NREL is a national laboratory of the U.S. Department of Energy, Office of Energy 

Efficiency & Renewable Energy, operated by the Alliance for Sustainable Energy, LLC.

Contract No. DE-AC36-08GO28308

Exploring High-Dimensional 

Data Space: Identifying Optimal 

Process Conditions in 

Photovoltaics

Preprint

Changwon Suh, Stephen Glynn, John Scharf, 

Miguel A. Contreras, Rommel Noufi, and 

Wesley B. Jones 

National Renewable Energy Laboratory

David Biagioni 

University of Colorado, Boulder

Presented at the 37th IEEE Photovoltaic Specialists Conference 

(PVSC 37)

Seattle, Washington

June 19-24, 2011

Conference Paper

NREL/CP-2C00-50693

July 2011
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1 

EXPLORING HIGH-DIMENSIONAL DATA SPACE: IDENTIFYING OPTIMAL PROCESS 

CONDITIONS IN PHOTOVOLTAICS

Changwon Suh1, David Biagioni2, Stephen Glynn1, John Scharf1, 

Miguel A. Contreras1

, Rommel Noufi1, and Wesley B. Jones1

1

National Renewable Energy Laboratory, 1617 Cole Blvd., Golden, CO 80401, United States 2

Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, United States

ABSTRACT

We demonstrate how advanced exploratory data analysis 

coupled to data-mining techniques can be used to 

scrutinize the high-dimensional data space of 

photovoltaics in the context of thin films of Al-doped ZnO 

(AZO), which are essential materials as a transparent 

conducting oxide (TCO) layer in CuInxGa1-xSe2 (CIGS) 

solar cells. AZO data space, wherein each sample is 

synthesized from a different process history and assessed 

with various characterizations, is transformed, 

reorganized, and visualized in order to extract optimal 

process conditions. The data-analysis methods used 

include parallel coordinates, diffusion maps, and 

hierarchical agglomerative clustering algorithms combined 

with diffusion map embedding.

INTRODUCTION

Recent advances in wide variety of synthetic routes and 

characterizations in photovoltaics (PV) have created a 

critical need to understand complex synthesis/growthmechanism relationships in terms of process-propertyperformance relationships (PPPRs.) Inherent in this 

process is dealing with the high-dimensional (i.e., having 

myriad variables) PV data space, which creates greater 

challenges for visualizing and exploring such relationships.

In TCO development, for instance, process parameters in 

sputtering processes include multiple film growth 

conditions such as target composition, substrate 

temperature, and atmosphere of the chamber. 

Characterizing TCOs as a baseline mainly involves 

various structural, transport, and optical parameters 

termed as primary descriptors, which often comprise 

thickness of structure, Hall measurement for transport 

mechanisms, and analysis such as refractive 

index/extinction coefficient for the materials’ optical 

responses. Further advanced analysis for a more physical 

interpretation can be performed with other metrology tools 

such as photoluminescence.

On the other hand, it is common that the information from 

the above-mentioned characterizations performed 

independently is not sufficient to fully understand growth 

mechanisms in terms of synthesis history. For this reason, 

it is often necessary to include non-measurable 

characteristics such as structural disorder or figures of 

merit, which are termed secondary descriptors. Here, the 

secondary descriptors are combinations achieved through 

mathematical and theoretical transformations of the 

primary descriptors.

Therefore, it is highly desirable to develop tools for highdimensional data explorations, merging process 

conditions, various characterization data, and properties to 

gain a complete insight into physical phenomena such as 

growth mechanisms of complex PV materials. Augmenting 

this type of knowledge could inform a functional 

understanding of PV materials in terms of targeted 

properties, optimize process controls for desired 

performance, and contribute to shifting the current PV 

manufacturing industry to implementing better low-cost 

materials and processes.

In this paper, we demonstrate approaches to exploring PV 

data in a more meaningful way for better PV design. This 

is done in the context of a case study on the data of AZO 

films generated from different process histories involving 

sputtering methods.

EXPERIMENTAL

An example of the challenges faced in PV research is the 

development of the CIGS solar cell. The immediate goal of 

CIGS device designers is to achieve bench-scale results 

(~20% efficiency), but increasing the efficiency of CIGS is 

difficult based on the complexity of its multiple layers. 

