of an alkali-fluoride post-deposition treatment (PDT), e.g. initially by sodium-fluoride (NaF), of the Cu(In,Ga)Se 2 (CIGSe) absorber surface. [START_REF] Rudmann | Sodium Incorporation Strategies for CIGS Growth at Different Temperatures[END_REF] The application of a similar PDT procedure to the CIGSe absorber using fluorides of heavier alkali elements, e.g., potassium-fluoride (KF), rubidium-fluoride (RbF) and cesiumfluoride PDTs led to a series of record efficiencies in the past few years. [START_REF] Nakamura | Cd-Free Cu(In,Ga)(Se,S) 2 Thin-Film Solar Cell with Record Efficiency of 23.35%[END_REF][START_REF] Chirilă | Potassium-Induced Surface Modification of Cu(In,Ga)Se 2 Thin Films for High-Efficiency Solar Cells[END_REF][START_REF] Jackson | Compositional Investigation of Potassium Doped Cu(In,Ga)Se 2 Solar Cells with Efficiencies up to 20.8%[END_REF][START_REF] Jackson | Properties of Cu(In,Ga)Se 2 Solar Cells with New Record Efficiencies up to 21.7%[END_REF][START_REF] Jackson | Effects of Heavy Alkali Elements in Cu(In,Ga)Se 2 Solar Cells with Efficiencies up to 22.6%[END_REF][START_REF] Kato | Record Efficiency for Thin-Film Polycrystalline Solar Cells up to 22.9% Achieved by Cs-Treated Cu[END_REF] The application of alkali-PDTs in most of the studies resulted in absorber thin-films with improved bulk and surface electronic properties related to the alkali incorporation into the CIGSe (e.g., diffusion along grain boundaries and into grain interior [START_REF] Ruckh | Influence of Substrates on the Electrical Properties of Cu(In,Ga)Se 2 Thin Films[END_REF][START_REF] Reinhard | Features of KF and NaF Postdeposition Treatments of Cu(In,Ga)Se 2 Absorbers for High Efficiency Thin Film Solar Cells[END_REF] associated with the effects on the doping concentration, charge carrier life time and mobility) [START_REF] Wuerz | Diffusion of Rb in Polycrystalline Cu(In,Ga)Se 2 Layers and Effect of Rb on Solar Cell Parameters of Cu(In,Ga)Se 2 Thin-Film Solar Cells[END_REF][START_REF] Nicoara | Direct Evidence for Grain Boundary Passivation in Cu(In,Ga)Se 2 Solar Cells Through Alkali-Fluoride Post-Deposition Treatments[END_REF][START_REF] Nicoara | Effect of the KF Post-Deposition Treatment on Grain Boundary Properties in Cu(In,Ga)Se 2 Thin Films[END_REF][START_REF] Pianezzi | Unveiling the Effects of Post-Deposition Treatment with Different Alkaline Elements on the Electronic Properties of CIGS Thin Film Solar Cells[END_REF][START_REF] Laemmle | Efficiency Enhancement of Cu(In,Ga) Se 2 Thin-Film Solar Cells by a Post-Deposition Treatment with Potassium Fluoride[END_REF] and chemical changes of the thin-film top surface. [START_REF] Weinhardt | Surface and Interface Properties in Thin-Film Solar Cells: Using Soft X-Rays and Electrons to Unravel the Electronic and Chemical Structure[END_REF] As already widely reported for most commonly used growth approaches, an untreated CIGSe thinfilm surface has a Cu-depleted stoichiometry [START_REF] Schmid | Chalcopyrite/Defect Chalcopyrite Heterojunctions on the Basis of CuInSe 2[END_REF][START_REF] Schmid | Photoemission Studies on Cu(In,Ga)Se 2 Thin Films and Related Binary Selenides[END_REF][START_REF] Morkel | Flat Conduction-Band Alignment at the Thin-Film Solar-Cell Heterojunction[END_REF] which leads to a surface bandgap widening due to the downward shift of the valence band maximum (VBM) at a slight change of the position of the conduction band minimum (CBM). Such a surface band gap widening at the chalcopyrite thin-film absorbers is regarded as a crucial property for high-efficiency CIGSSe-based photovoltaic (PV) devices. However, along with this natural copper-depleted stoichiometry, the chalcopyrite thin-film surfaces show presence of sodium, which arises by diffusion from the applied soda-lime glass (SLG) substrates. [START_REF] Heske | Na-Induced Effects on the Electronic Structure and Composition of Cu(In,Ga)Se 2 Thin-Film Surfaces[END_REF][START_REF] Rusu | CuGaSe 2 Thin Films Prepared by a Novel CCSVT Technique for Photovoltaic Application[END_REF][START_REF] Würz | In Situ X-Ray Photoelectron Spectroscopy Study of the Oxidation of CuGaSe 2[END_REF][START_REF] Rusu | Three-Dimensional Structure of the Buffer/Absorber Interface in CdS/CuGaSe 2 Based Thin Film Solar Cells[END_REF] Moreover, on air-exposed, untreated (no PDT) absorber surfaces on SLG substrates Ga 2 O 3 , SeO 2 , Na x O, Na 2 CO 3 , and InO x were observed. [START_REF] Ruckh | Influence of Substrates on the Electrical Properties of Cu(In,Ga)Se 2 Thin Films[END_REF][START_REF] Würz | In Situ X-Ray Photoelectron Spectroscopy Study of the Oxidation of CuGaSe 2[END_REF][START_REF] Hauschild | Impact of a RbF Postdeposition Treatment on the Electronic Structure of the CdS/Cu(In,Ga)Se 2 Heterojunction in High-Efficiency Thin-Film Solar Cells[END_REF] Furthermore, treatment by different alkali elements led to formation of different surface phases.

For CIGSe exposed to a KF-PDT, surface phases such as GaF 3 , KInSe 2 , In 2 Se 3 or In 2 Se 3 :K were deduced and a larger widening of the surface band gap to over 2.0 eV was reported along with the changes in positions of both CBM and VBM. [START_REF] Reinhard | Potassium Postdeposition Treatment-Induced Band Gap Widening at Cu(In,Ga)Se 2 Surfaces -Reason for Performance Leap?[END_REF][25][START_REF] Lepetit | Post Deposition Treatment in Co-Evaporated Cu(In,Ga)Se 2 Thin Film Solar Cells: Beneficial or Detrimental Effect Induced by the Absorber Characteristics[END_REF][START_REF] Pistor | Experimental Indication for Band Gap Widening of Chalcopyrite Solar Cell Absorbers After Potassium Fluoride Treatment[END_REF] A common finding in these investigations is a further reduced Cu content at the absorber surface. [START_REF] Weinhardt | Surface and Interface Properties in Thin-Film Solar Cells: Using Soft X-Rays and Electrons to Unravel the Electronic and Chemical Structure[END_REF] Note that presence of potassium and formation of potassium-containing phases is conditioned by the presence of a Cu-poor CIGSe surface composition. [START_REF] Lepetit | Post Deposition Treatment in Co-Evaporated Cu(In,Ga)Se 2 Thin Film Solar Cells: Beneficial or Detrimental Effect Induced by the Absorber Characteristics[END_REF] For RbF-PDT samples, InO x and GaO x phases were found. [START_REF] Maticiuc | In Vacuo XPS Investigation of Cu(In,Ga)Se 2 Surface After RbF Post-Deposition Treatment[END_REF][START_REF] Kodalle | Elucidating the Mechanism of an RbF post Deposition Treatment in CIGS Thin Film Solar Cells[END_REF][START_REF] Kreikemeyer-Lorenzo | Rubidium Fluoride Post-Deposition Treatment: Impact on the Chemical Structure of the Cu(In,Ga)Se 2 Surface and CdS/Cu(In,Ga)Se 2 Interface in Thin-Film Solar Cells[END_REF] In addition NaF, RbF, In(OH) 3, GaF 3 and secondary phases of Rb-In-Se, and In x Se y :Rb,Na were detected. [START_REF] Maticiuc | In Vacuo XPS Investigation of Cu(In,Ga)Se 2 Surface After RbF Post-Deposition Treatment[END_REF][START_REF] Kodalle | Elucidating the Mechanism of an RbF post Deposition Treatment in CIGS Thin Film Solar Cells[END_REF][START_REF] Kodalle | Effects of KF and RbF Post Deposition Treatments on the Growth of the CdS Buffer Layer on CIGS Thin Films -a Comparative Study[END_REF] Only a small downward shift of both CBM and VBM was found, while a minor change of the surface band gap from 1.46 eV to 1.51 eV was recorded. [START_REF] Hauschild | Impact of a RbF Postdeposition Treatment on the Electronic Structure of the CdS/Cu(In,Ga)Se 2 Heterojunction in High-Efficiency Thin-Film Solar Cells[END_REF] The Cu content at the CIGSe surface was found to be reduced as in the case of the KF-PDT.

The subsequent growth of the CdS buffer layer on top of the CIGSe absorber by chemical bath deposition (CBD), in view of forming the active CdS/CIGSe heterojunction of the PV devices, is also influenced by the kind of absorber surface conditioning. During the CBD process, oxide phases as well as surplus NaF, KF and RbF are etched off from the CIGSe surface. [START_REF] Weinhardt | Surface and Interface Properties in Thin-Film Solar Cells: Using Soft X-Rays and Electrons to Unravel the Electronic and Chemical Structure[END_REF][START_REF] Würz | In Situ X-Ray Photoelectron Spectroscopy Study of the Oxidation of CuGaSe 2[END_REF][START_REF] Kodalle | Elucidating the Mechanism of an RbF post Deposition Treatment in CIGS Thin Film Solar Cells[END_REF][START_REF] Kreikemeyer-Lorenzo | Rubidium Fluoride Post-Deposition Treatment: Impact on the Chemical Structure of the Cu(In,Ga)Se 2 Surface and CdS/Cu(In,Ga)Se 2 Interface in Thin-Film Solar Cells[END_REF] The etching effect is attributed to ammonia (NH 3 ), which is a constituent of the CBD solution. The CdS films grow denser and more homogeneously after the PDT treatment of the absorber surface. [START_REF] Kodalle | Effects of KF and RbF Post Deposition Treatments on the Growth of the CdS Buffer Layer on CIGS Thin Films -a Comparative Study[END_REF] Upon formation of the CdS/RbF-PDT CIGSe interface, the bands bend downward. However, a flat CBM alignment at approximately 0.51 eV from the Fermi level (E F ) was found between the CdS buffer and the RbF-PDT CIGSe layer. [START_REF] Hauschild | Impact of a RbF Postdeposition Treatment on the Electronic Structure of the CdS/Cu(In,Ga)Se 2 Heterojunction in High-Efficiency Thin-Film Solar Cells[END_REF] A similar CBM alignment was found for the case of the CdS/CIGSSe interface. [START_REF] Weinhardt | Band Alignment at the i-ZnO/CdS Interface in Cu(In,Ga)(S,Se) 2 Thin-Film Solar Cells[END_REF] The bandgap of the bulk CdS in the latter case was reported of 2.46 eV against 2.60 eV in the case of CdS grown on RbF-PDT CIGSe. [START_REF] Hauschild | Impact of a RbF Postdeposition Treatment on the Electronic Structure of the CdS/Cu(In,Ga)Se 2 Heterojunction in High-Efficiency Thin-Film Solar Cells[END_REF] The former value is in agreement with the previously published data, [START_REF] Rusu | Optimisation of the CBD CdS Deposition Parameters for ZnO/CdS/CuGaSe 2 /Mo Solar Cells[END_REF][START_REF] Mahanty | Optical Properties of Chemical Bath Deposited CdS Thin Films[END_REF][START_REF] Zelaya-Angel | Band-Gap Shift in CdS Semiconductor by Photoacoustic Spectroscopy: Evidence of a Cubic to Hexagonal Lattice Transition[END_REF] while the significantly larger latter one is attributed to CdS with inclusions of Cd(OH) 2 and/or CdSO 4 . [START_REF] Hauschild | Impact of a RbF Postdeposition Treatment on the Electronic Structure of the CdS/Cu(In,Ga)Se 2 Heterojunction in High-Efficiency Thin-Film Solar Cells[END_REF] It was found in addition that Rb and Na atoms diffuse toward the CdS surface. [START_REF] Kreikemeyer-Lorenzo | Rubidium Fluoride Post-Deposition Treatment: Impact on the Chemical Structure of the Cu(In,Ga)Se 2 Surface and CdS/Cu(In,Ga)Se 2 Interface in Thin-Film Solar Cells[END_REF] The results reviewed above show an impressive work realized and a large amount of data collected. However, the full impact of the alkali-fluoride PDTs on the electronic structure of the CdS/CIGSe heterojunction used in solar cells and the correlation to their photovoltaic parameters

are not yet fully understood. Moreover, due to the application of methods which require ultrahigh vacuum, e.g., X-ray and UV photoelectron spectroscopy as well as inverse photoemission spectroscopy, the studies have been conducted on a limited number of samples. It is the aim of this work to determine the electronic parameters and propose detailed energy level diagrams for the CdS/CIGSe interface in high-efficiency solar cells based on PDT CIGSe absorber thin films using heavy alkali metal fluorides such as KF and RbF. For this purpose, we investigate (i) the impact of the KF-PDT and RbF-PDT on the electronic structure of the CIGSe layer and (ii) the CdS/CIGSe interface in a sequential time-dependent CdS thickness evolvement during the CBD process. We combine Kelvin probe (KP) and photoelectron yield spectroscopy (PYS) methods at ambient conditions to determine the work function () and the energetic position of the valence band maximum (E VBM ) of surfaces and interfaces. We complement the obtained  and E VBM results by the data on the conduction band minimum energy (E CBM ) and propose, by considering the thin film band gaps obtained from internal quantum efficiency (IQE) measurements on solar cells, the detailed band diagrams. Finally, the obtained energy band diagrams are correlated to the PV parameters of the corresponding solar cells.

EXPERIMENTAL SECTION

2.1. Samples. In order to investigate the impact of the KF-and RbF-PDTs on the electronic properties of the CIGSe thin-film surface as well as the formation of the CBD-CdS/CIGSe interface during the growth of the CdS buffer layer in dependence of the CdS thickness, three sample sets (10 pieces each) for (i) no-PDT (ii) KF-PDT and (iii) RbF-PDT respectively were made from CIGSe thin-films deposited by otherwise nominally identical fabrication processes.