Therefore, it is critical to use components of the PPPRs for 

each layer—from the molybdenum back contact to the 

AZO transparent front contact—and fuse them together for 

an entire device. An AZO layer with its superior optical and 

electric properties is a key component as the TCO layer 

for CIGS. The multiple requirements for good AZO, based 

on a 20%-efficient CIGS device, are: (i) carrier density of 

5.6×1020 cm-3, (ii) mobility of 13 cm2

/V⋅sec, and (iii) 

thickness 1200 Å [1]. 

Syntheses of AZO Layer 

AZO thin films were synthesized onto Corning Eagle 2000 

glass with various RF sputtering conditions. From the 

syntheses, we chose one isolated set of AZO data, which 

consists of 40 samples. The selected samples do not 

represent best properties for reference AZO. Rather, they 

possess the wide variety of TCO properties that needs to 

be enhanced via fine-tuning of process conditions. For 

baseline measurements, each film was analyzed with a 

four-point probe and Hall measurement for electric 

properties and n&k analysis for optical properties.
2 

High-Dimensional AZO Data Generation

With an eye toward meeting above-mentioned multifunctional needs of the AZO layer, numerous studies have 

examined the effects of process conditions on the 

properties of thin films (e.g., [2, 3].) However, most 

attempts describe correlations between only two or three 

variables to circumvent the difficulty of simultaneously 

exploring multiple properties of AZO materials.

Our current research on TCO layers in CIGS devices aims 

at practically optimizing the properties and then finding 

standard recipes for synthesizing the layer by tuning 

process variables to achieve the final desired optical and 

electrical requirements simultaneously. Therefore, it is 

valuable to evaluate AZO qualities with a wide variety of 

characterizations as well as secondary descriptors 

generated from miscellaneous physical theories. For 

example, it is typical to investigate defects, structural 

disorder, or optical energy gap with the absorption edge of 

distinct regions based on the single oscillator model [4]. 

Having baseline film properties, we chose and compiled 

various secondary descriptors of TCOs (Table 1). The 

details of secondary descriptors are not described here, 

but the reader is referred to numerous texts in the field as 

shown in Table 1. High-dimensional AZO data incorporate 

various theoretical and phenomenological secondary 

descriptors into primary descriptors and process 

conditions to delineate characteristics of engineered TCO 

materials in minute detail.

APPROACH

High-dimensional AZO data are explored with various 

exploratory data analysis tools to uncover PPPRs between 

multiple variables in Table 1. Assuming that variables in 

the AZO data are intimately correlated, the exploration 

used to extract meaningful information is threefold:

• Direct visualization without data treatment

• Non-linear dimensionality reduction

• Data reorganization for revealing hierarchical data 

structure.

Direct Data Visualization Using Parallel Coordinates

The AZO data space can be explored using parallel 

coordinates that visualize high-dimensional information 

directly into two-dimensional space [5]. The advantage of

using a simple yet powerful algorithm is that it extracts 

hidden correlations without transforming the original 

multivariate data set, and therefore leads to uncomplicated 

data interpretation. It maps an N-dimensional data space 

onto the two-dimensional space by drawing N parallel 

axes. The abscissa in parallel coordinates represents 

each variable, whereas the ordinate represents different 

levels of variables (e.g., ten different levels in this study).

Factor Description

Process 

Condition

Base pressure (PBase), forward power of 

target (WTarget), process pressure 

(PProcess), heater temperature (TEHeater), 

Ar flow rate (FLAr), 2% O2/Ar flow rate 

(FL2% O2/Ar), scan speed (νScan), number 

of pass (NOPass), and crystal rate 

(XtalRate) 

Primary 

Descriptor

Thickness of film (t), sheet resistance by 

4-point probe (Rs-4pp), sheet resistance 

by Hall measurement (Rs-Hall), electrical 

resistivity by 4-point probe (ρ4pp), 

electrical resistivity by Hall measurement 

(ρHall), mobility (μ), free carrier 

concentration (ne), and optical responses

as spectral data (i.e. transmittance, 

reflectance, refractive index, and 

extinction coefficient) 