The CIGSe thin-films were prepared by a three-stage co-evaporation process [START_REF] Repins | 9%-Efficient ZnO/CdS/CuInGaSe 2 Solar Cell with 81.2% Fill Factor[END_REF] on 1 mm thick SLG substrates coated with 400 nm thick DC-sputtered molybdenum. [START_REF] Kodalle | Effects of KF and RbF Post Deposition Treatments on the Growth of the CdS Buffer Layer on CIGS Thin Films -a Comparative Study[END_REF] The SLG substrates were used without a diffusion barrier coating, so that Na can diffuse from the substrates into the CIGSe films during growth. The integral composition, as measured by X-ray fluorescence spectroscopy (XRF), exhibited x Cu /( x Ga + x In ) = CGI  0.80 and x Ga /(x Ga + x In ) = GGI  0.35, x denoting the molar fraction of the respective element. Details about the CIGSe deposition process can be found elsewhere. [START_REF] Kodalle | Effects of KF and RbF Post Deposition Treatments on the Growth of the CdS Buffer Layer on CIGS Thin Films -a Comparative Study[END_REF][START_REF] Barreau | Recrystallization of CIGSe Layers Grown by Three-Step Processes: A Model Based on Grain Boundary Migration[END_REF] After the CIGSe deposition, the samples for the second and third sets were immediately transferred through air to vacuum deposition tools for the KF-PDT and RbF-PDT treatments. Both PDTs consisted of evaporating the respective alkali-fluoride at rates between 0.1-0.2 Å/s under Se atmosphere at substrate temperatures between 280-350°C for 15 minutes. [START_REF] Lepetit | Post Deposition Treatment in Co-Evaporated Cu(In,Ga)Se 2 Thin Film Solar Cells: Beneficial or Detrimental Effect Induced by the Absorber Characteristics[END_REF][START_REF] Kodalle | Effects of KF and RbF Post Deposition Treatments on the Growth of the CdS Buffer Layer on CIGS Thin Films -a Comparative Study[END_REF] One sample from each of the three sets was kept as reference for further investigations. The subsequent CdS deposition on the other samples was performed in an aqueous CBD solution consisting of Cd-acetate [2.6 mM], thiourea [0.1 M] and ammonia [1 M] at 58°C. [START_REF] Kodalle | Effects of KF and RbF Post Deposition Treatments on the Growth of the CdS Buffer Layer on CIGS Thin Films -a Comparative Study[END_REF] To avoid cross-contamination, the untreated, KF-treated, and RbF-treated samples were immersed in separate glass beakers. The time counting was started immediately after the samples' immersion and, starting after 1 min, one sample was taken out of the bath per minute resulting in a variation of the CBD-duration from 0 min to 9 min. The XRF data show that CdS deposition starts at 1.8  0.3 min, 1.7  0.3 min and 2.3  0.4 min for the no-PDT, KF-PDT and RbF-PDT samples, respectively (see Figure S1). The growth rates determined for the deposition time between 3-7 min achieve 6.6  1.0 nm/min, 7.8  1.2 nm/min and 7.0  1.8 nm/min for the no-PDT, KF-PDT and RbF-PDT samples, correspondingly. The thin-film thicknesses after 9 min CBD deposition time are: 42 ± 6 nm, 48 ± 5 nm, 49 ± 6 nm for the no-PDT, KF-PDT and RbF-PDT treated CIGSe absorbers, respectively. All samples were stored in a vacuum desiccator and air-exposure was kept to a minimum. Further details on these samples and a detailed discussion of the CdS growth on the differently treated CIGSe surfaces can be found here. [START_REF] Kodalle | Effects of KF and RbF Post Deposition Treatments on the Growth of the CdS Buffer Layer on CIGS Thin Films -a Comparative Study[END_REF] The KP-PYS measurements were conducted on single samples after being removed from the desiccator.

Characterization methods.

The KP [START_REF] Lord | Contact Electricity of Metals[END_REF][START_REF] Baumgärtner | Micro Kelvin Probe for Local Work-Function Measurements[END_REF] and the PYS [START_REF] Gobeli | Direct and Indirect Excitation Processes in Photoelectric Emission from Silicon[END_REF][START_REF] Sebenne | High-Resolution Photoemission Yield and Surface States in Semiconductors[END_REF][START_REF] Sebastiani | Low-Energy Yield Spectroscopy as a Novel Technique for Determining Band Offsets: Application to the c-Si(100)/a-Si:H Heterostructure[END_REF] methods were combined in an ambient pressure KP Technology SKP5050-APS02 instrument [START_REF] Baikie | Dual Mode Kelvin Probe: Featuring Ambient Pressure Photoemission Spectroscopy and Contact Potential Difference[END_REF][START_REF] Baikie | Ambient Pressure Photoemission Spectroscopy of Metal Surfaces[END_REF] to determine the materials surface electronic properties such as the work function,  , and the ionization energy (E i ). The advantage of combining these methods is that (i) the same tip electrode is used as the current collector and (ii) the work-function of samples, including that of the reference sample for KP measurements, can be directly checked by the PYS measurements which are independent of the  tip .

Kelvin Probe. The KP method for determining the materials work-function is based on measuring the contact potential difference (CPD) between the Kelvin probe as reference (Kelvin electrode) and the investigated sample. The CPD can be found by compensating it by an external dc bias voltage (V dc ) until V CPD + V dc = 0. Since -eV CPD =  tip - sample , where e is the elementary charge, with the known  tip the work function of the investigated sample is calculated. The precise determination of the V CPD is based on measuring the ac current (i ac ) which flows through the capacitor (formed between the Kelvin electrode and the investigated sample) if the distance between the two plates is harmonically varied. [START_REF] Zisman | A New Method of Measuring Contact Potential Differences in Metals[END_REF][START_REF] Anderson | Study of Adsorption on Metals by the Contact Potential Technique[END_REF][START_REF] Macdonald | Exact Solution of a Time-Varying Capacitance Problem[END_REF][START_REF] Baikie | Noise and the Kelvin Method[END_REF] The dependence of the i ac on the modulation of the capacitance was analyzed in terms of a simple equivalent circuit (Figure S2a) in which the built-in potential, V CPD , is represented as a battery which is connected in series with an external battery of an adjustable dc bias, V dc . For the actual measurements of the V CPD , an additional nullcurrent detector is inserted in series in the circuit. Then, for a certain voltage (V CPD + V dc ) and a change in capacitance due to the Kelvin electrode's oscillation, the current is given by i ac (t) = (V CPD + V dc ) (dC(t)/dt). In this equation, the capacitance C(t) of the capacitor with two parallel plates of an area A varies due to the variable distance d(t) = d 0 + d 1 sin(t), where d 0 is the capacitor mean spacing and d 1 is the amplitude of oscillation with the angular frequency . The voltage V dc is changed until the null-current condition in the circuit is detected by a sensitive current amplifier within the null-method approach. This compensating voltage V dc = -V CPD is recorded as the contact potential difference between the two plates of the capacitor. Another approach of recording the V CPD is based on the off-null detection method [START_REF] Baikie | Ambient Pressure Photoemission Spectroscopy of Metal Surfaces[END_REF][START_REF] Baikie | Low Cost PC Based Scanning Kelvin Probe[END_REF] which uses the dependence of the peak-to-peak output voltage (V ptp ) on V dc since the V ptp is proportional to (V CPD + V dc ). In this case, the V CPD is determined from the intercept of the V ptp = f(V dc ) curve at V ptp = 0 V from two high-signal level measurements (Figure S2b). The determined gradient (slope) G = V ptp /V dc is used in a feedback control loop to maintain a constant mean tip-to-sample distance, i.e., the probe to sample spacing can be controlled by pre-setting G.

The KP system used in this study employs a 2.0 mm diameter tip with a gold-alloy coating.

In the CPD mode, the tip is vibrated at 75 Hz with an amplitude of approximately 0.2 mm at a pre-set gradient G = 300 resulting in an average tip-sample spacing of about 1.0 mm. The KP setup is placed in a Faraday cage which screens the eventual external electrical fields and allows for a controlled sample illumination. The samples with the size of up to 5×5 cm 2 are mounted on a motorized 3-axis stage with a transitional position control of <300 nm which allows for selective as well as mapping options. Reference measurements for the calibration of the work function of the tip,  tip , were conducted on an Au thin-film. Typical  tip values between 4.92 eV -5.00 eV were determined by means of the photoelectron yield spectroscopy (see next chapter).

Subsequently, the work function of the sample,  sample , was calculated from the CPD data according to  sample =  tip + eCPD. The measurements of the surface photovoltage (SPV) [START_REF] Kronik | Surface Photovoltage Phenomena: Theory, Experiment, and Applications[END_REF] defined as SPV = CPD light -CPD dark were performed using laser diodes with wavelength/energy of 405 nm/3.06 eV, 450 nm/2.75 eV, 532 nm/2.33 eV, 670 nm/1.85 eV, and 980 nm/1.26 eV with a light power density of 100 mW/cm 2 . The CPD resolution, as determined on an Au thin film, is better than 2 mV (see Figure S3). The standard CPD deviation depends yet on the material surface state and was therefore determined in each measurement separately.

Photoelectron yield spectroscopy. The PYS method was first established in vacuum [START_REF] Sebenne | High-Resolution Photoemission Yield and Surface States in Semiconductors[END_REF][START_REF] Sebastiani | Low-Energy Yield Spectroscopy as a Novel Technique for Determining Band Offsets: Application to the c-Si(100)/a-Si:H Heterostructure[END_REF][START_REF] Honda | Atmospheric Effect of Air, N 2 , O 2 , and Water Vapor on the Ionization Energy of Titanyl Phthalocyanine Thin Film Studied by Photoemission Yield Spectroscopy[END_REF] and then steadily developed for the application in non-vacuum ambiance, e.g., in air. [START_REF] Baikie | Dual Mode Kelvin Probe: Featuring Ambient Pressure Photoemission Spectroscopy and Contact Potential Difference[END_REF][START_REF] Baikie | Ambient Pressure Photoemission Spectroscopy of Metal Surfaces[END_REF][START_REF] Kirihata | Externally Quenched Air Counter for Low-Energy Electron Emission Measurements[END_REF][START_REF] Miyamoto | Ionization Potential of Organic Pigment Film by Atmospheric Photoelectron Emission Analysis[END_REF][START_REF] Grigalevicius | 3,6-Di(N-Diphenylamino)-9-Phenylcarbozole and its Methyl-Substituted Derivative as Novel Hole-Transporting Amorphous Molecular Materials[END_REF][START_REF] Honda | Atmospheric Effect of Air, N 2 , O 2 , and Water Vapor on the Ionization Energy of Titanyl Phthalocyanine Thin Film Studied by Photoemission Yield Spectroscopy[END_REF][START_REF] Nakayama | Direct Observation of the Electronic States of Single Crystalline Rubrene under Ambient Condition by Photoelectron Yield Spectroscopy[END_REF][START_REF] Harwell | Probing the Energy Levels of Perovskite Solar Cells via Kelvin Probe and UV Ambient Pressure Photoemission Spectroscopy[END_REF] The photoemission process in vacuum consists of the following steps: (i) optical excitation of electrons into states of higher energy, (ii) electron transport towards the surface, and (iii) escape through the surface to vacuum. [START_REF] Berglund | Photoemission Studies of Copper and Silver: Theory[END_REF] During the steps (ii) and (iii), primary photoelectrons are inelastically scattering and secondary electrons with lower kinetic energies are excited. For the low kinetic energy electrons an escape depth <50 Å was reported. [START_REF] Sebastiani | Low-Energy Yield Spectroscopy as a Novel Technique for Determining Band Offsets: Application to the c-Si(100)/a-Si:H Heterostructure[END_REF][START_REF] Honda | Atmospheric Effect of Air, N 2 , O 2 , and Water Vapor on the Ionization Energy of Titanyl Phthalocyanine Thin Film Studied by Photoemission Yield Spectroscopy[END_REF] The escaped electrons are easily collected under the application of a moderate electrical field in order to avoid the formation of the negative space charge near the sample surface. Under ambient conditions, an ionic current is recorded instead of an electronic one. In air, the ejected electrons with a mean free path of less than 1 µm attach to the air constituent molecules and form anions such as O 2  , N 2  and OH  . [START_REF] Baikie | Ambient Pressure Photoemission Spectroscopy of Metal Surfaces[END_REF][START_REF] Nakayama | Direct Observation of the Electronic States of Single Crystalline Rubrene under Ambient Condition by Photoelectron Yield Spectroscopy[END_REF][START_REF] Gobeli | Direct and Indirect Excitation Processes in Photoelectric Emission from Silicon[END_REF] Having a much higher mean free pass than the electrons, these anions drift to the positively charged electrode, thus allowing to record the photoelectron current indirectly.

The PYS method for determining the work-function of metals and the ionization energy of semiconductors is based on measuring the photoelectron yield Y(h) as a function of photon energy h, Y(h) being defined as the number of photoemitted electrons per incident photon, at a given photon energy h. If the photon energy becomes greater than the threshold ionization energy, E i , the Y(h) value starts to increase according to

The power index n = 2 for metal samples [START_REF] Fowler | The Analysis of Photoelectric Sensitivity Curves for Clean Metals at Various Temperatures[END_REF] and degenerate semiconductors. For most of inorganic semiconductors n = 3 when considering the volume excitation, [START_REF] Ballantyne | Effect of Phonon Energy Loss on Photoemissive Yield Near Threshold[END_REF] while n = 1, 3/2, 5/2 depending on the contribution of the surface imperfection and band states. [START_REF] Kane | Theory of Photoelectric Emission from Semiconductors[END_REF][START_REF] Gobeli | Direct and Indirect Excitation Processes in Photoelectric Emission from Silicon[END_REF] For a large number of organic materials n = 3 is observed, [START_REF]Electronic Processes in Organic Electronics[END_REF] as in the case of inorganic semiconductors.

Thus, the E i value is determined as the threshold in the h Y n plots. In addition to the ionization energy, the PYS spectra contain the information on the effective density of states spreading from the Fermi level (E F ) down to h- below E F . [START_REF] Sebenne | High-Resolution Photoemission Yield and Surface States in Semiconductors[END_REF] In our work, the PYS setup uses the same Kelvin probe (static) to detect the photoemission currents as a function of the incident photon energy. The light source comprises a deuterium (D 2 ) lamp coupled with a grating monochromator. The range of the incident photon energy is 3.4-7.0

eV. The light is guided via a DUV optical fiber forming on the sample surface an elliptical spot with a size of approximately 3 × 4 mm 2 . A bias potential of +10 V is applied to the tip. The photoemission threshold is determined with a resolution of 30 meV.



The information depth (ID) of the PYS measurements is estimated by considering the exponential attenuation of photoelectrons in matter according to the kinetic energy-dependent inelastic mean free paths (IMFPs) (see Figure S4 and related explanations). Thus, if photoelectrons are detected as generated at a distance of 3×IMFP below the sample's surface, the signal will be attenuated to 5%. associated with the Cd(OH) 2 adsorption. Notice that during this stage no CdS is deposited and that its duration depends strongly on the chemical structure of the surface and its morphology. [START_REF] Kodalle | Effects of KF and RbF Post Deposition Treatments on the Growth of the CdS Buffer Layer on CIGS Thin Films -a Comparative Study[END_REF][START_REF] Lincot | Situ Growth and Structural Studies by Combined Quartz Crystal Microbalance and Electrochemical Impedance Techniques[END_REF] (iii) The ion-by-ion CdS growth and (iv) cluster-by-cluster deposition. [START_REF] Lincot | Situ Growth and Structural Studies by Combined Quartz Crystal Microbalance and Electrochemical Impedance Techniques[END_REF][START_REF] Choubrac | 7.6% CZGSe Solar Cells Thanks to Optimized CdS Chemical Bath Deposition[END_REF] Thus, the first CBD stage can be depicted for a duration of up to 1 min, followed by a second stage for the duration 1 min  t CBD  3(4) min and by the third CBD stage for the duration 3(4) min  t CBD  9 min. The fourth CBD stage seems to have a minor contribution for the t CBD  9 min since no essential changes in the ionization energy are observed. Therefore, this last CBD stage will be omitted from further discussions. Thus, by applying the above correlation, we claim that the specific minimum of the ionization energy at t CBD = 1 min may be assigned to the end phase of the CBD etching process of the CIGSe surface. The t CBD = 1 min time is found below the starting time of the CdS growth between 1.7-2.3 min (see Experimental Section) which eV, practically coincide within the error bars. It can be thus concluded that in both cases the CIGSe surface has a similar Cu-poor surface composition. The deviations in E i for the RbF-PDT sample may be attributed to the effects of Rb on the CIGSe surface chemistry and electronics.