Secondary 

Descriptor

Electronic mean free path (l1 [6] and l2

[7]), Urbach tail energy (E0), dispersion 

energy (Ed), single oscillator energy 

(EOsc), long wavelength refractive index 

(n∞), n∞ squared (ε∞), average oscillator 

wavelength (λ0), average oscillator 

strength (S0), optical band gap energy 

from Tauc’s plot (Eg(Tauc)), refractive 

index at 500 nm (n500nm), extinction 

coefficient at 500 nm (k500nm), real 

dielectric constant at 500 nm (ε1-500nm), 

imaginary part of dielectric constant at 

500 nm (ε2-500nm), position of the maxima 

of the first derivative of the transmission 

((dT/dλ)Max-eV), absorption coefficient at 

(dT/dλ)Max-eV (α(dT/dλ)Max-eV), overall 

transmittance between 400 and 900 nm 

(TAvg. (400-900nm)), degree of inhomogeneity 

of refractive index ((∆n/ñ)%), measure of 

TCO performance at 500 nm ((σ/α)500nm), 

and figure of merit of TCO at 550 nm 

(φTC-550nm) [8-13]

Table 1. High-dimensional data space of AZO films.

The parallel-coordinates system is particularly suitable for 

mapping AZO data. First, it enables one to discriminate 

among samples having similar process conditions in the 

space of nine process variables, because the AZO data 

set contains many samples repeatedly synthesized with 

process conditions that are quite similar, but not identical. 

Second, the point of view to explore the data set is 

interchangeable among processing conditions, properties, 

and performance. For instance, the effects of processing 

conditions on properties of AZO films are shown in Fig. 1, 

in which it is also possible to visualize the effects of 

properties such as ρHall on processing conditions. This 

approach of changing viewpoints is often far more 

effective than other approaches for choosing targeted 
3 

levels of specific properties to scrutinize the control of 

processing conditions in inverse engineering.

Diffusion Maps for Dimensionality Reduction

Because of the limitations of human perception, lowdimensional representation of high-dimensional data is of 

great importance to science, including the PV area. For 

instance, principal component analysis (PCA) is a global 

method of characterizing covariance in a high-dimensional 

data set to find the low-dimensional space that delineates 

high-dimensional data with minimal loss of information. It 

uses linear projections that minimize the least-squares 

error for a given number of components by rotating the 

original coordinate system for high-dimensional data so as 

to maximize the variance in the data, while minimizing the 

loss of information. However, the dependence of PCA on 

linear projections is not suitable for all data sets. In 

particular, if the data points lie on a nonlinear surface, 

linear projections may not lead to proper dimensionality 

compression because the projection planes may not 

coincide with the surface of interest. Linear projectionbased analysis is often inadequate for TCO data, because 

the PV data may be interrelated nonlinearly.

A recently developed alternative to PCA is diffusion map 

embedding [14,15]. Unlike PCA, which projects the data 

onto linear subspaces, diffusion maps construct a graph 

(or network) on the data in which features are connected 

by weighted edges. A common choice of weight function is 

the Gaussian or heat kernel,

wij = exp{−ρ(xi

, xj)

2

ε

} (1) 

where ρ is a function measuring the distance between 

features xi and xj, and ε is a non-negative parameter. 

Because the Gaussian kernel decays quickly, the resulting 

graph places a high emphasis on local distances and can 

therefore lead to discovery of nonlinear relationships 

between the features. Computationally, the weight matrix 

W is row normalized, yielding a row-stochastic Markov 

matrix 

P = D−1W (2) 

where D is diagonal with elements 

Dii = wij ∑j (3) 

The eigenvalues (λk) and eigenvectors (φk) of P are then 

computed as a coordinate system of the original data set, 

and the data are embedded into a low-dimensional 

subspace corresponding to the first, say r, eigenvectors 

that are considered efficient data representatives. The 

eigenbasis corresponds to modes of a random walk (or 

diffusion) on the data, so points that are nearby on the 

graph will also be close in the embedded space. Like 

PCA, this procedure can lead to drastic dimension 

reduction when the number of eigenvectors needed for the 

embedding is small. It differs, however, in the sense that 

local, and potentially nonlinear, relationships between 

features may become apparent in the embedding that 

would be missed if only linear projections were used.

Diffusion maps are commonly used for clustering: a typical 

algorithm embeds the data into a low-dimensional 

subspace, and then applies a standard clustering 

algorithm to the embedded data.

Hierarchical Clustering with Diffusion Maps

It is often the case that scientific data are obtained in such 

a way that no clear ordering of either the variables or the 

samples is apparent at first glance. In AZO design, for 

example, it is far more useful to have a clearer 

organization of the data matrix (and hence visualization of 

the data) in order of “similarity,” which elucidates 

relationships between both the processing conditions and 

properties via a hierarchical tree structure known as a 

dendrogram. To this end, cluster analysis (CA) can be 

applied to explore similarities of the data space and 

partition them into a number of clusters where 

observations or variables in the same cluster are “similar,” 

whereas dissimilar samples/variables are in different 

clusters.

Among CA approaches, hierarchical clustering algorithms 

provide a simple and systematic way to organize and 

visualize discrete data. In this study, we consider only 

bottom-up algorithms in which smaller clusters are 

successively combined in larger ones. Given a data matrix 

with m samples or observations (rows) and n features, 

characteristics, or responses (columns), the goal of 

agglomerative algorithms is to iteratively merge clusters 

according to their similarity. The way in which this is 

achieved depends fundamentally on the choice of:

• A non-negative distance or dissimilarity function 

between two clusters A and B, denoted by ρ(A,B) 

• A linkage rule for recomputing the distance between 

old and new clusters.

In particular, given the distance function ρ(A,B), we first 

compute the pairwise distances between all of the 

features. Next, the minimal pairwise distance is identified 

and recorded, the corresponding features are merged, and 

the distance between this new cluster and the remaining 

clusters is computed according to the linkage rule 

selected. Three common linkage rules are single, 

complete, and average [16]. After (n-1) steps, all features 

are contained in a single cluster, and the algorithm stops.

Diffusion mapping can be incorporated for more clearly 

understanding cluster structures by placing emphasis on 

local distances between points in the feature space. In 

particular, given a distance function ρ(A,B), the similarity 

between clusters is described by weights in a data-derived 

network as shown in Eq. 1. This formalism is easily 
4 

adapted to the hierarchical clustering schemes described 

above. In particular, given a probability of transition 

between A and B (i.e., element P(A,B) of the Markov 

matrix), we define a new distance function by, for 

example, 

ρ(A,B) = 1− PAB (4) 

where 0≤ρ≤1 because 0≤P≤1. The advantage of this 

approach compared to CA is its emphasis on local 

distances, which intuitively may be a more natural setting 

to search for meaningful clusters. Given the diffusion map 

distance defined above, any of the linkage rules can be 

applied.

RESULTS AND DISCUSSION

Parallel Coordinates

As shown in Fig.1, associations of processing conditions 

and TCO responses are visualized by the brushing 

technique in parallel coordinates. In this way, once specific 

levels of a variable are selected or brushed with color 

(e.g., red), it automatically provides all the levels for other 

variables, which allows one to quickly monitor qualities of 

films in terms of the processing conditions. Note that blue 

polygons in Fig. 1 represent the reference AZO film for a 

20%-efficient CIGS device [1], and the discontinuous 

polygons imply the missing values of corresponding 

variables. Prior to applying an algorithm for parallel 

coordinates, each variable of the compiled AZO data is 

scaled from maximum and minimum between 0 and 1.

In Fig. 1 (a), the selected low levels of WTarget

automatically provide the corresponding process 

conditions performed, such as low PProcess and high 

TEHeater. For the given conditions of low values of WTarget

and PProcess as in Fig. 1 (a), the increase in t and μ with 

decreasing resistivities is noticeable and therefore results 

in increased performance, (σ/α)500nm and φTC-550nm, of the 

AZO films. Under the reverse conditions, Fig. 1(b) shows 

that the behaviors of t, ρ4pp, ρHall, and μ are somewhat 

opposite to the identified results in Fig. 1(a). In addition, 

the values of E0 in Fig. 1(b) show that films synthesized 

with high values of WTarget and PProcess have more 

structural disorder [17]. Moreover, it is obvious that long l1

and l2 in Fig. 1 (a) implies good-quality films, which 

corresponds to the decrease in values of resistivities [18], 

because the physical interpretation of l1 and l2 is typically 

made in terms of scattering mechanisms between grains 

and grain size that is correlated with t [19]. 

According to the results of Das et al. [20], low WTarget gives 

rise to slow deposition rates and more stoichiometric films, 

which lead to more resistive films. The same effects are 

provided at a high PProcess, which results from increased 

scattering between particles and a strong decrease in μ 

[21]. Likewise, it is known that the increase in WTarget as 

the PProcess decreases makes the deposition rate fast 

enough, non-stoichiometric, and less resistive [20]. 