Note that after a RbF-PDT a surface band gap of 1.51 eV was found on CIGSe thin films [START_REF] Hauschild | Impact of a RbF Postdeposition Treatment on the Electronic Structure of the CdS/Cu(In,Ga)Se 2 Heterojunction in High-Efficiency Thin-Film Solar Cells[END_REF] which fits to the bandgaps of the Cu-poor surface composition between 1.4 eV for CuInSe 2 [START_REF] Morkel | Flat Conduction-Band Alignment at the Thin-Film Solar-Cell Heterojunction[END_REF] and 1.61

eV for Cu(In,Ga)Se 2 . [START_REF] Reinhard | Potassium Postdeposition Treatment-Induced Band Gap Widening at Cu(In,Ga)Se 2 Surfaces -Reason for Performance Leap?[END_REF] In spite of the large E i values observed on bare CIGSe samples, after 1 min dipping in the CBD solution, the E i of all samples decreases significantly, as previously observed, to 5.24 eV, 5.22 eV and 5.39 eV for the no-PDT, RbF-PDT and KF-PDT samples, accordingly. These ionization energy changes indicate chemical modifications of the surface of the thin films agreeing with the etching effects of the CBD mentioned above. [START_REF] Weinhardt | Surface and Interface Properties in Thin-Film Solar Cells: Using Soft X-Rays and Electrons to Unravel the Electronic and Chemical Structure[END_REF][START_REF] Kodalle | Elucidating the Mechanism of an RbF post Deposition Treatment in CIGS Thin Film Solar Cells[END_REF][START_REF] Kreikemeyer-Lorenzo | Rubidium Fluoride Post-Deposition Treatment: Impact on the Chemical Structure of the Cu(In,Ga)Se 2 Surface and CdS/Cu(In,Ga)Se 2 Interface in Thin-Film Solar Cells[END_REF] The significant decrease of E i is followed by a gradual increase of the ionization energy of all samples for the deposition times up to 3-4 minutes. This indicates a gradual change in the physico-chemical properties of the samples surface, which we attribute to the nucleation of the Cd(OH) 2 molecules on the CIGSe surface during the induction/coalescence phase of the CBD-CdS growth process [START_REF] Lincot | Situ Growth and Structural Studies by Combined Quartz Crystal Microbalance and Electrochemical Impedance Techniques[END_REF][START_REF] Choubrac | 7.6% CZGSe Solar Cells Thanks to Optimized CdS Chemical Bath Deposition[END_REF] and to the diffusion of Cd-atoms into the chalcopyrite lattice creating Cd Cu anti-sites from the available Cu vacancies, V Cu . [START_REF] Kodalle | Effects of KF and RbF Post Deposition Treatments on the Growth of the CdS Buffer Layer on CIGS Thin Films -a Comparative Study[END_REF][START_REF] Nakada | Direct Evidence of Cd Diffusion Into Cu(In,Ga)Se 2 Thin Films During Chemical-Bath Deposition Process of CdS Films[END_REF] We emphasize that there might be more complex reactions taking place on a smaller time scale than we probed, e.g., formation of CdSe at 1 s CBD immersion time. [START_REF] Nicoara | Evidence for Chemical and Electronic Nonuniformities in the Formation of the Interface of RbF-Treated Cu(In,Ga)Se 2 with CdS[END_REF] Note however that no CdSe was detected for any of the CBD-CdS deposition times >1 s investigated here. [START_REF] Nicoara | Evidence for Chemical and Electronic Nonuniformities in the Formation of the Interface of RbF-Treated Cu(In,Ga)Se 2 with CdS[END_REF] The further increase of the CBD-CdS deposition time results in a value of about 6.05 ± 0.03 eV for all samples. The fluctuations observed are explained by slightly different surface dipoles on different samples. The latter E i value coincides in fact within the error bars with the one measured on CdS reference layers prepared by the same CBD process on ITO/glass substrates (Figure S8). Thus, for the CBD-CdS deposition times >4 min closed CdS films with similar chemical properties are prepared independently of the PDT.

Using the work function data in Figure 1d and the ionization energy results in Figure 3a, the development of the VBM at the CdS/CIGSe interface relative to the Fermi level is plotted in Figure 3b (according to the equation 2). The fluctuations of E VBM are reduced compared to the variations observed for the work function and ionization energy, since for each pair of  and E i data the specific local vacuum level is used. Thus, the influence of the native surface dipole is excluded (consult Figure S5). The VBM values of -0.85 eV, -0.91 eV and -1.18 eV for the no-PDT, RbF-PDT and KF-PDT reference samples, respectively, are similar within the error bars to the published data. [START_REF] Hauschild | Impact of a RbF Postdeposition Treatment on the Electronic Structure of the CdS/Cu(In,Ga)Se 2 Heterojunction in High-Efficiency Thin-Film Solar Cells[END_REF][START_REF] Reinhard | Potassium Postdeposition Treatment-Induced Band Gap Widening at Cu(In,Ga)Se 2 Surfaces -Reason for Performance Leap?[END_REF][START_REF] Pistor | Experimental Indication for Band Gap Widening of Chalcopyrite Solar Cell Absorbers After Potassium Fluoride Treatment[END_REF] After 1 min immersion of samples in the CBD-CdS solution, the VBM positions change significantly to values of -0.67 eV, -0.62 eV and -0.56 eV for the no-PDT, RbF-PDT and KF-PDT samples, correspondingly. Good agreement with the available data is found for the KF-PDT sample. [START_REF] Reinhard | Potassium Postdeposition Treatment-Induced Band Gap Widening at Cu(In,Ga)Se 2 Surfaces -Reason for Performance Leap?[END_REF] The E VBM = -1.79 eV of a thick CdS thin film deposited on a no-PDT CIGSe sample is found in good agreement with the literature data. [START_REF] Hauschild | Impact of a RbF Postdeposition Treatment on the Electronic Structure of the CdS/Cu(In,Ga)Se 2 Heterojunction in High-Efficiency Thin-Film Solar Cells[END_REF][START_REF] Weinhardt | Band Alignment at the i-ZnO/CdS Interface in Cu(In,Ga)(S,Se) 2 Thin-Film Solar Cells[END_REF] Considering the obtained VBM graphs in Figure 3b, we now plot the CBM edges in Figure 4 by considering the relation E CBM = E VBM + E g . The respective bandgaps are considered based on the chemical and electronic structure of the CIGSe surface state as well as the CBD-CdS deposition phases. The following scheme related to the CBD deposition time, t CBD , is applied:  3(4) min  t CBD  9 min & CBD-CdS ion-by-ion growth regime: 66,67 E g-CdS = 2.42 ± 0.02 eV as determined from internal quantum efficiency measurements on complete solar cells. We consider here the constant E g being aware of the fact that at a longer CBD duration the CdS ion-by-ion growth phase may have been complemented by the clusterby-cluster (colloidal) growth regime, which still however does not show a visible influence on the CdS opto-electronical properties.

 t CBD = 0 min & PDT-
A summary of the considered and calculated data is presented in Table 1.

The obtained results for the VBM and the CBM development as a function of the CBD-CdS deposition time are presented in Figure 4 for each PDT separately. For the sample with no-PDT in Figure 4a, we assume a surface band gap of 1.46 eV corresponding to a Cu-poor, vacancy compound or reconstructed surface, in accordance with Hauschild et al. [START_REF] Hauschild | Impact of a RbF Postdeposition Treatment on the Electronic Structure of the CdS/Cu(In,Ga)Se 2 Heterojunction in High-Efficiency Thin-Film Solar Cells[END_REF] For the cleaned surface -after 1 min CBD -the same surface E g is considered as that is attributed to the same Cu-poor surface composition as in the previous case, however after oxide removal.

The oxide removal is electronically equivalent to the surface dipole removal, thus shifting both the VBM and CBM edges upward without changing E g . The CBM position at the beginning of the Cd(OH) 2 nucleation phase, i.e. for the 2 minutes CBD duration, is still kept constant at the level of the cleaning stage since for the CdS/CIGSe interface a flat CBM band alignment was reported. [START_REF] Hauschild | Impact of a RbF Postdeposition Treatment on the Electronic Structure of the CdS/Cu(In,Ga)Se 2 Heterojunction in High-Efficiency Thin-Film Solar Cells[END_REF][START_REF] Weinhardt | Band Alignment at the i-ZnO/CdS Interface in Cu(In,Ga)(S,Se) 2 Thin-Film Solar Cells[END_REF] Note that there are numerical simulation studies as well as some experimental The ECBM values for the latter CBD duration were assumed similar to those at tCBD = 1 min since most of studies performed by photoelectron spectroscopy measurements have found a flat band CBM alignment at the CdS/CIGSe interface (with Cu-poor CIGSe surface composition). [START_REF] Hauschild | Impact of a RbF Postdeposition Treatment on the Electronic Structure of the CdS/Cu(In,Ga)Se 2 Heterojunction in High-Efficiency Thin-Film Solar Cells[END_REF][START_REF] Weinhardt | Band Alignment at the i-ZnO/CdS Interface in Cu(In,Ga)(S,Se) 2 Thin-Film Solar Cells[END_REF] The ECBM data for tCBD = 3 min is determined from extrapolated values between the CIGSe surface at tCBD = 2 min and the formed CdS thin film at tCBD = 4 min. The resultant band gaps were then accordingly calculated. a Band gap of the identified In2Se3 thin film in our PYS measurements.

b Band gap of CdS thin films determined from internal quantum efficiency measurements on complete ZnO/CdS/CIGSe/Mo/SLG solar cells. findings that report a spike-like CBM alignment at the abrupt absorber/buffer interface. [START_REF] Sozzi | On the Combined Effects of Window/Buffer and Buffer/Absorber Conduction-Band Offsets, Buffer Thickness and Doping on Thin-Film Solar Cell Performance[END_REF][START_REF] Schmid | A Comprehensive Characterization of the Interfaces in Mo/CIS/CdS/ZnO Solar Cell Structures[END_REF][START_REF] Park | Junction Formation at the Interface of CdS/CuIn x Ga (1-x) Se 2[END_REF] For this CBD deposition time, however, the CdS growth still does not occur, as concluded above.

Since the VBM shows constant values starting from approximately 3 min CdS deposition time, we consider this time point as the beginning of the ion-by-ion growth phase of the CdS layer [START_REF] Kodalle | Effects of KF and RbF Post Deposition Treatments on the Growth of the CdS Buffer Layer on CIGS Thin Films -a Comparative Study[END_REF][START_REF] Choubrac | 7.6% CZGSe Solar Cells Thanks to Optimized CdS Chemical Bath Deposition[END_REF] with an E g-CdS = 2.42 ± 0.02 eV determined from corresponding internal quantum efficiency measurements on solar cells. [START_REF] Kodalle | Effects of KF and RbF Post Deposition Treatments on the Growth of the CdS Buffer Layer on CIGS Thin Films -a Comparative Study[END_REF] For the bare KF-PDT reference sample, a surface band gap of 2.0 eV is considered based on our PYS results, which suggest the presence of an In 2 Se 3 :Na layer, and reports in literature that have proposed an In 2 Se 3 :K layer at the CIGSe surface. [START_REF] Reinhard | Potassium Postdeposition Treatment-Induced Band Gap Widening at Cu(In,Ga)Se 2 Surfaces -Reason for Performance Leap?[END_REF][25][START_REF] Lepetit | Post Deposition Treatment in Co-Evaporated Cu(In,Ga)Se 2 Thin Film Solar Cells: Beneficial or Detrimental Effect Induced by the Absorber Characteristics[END_REF][START_REF] Pistor | Experimental Indication for Band Gap Widening of Chalcopyrite Solar Cell Absorbers After Potassium Fluoride Treatment[END_REF] For the cleaned surface after 1 min CBD, a minimum surface band gap of 1.58 eV is considered, as reported by Handick et al. [START_REF] Reinhard | Potassium Postdeposition Treatment-Induced Band Gap Widening at Cu(In,Ga)Se 2 Surfaces -Reason for Performance Leap?[END_REF] (being aware that these reference data are taken from films prepared by a low-temperature multistage process on molybdenum coated polyimide foils). We note that, as in the case of no-PDT, the CBM position at the beginning of the Cd(OH) 2 nucleation phase (i.e., for the 2 min CBD duration) is kept at the same level. Starting from approximately 3 min of the CdS deposition, a constant E g-CdS = 2.42 eV is considered. In the case of the RbF-PDT reference sample, a surface band gap of 1.51 eV is assumed, as determined by Hauschild et al. [START_REF] Hauschild | Impact of a RbF Postdeposition Treatment on the Electronic Structure of the CdS/Cu(In,Ga)Se 2 Heterojunction in High-Efficiency Thin-Film Solar Cells[END_REF] The same surface E g is presumed for the cleaned CIGSe surface after 1 min of the CBD processing, resulting in a E CBM = 0.89 eV. This latter value is assumed also for the CBM position at the beginning of the CdS induction/coalescence phase, i.e. at 2 min of the CBD-CdS deposition time. For the ion-by-ion CdS growth phase, an E g-CdS = 2.42 eV is considered, as in the previous cases.

From the above findings of the behaviour of the work function, ionization energy, the VBM and CBM positions, we conclude that the initial surface state of the no-PDT, KF-PDT and RbF-PDT samples, characterized by large VBM downward shifts, is not preserved after the immersion in the CBD-CdS solution. Due to the etching effect of ammonia during the first CBD minute, the CIGSe surface is cleaned from oxides and simultaneously from K-In-Se and Rb-In-Se phases formed during the KF-PDT and RbF-PDT. That is why the E VBM of the CIGSe surface changes to values between 0.56 eV -0.67 eV. The observed differences in the E VBM are attributed to doping by the K and Rb atoms diffused into the Cu-poor CIGSe surface layer. Thus, because during the CBD etching stage surface phases are removed from CIGSe, we will consider in the final band diagrams the electronic states of the CIGSe layers as after the 1 min immersion in the CdS chemical bath.

In the final energy band diagrams, we will consider in addition the CIGSe thin-film bulk band gap of 1.18 eV determined from internal quantum efficiency measurements on solar cells prepared from identically processed absorbers. [START_REF] Kodalle | Effects of KF and RbF Post Deposition Treatments on the Growth of the CdS Buffer Layer on CIGS Thin Films -a Comparative Study[END_REF] The VBM position relative to the Fermi level is found according to Fermi-Dirac statistics [START_REF] Sze | Physics of Semiconductor Devices[END_REF] by considering the density-of-state effective mass for holes in CIGSe m p = 0.235m 0 , [START_REF] Persson | Anisotropic Hole-Mass Tensor of CuIn 1-x Ga x (S,Se) 2 : Presence of Free Carriers Narrows the Energy Gap[END_REF] where m 0 is the electron rest mass, and the concentration of holes of 1.1 10 16 cm -3 determined from capacitance-voltage measurements on accompanying solar cells made from the same absorbers. We thus obtain the E F -E VBM = 0.14 eV.