Taken overall, it seems that the effect of lowering PProcess

is more significant than controlling WTarget to enhance 

conductivity as well as performance of AZO films. Under 

the condition of low WTarget and low PProcess with relatively 

low deposition rates (νScan and XtalRate) as in Fig. 1(a), 

therefore, the increase in t is attributed to relatively low 

PProcess. Although we do not show various examples of 

parallel coordinates as a way of finding PPPRs, we could 

identify various interesting behaviors of film properties and 

the effects of process conditions with parallel 

coordinates—behaviors and effects that are almost 

impossible to identify using a traditional bivariate 

approach.

(a)

(b)

Figure 1. Parallel coordinates representing the effects 

of (a) low regions and (b) high values of WTarget and 

PProcess on properties of AZO films.

Diffusion Maps

In diffusion mapping, variables or samples are visualized 

in low-dimensional diffusion space. As an example, 

grouping of variables is shown in Fig. 2. Processing 

conditions such as TEHeater, FLAr, NOPass, and PBase sit in 

the middle of two groups, upper left and lower right. The μ, 

l1, and l2 are grouped with AZO performance ((σ/α)500nm

and φTC-550nm) in the group at the upper left. We can see 

that processing conditions such as PProcess, WTarget, and 

νScan are grouped with film resistivities that are related 

negatively to performance of TCO materials. It is worth to 

compare Fig. 1 and Fig. 2 for thorough study.

Hierarchical Clustering

Hierarchical agglomerative clustering (HAC) algorithms 

combined with diffusion map embedding were performed 

in order to reorganize and transform the TCO data space 

based on similarities between variables/samples within 
5 

high-dimensional space. To elucidate linear/nonlinear 

process-property relationships of TCOs, the adapted HAC 

algorithm in this study is a much more robust and 

systematic data analysis technique than the conventional 

HAC approach. The reason is that the adapted algorithm 

emphasizes local distance between points in the highdimensional variable space and captures nonlinearities in 

the data via diffusion map embedding [14, 15]. The results 

are summarized with heat-map representation to identify 

interdependencies between all variables with each role of 

the variables (Fig. 3). Figure 3(a) is a heat-map 

representation of original AZO data with no clustering 

applied. The heat map here can be considered as a colorcoded data table for representing raw data. In heat maps, 

an intersection of a pair of rows and columns creates a 

grid associated with a pair of sample and property.

Figure 2. Low-dimensional embedding of AZO 

variables using the top three nontrivial eigenfunctions 

in diffusion space. Four processing conditions 

shaded with a blue ellipse form a boundary that 

separates the two groups.

Figure 3(a) provides raw AZO data, but they need to be 

organized in a more meaningful way. Figure 3(b) is a heat 

map after the order of the AZO samples and variables 

determined in terms of dissimilarity measure by 

hierarchical cluster analysis. With this data visualization, 

the domain scientist can quickly identify critical regions 

and interdependencies between AZO films or variables by 

combining heat-map representation with dendrograms 

(Fig. 3 (c)) from cluster analysis. We chose the average 

linkage rule and ρ(xi

, xj) = 2 − 2corr(xi, xj) as our 

dissimilarity function with ε=4. The color of each rectangle 

corresponds to a scaled value in the AZO films.

CONCLUSIONS AND FUTURE WORK

We have demonstrated how the merger of highdimensional visualization with complex PV data sets such 

as AZO data provides a significant “value added” level to 

simultaneously find hidden PPPRs for optimizing the AZO 

layer and enhancing process controls.

(a)

(b)

(c)

Figure 3. (a) Heat map of the original AZO data without 

CA; (b and c) heat map and dendrogram of the AZO 

data arranged by HAC according to the dissimilarity 

measure from the diffusion map, respectively. 
6 

Based on the identified correlations, we can expand our 

approaches to linear/nonlinear regression techniques, 

leading to explicit optimization of the desired features.

In addition, we will expand our discussion to the concept 

of virtual metrology as unique “information probes” for 

exploring characterization data sets across different length 

scales by exposing relationships between diverse types of 

data. This is data exploitation to correlate, predict, and 

ultimately control measurable parameters such as 

thickness, as well as non-measurable properties such as 

structural disorder for better process controls toward 

achieving high-efficiency solar devices.

ACKNOWLEDGEMENT

We would like to thank Kwiseon Kim for critical comments.

This work was supported by the U.S. Department of 

Energy under Contract No. DE-AC36-08-GO28308 with 

the National Renewable Energy Laboratory.

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