The VBM and CBM of the CIGSe bulk are shown in Figure S9 along with the surface electronics immediately after the respective PDTs and the changes that occurred after 1 min CBD duration. We note that the VBM of the CIGSe surface is still much lower than the VBM of the CIGSe bulk, however not as low as immediately after the PDT process. Considering this CIGSe bulk/surface energy level structure and completing it by the VBM and the CBM development for the CBD duration 1 min from Figure 4, we propose detailed energy band diagrams in Figure 5. We observe features which were not previously reported. The CIGSe surface band gap varies, depending on the kind of conditioning, from 1.46 eV to 1.51 eV and 1.58 eV for the no-PDT, RbF-PDT and KF-PDT, respectively (see also data for 1 min CBD duration in Table 1). This may be a hint for different effects of the Na, Rb and K on the electronic properties of the Cu-poor CIGSe surface layer due to the in-diffusion of those species.

At the CIGSe/CdS interface the lowest VBM-discontinuity of 0.17 eV is observed for the RbF-PDT sample. That could be an indication for beneficially lower saturation currents of illuminated solar cells. In addition, at the CIGSe/CdS interface a gradual change of the band gap between the CIGSe surface and the CdS thick layer is observed. Thus, interfacial band gaps of 1.88 eV, 1.95 eV and 1.68 eV are observed for no-PDT, KF-PDT and RbF-PDT, respectively. These band gaps increase then gradually to the band gap of CdS thick layer of 2.42 eV. Hence, the highest band gap gradient, related mostly to the VBM variation, is observed for the RbF-PDT sample. We attribute this gradual increase of the interfacial band gap to a continuous change in the CdS thin film composition. The differences observed for the starting band gaps can be explained by differences in the starting layer stoichiometry influenced by the CIGSe absorber surface chemistry. This interfacial model is supported by the data on the [Cd]:[S] ratio of about 3:1 reported for 1-5 min CBD deposition time. [START_REF] Nicoara | Evidence for Chemical and Electronic Nonuniformities in the Formation of the Interface of RbF-Treated Cu(In,Ga)Se 2 with CdS[END_REF] The previously estimated CdS thickness for the capping layer on no-PDT sample of about 11 nm may be attributed to such an interfacial layer.

For the RbF-PDT and KF-PDT samples we find interfacial layer thicknesses of 12 nm and 18 nm, respectively. Thus, we propose that, due to compositional changes, the band gap of the growing CdS buffer layer gradually changes from non-stoichiometric to stoichiometric CdS within the formation of the first 11-18 nm of the buffer. Note that the contribution of the OH groups and alkali, C or In atoms [START_REF] Lepetit | Post Deposition Treatment in Co-Evaporated Cu(In,Ga)Se 2 Thin Film Solar Cells: Beneficial or Detrimental Effect Induced by the Absorber Characteristics[END_REF][START_REF] Nicoara | Evidence for Chemical and Electronic Nonuniformities in the Formation of the Interface of RbF-Treated Cu(In,Ga)Se 2 with CdS[END_REF] cm -3 , 2.5  10 5 cm -3 and 3.6  10 4 cm -3 determined for the no-PDT, RbF-PDT and KF-PDT cases, respectively. Thus, differences in charge carrier densities amount to orders of magnitude implying strong variations in the CdS layer resistivity. We relate these findings to compensating effects of Rb and K atoms in CdS, similarly to that of the Na atoms in CdS which cause acceptor type defects due to a single valence electron. [START_REF] Varley | First Principles Calculations of Point Defect Diffusion in CdS Buffer Layers: Implications for Cu(In,Ga)(Se,S) 2 and Cu 2 (Zn,Sn)(Se, S) 4 -Based Thin-Film Photovoltaics[END_REF] We explain these findings by the diffusion of the Rb and K atoms from the CIGSe surface into the CdS layer as well as by the presence of these alkali atoms in the chemical bath after having been etched off from films surface during a first stage of the CBD process. Note in addition that in previous works Rb and Na atoms were detected on the CdS surface. [START_REF] Kreikemeyer-Lorenzo | Rubidium Fluoride Post-Deposition Treatment: Impact on the Chemical Structure of the Cu(In,Ga)Se 2 Surface and CdS/Cu(In,Ga)Se 2 Interface in Thin-Film Solar Cells[END_REF] Thus, Na, K and Rb atoms act in CdS as compensating acceptors, the K atoms having a stronger compensation impact which may also affect the optical properties detrimentally. This finding is supported by theIQE measurements which show the strongest CdSabsorption for KF-treated absorber for all CBD durations. [START_REF] Kodalle | Effects of KF and RbF Post Deposition Treatments on the Growth of the CdS Buffer Layer on CIGS Thin Films -a Comparative Study[END_REF] While the authors claim that this should be attributed to an alkali induced thickness effect of the deposited CBD CdS buffer, this additional compensation effect may also contribute to why the solar cells from KF-PDT absorbers show lower fill factors and short-circuit currents compared to their counterparts from RbF-PDT absorbers.

3 .Figure 1 .Figure 2 .Figure 3 .

 3123 Figure 1. (a-c) Selected time dependent work function curves of the (a) no-PDT, (b) KF-PDT and (c) RbF-PDT conditioned CIGSe surface as a function of the CdS chemical bath deposition (CBD) time

Figure 4 .

 4 Figure 4. Evolution of the valence and conduction band edges (VBM, CBM) of the (a) no-PDT, (b) KF-PDT and (c) RbF-PDT conditioned CIGSe surface.

  is not excluded. For the thick CdS layer we observe that the CBM positions of the KF-PDT and RbF-PDT samples shift upward compared to the no-PDT sample which shows the lowest E CBM -E F = 0.63 eV. The E CBM -E F increases to 0.78 eV for the RbF-PDT and to 0.83 eV for the KF-PDT samples. The increased E CBM -E F of CdS layers point at a decreased charge carrier concentration. The magnitude of

Figure 5 .

 5 Figure 5. Energy band diagrams of the CIGSe/CdS heterojunction with (a) no-PDT, (b) KF-PDT and (c) RbF-PDT conditioned CIGSe surface.

  carrier concentration are found from estimations on the CdS surface. By applying the Fermi-Dirac statistics for the n-type CdS with the m n = 0.25m 0[START_REF] Ekuma | An-Initio Local Density Approximation Description of the Electronic Properties of Zinc Blende Cadmium Sulfide (zb-CdS)[END_REF] and using the E CBM -E F values, we obtain the concentration of the free electrons on the CdS surface of 8.2  10 7

31

 31

  the Kelvin probe and the photoelectron yield spectroscopy methods at ambient pressure to investigate the formation of the CdS/CIGSe interface with differently conditioned CIGSe surface by alkali-fluoride post-deposition treatments such as KF-PDT and RbF-PDT. The results were compared to reference samples without PDT. An overall good agreement to previously published data has been found. In addition, we observe distinct features related to an initial CBD-CdS growth stage associated with a CIGSe surface etching phase. We found in particular that the initial surface state of the PDT-treated CIGSe surface is not preserved after immersion in the CBD. After the initial CBD surface etching stage, the VBM of the CIGSe surface is significantly shifted towards the Fermi level, e.g. by about 180 meV, 290 meV and 620 meV for the no-PDT, RbF-PDT and KF-PDT treated CIGSe surfaces, respectively. However, the CIGSe surface VBM positions are still much below the VBM of the CIGSe bulk. We propose that the native oxides and secondary phases are removed, resulting in a CIGSe surface with a band gap of 1.46 eV -1.58 eV characteristic for copper-poor CIGSe surface composition. At the CdS/CIGSe interface a lowest VBM discontinuity is found for the RbF-PDT sample. At this interface, during the CdS induction/coalescence phase, a thin layer with a thickness 11-18 nm and a graded band gap is formed due to a gradual variation of the CdS composition. This layer is electronically characterized by a gradual VBM downward shift. During the ion-by-ion CBD growth stage the CdS thin films incorporate the alkali Na, K and Rb atoms which were found to act in CdS as compensating acceptors, leading to an increased thin film resistivity. The observed electronic characteristics correlate well with the behavior of the solar cells prepared from the same CIGSe absorbers.

  Therefore, the ID of our PYS measurements is estimated to approximately 3×IMFP, and hence, the PYS spectra measured from closed layers with thicknesses exceeding 3×IMFP can be accurately recorded without being influenced by the dipole" show no or negligible development. If these assumptions are fulfilled, the results obtained under the ambient measurement conditions shall agree well with the  and E i values obtained by measurements conducted under non air-exposed conditions, e.g., in vacuum.

	Notice that for each measured surface,  defines the local vacuum level (see Figure S5). The
	VBM development relative to the Fermi level, E F , can be then found as
	(2)

substrate. The ID values were estimated for Cu, In, Cd, Se and Na electrons. The highest ID of approximately 3.8  0.8 nm is found for In electrons. The maximum values of IDs for Cu, Cd and Se may be approximated to 1.2  0.3 nm, 1.5  0.3 nm, and 1.4  0.3 nm, respectively. The ID for Na is 2.1  0.4 nm. Thus, the conducted PYS measurements will not record any contribution from the CIGSe substrate after its coverage by a continuous layer with a thickness >3.8 nm. The PYS spectra from subsequently deposited films which contain Cd will be measured with an ID of approximately 1.5 nm.

2.3. Evaluation of electronic parameters.

As the measurements were performed in ambient conditions, assumptions are made concerning possible surface dipoles due to air exposure. (i) Molecules adsorbed from air, e.g., oxygen and/or adventitious hydrocarbon (C-C, C-H bonds) promote formation of a surface dipole ( s ) (see Figure

S5

), called thereafter the "adsorbed surface dipole", on both sample and tip surfaces thus resulting in no influence to the measured CPD and, consequently, to the calculated work function values (since the  sample =  tip + eV CPD ). (ii) The sample surface is chemically stable during the short time exposure to the ambience so that the native oxides potentially present at the film surface and thus the related "native surface For measurements in the dark, the E VBM includes the surface band bending contribution as comprised in . The band bending can be eliminated, if the Fermi level at the surface is not pinned, by performing KP measurements under sufficiently intense illumination with photons of energies higher than E g . Thus, for the obtained E VBM the bulk values may be assumed if the contribution of the native surface dipole is neglectable. With the E VBM data, the CBM development can be calculated as E CBM = E VBM + E g by considering the E g values obtained from complementary measurements and literature data. The complementary measurements include photoelectron yield spectra on reference samples and quantum efficiency measurements on complete solar cells.

  CIGSe surface: E g of the no-PDT, KF-PDT and RbF-PDT CIGSe as identified from literature by comparing the surface chemical composition and electronic properties which fit to our data, e.g., to E VBM results. t CBD = 1 min & CBD-etched CIGSe surface: available E g values for the Cu-poor surface composition after the CBD etching phase. t CBD = 2 min & beginning of the CBD induction/coalescence regime (initiation of the Cd(OH) 2 nucleation): E g which results from considering the measured in this work VBM position and the same E CBM as for the t CBD = 1 min, since a flat band CBM alignment was determined at the CIGSe/CdS interface in most of the studies conducted on the overall Cu-poor CIGSe thin films.[START_REF] Hauschild | Impact of a RbF Postdeposition Treatment on the Electronic Structure of the CdS/Cu(In,Ga)Se 2 Heterojunction in High-Efficiency Thin-Film Solar Cells[END_REF][START_REF] Weinhardt | Band Alignment at the i-ZnO/CdS Interface in Cu(In,Ga)(S,Se) 2 Thin-Film Solar Cells[END_REF]  2 min  t CBD  3(4) min & CBD-CdS induction/coalescence regime (Cd(OH) 2 nucleation): graded E g resulting from the VBM and CBM changes. Note that during this CBD phase, with a duration dependent on the substrate surface condition, no CdS was found.[START_REF] Kodalle | Effects of KF and RbF Post Deposition Treatments on the Growth of the CdS Buffer Layer on CIGS Thin Films -a Comparative Study[END_REF][START_REF] Choubrac | 7.6% CZGSe Solar Cells Thanks to Optimized CdS Chemical Bath Deposition[END_REF]

Table 1 . VBM, CBM and Electronic Surface Band Gap Energies of the no-PDT, KF-PDT and RbF-PDT CIGSe Absorbers and CIGSe/CdS Interface after Different CBD Duration §

 1

§ EVBM data are derived from our KP-PYS measurements. The ECBM values are determined as EVBM + Eg except those for the 2 min CBD duration.

  [START_REF] Kodalle | Effects of KF and RbF Post Deposition Treatments on the Growth of the CdS Buffer Layer on CIGS Thin Films -a Comparative Study[END_REF]

shown in the inset. (d) Work function dependences of the no-PDT, KF-PDT and RbF-PDT conditioned

appears reasonable when considering the second induction/coalescence deposition stage. Above this time, as can be seen on the graph in Figure3a, the ionization energies achieve values characteristic of thick CdS films at about 3 min for no-PDT and 4-5 min for RbF-PDT and KF-PDT samples. The PYS spectra in Figure 2 and their enlarged versions in Figure S6 show no signatures from the CIGSe substrate for the times found for RbF-PDT and KF-PDT, meaning that the CdS thickness is at least above the ID related to In electrons, i.e., of 3.8  0.8 nm. For the no-PDT sample at 3 min deposition time, the PYS spectrum still shows a contribution from the CIGSe substrate (see Figure S6a). We correct therefore the time corresponding to the PYS information depth for CIGSe to 3.5 min (with an error of approximately 0.5 min), corresponding to a CdS capping layer thickness of 11.2  1.6 nm. Thus, we find an experimental information depth for the PYS measurements on CIGSe thin-films of 11.2  1.6 nm, in good agreement with the ID found for In electrons.

The surface ionization energy of the reference no-PDT sample (corresponding to t CBD = 0 min in Figure 3a) amounts to 5.41 eV. An E i = 5.50 eV close to the latter value is observed for the RbF-PDT sample, while the ionization energy of the KF-PDT sample significantly increases to 5.70 eV. In the case of the KF-PDT treated CIGSe surface, the E i = 5.70 eV lies between values of 5.60 eV and 5.84 eV measured on the reference In 2 Se 3 thin films deposited on Mo/SLG and SLG substrates, correspondingly (see Figure S7). This result points to the formation on the KF-treated CIGSe surface of an In 2 Se 3 phase (with a bandgap of 2.0 eV), [START_REF] Kodalle | Properties of Co-Evaporated RbInSe 2 Thin Films[END_REF] which is in agreement with previous studies that have reported presence of K-In-Se and In 2 Se 3 :K surface phases. 25,[START_REF] Lepetit | Post Deposition Treatment in Co-Evaporated Cu(In,Ga)Se 2 Thin Film Solar Cells: Beneficial or Detrimental Effect Induced by the Absorber Characteristics[END_REF] The surface E i = 5.50 eV of the RbF-PDT CIGSe reference is close to 5.41 eV of the no-PDT reference sample with the Cu-poor surface composition. Moreover, after the CBD surface etching stage the ionization energy of the RbF-PDT and no-PDT samples, 5.

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENT

The authors gratefully acknowledge S. Levcenco for conduction of independent capacitancevoltage measurements on reference samples.