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Thin Solid Films 511–5

Band-gap engineering in CIGS solar cells using Nelder–Mead simplex

optimization algorithm

G. Cernivec *, J. Krc, F. Smole, M. Topic

Faculty of Electrical Engineering, University of Ljubljana, Trzaxka 25, 1000 Ljubljana, Slovenia

Available online 18 January 2006

Abstract

Band-gap grading in a CIGS absorber and a conduction band offset at n/p hetero-interface are two important parameters of band-gap

engineering aiming at high efficient CIGS solar cells. To obtain optimal CIGS absorber’s band-gap grading profile an automatic optimization loop

based on Nelder–Mead simplex optimization algorithm has been implemented. The optimization problem is described with an objective function,

which—by varying the input parameters—is minimized or maximized. In our study two types of objective functions are used; optical and

electrical. As the most optimal profile a parabolic, double graded band-gap profile with a positive or nearly zero conduction band offset at n/p

hetero-interface is calculated. Structures with different CIGS absorber thicknesses and bulk and/or hetero-interface recombination lifetimes are

examined and their optimized parameters are discussed in the light of experimental achievements.

D 2005 Elsevier B.V. All rights reserved.

Keywords: CIGS solar cell; Graded absorber; Window/absorber hetero-interface; Optimization algorithm

1. Introduction

Minority carrier lifetime of a CIGS absorber, quality of an n-

type window layer/p-type absorber layer hetero-interface and

the CIGS absorber’s band-gap profile are parameters that have

major influence on the performance of the CIGS solar cell.

Both, analytical [1] and numerical [2] approaches are used to

analyze the influence of the CIGS absorber and hetero-interface

as limiting factors of the conversion efficiency. The band-gap

profiles of the CIGS absorber can be realized with the spatial

variation of gallium (Ga) in the co-evaporation process [3]. In

band-gap engineering [4,5] main focus is on the effect of the

CIGS grading (normal, reverse, double), while keeping the

absorber’s and hetero-interface’s quality at best known

experimentally determined values.

In our work we investigate the effect of the minority carrier

recombination lifetime in the CIGS absorber and the hetero-

interface recombination velocity on the optimal band-gap

engineered CIGS absorber under standard test conditions—

STC (AM1.5, 1000 W/m2, 25 -C). Local fluctuations in

polycrystalline CIGS absorber [1] affect the transport proper-

ties. In our numerical modeling approach band-gap and

0040-6090/$ - see front matter D 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.tsf.2005.11.095

* Corresponding author. Tel.: +386 1 4768 848; fax: +386 1 4264 630.

E-mail address: gregor.cernivec@fe.uni-lj.si (G. Cernivec).

electrostatic fluctuations eventually result in increased recom-

bination, which can be modeled as decreased bulk and/or

hetero-interface recombination lifetime.

We developed a numerical procedure for the automatic

optimization, which is explained in Section 2. Several

objective functions, needed in the automatic optimization

process were implemented to optimize the band-gap profile of

Cu(In1�xGax)Se2 absorber. Firstly, the minimization of the

thermalization power loss and the heat power loss is

investigated in graded band-gap profile optimization process,

in order to obtain solar cell with the overall maximum open-

circuit voltage and/or maximum output power. Secondly, the

CIGS absorbers with non-graded or graded band-gap profiles

of different thicknesses are optimized for the overall

maximum output power and conversion efficiency limitations

are discussed.

2. Optimization procedure

To describe an optimization problem an objective function,

i.e. functional, has to be constructed. It takes a vector input and

returns a scalar value. A Nelder–Mead simplex optimization

algorithm [6] is used in order to optimize the objective

function. The simplex algorithm does not need a derivative;

only a numerical evaluation of the objective function is

12 (2006) 60 – 65

ww

x1

x2

F1

F2

F3

Fnew

F2 > F1, F3

Fig. 1. Geometrical interpretation of Nelder–Mead optimization algorithm in

two parameter optimization space. F1, F2, and F3 are functional evaluations of

original simplex. F1, F3 and Fnew are functional evaluations of new simplex.

New simplex and old simplex are generally non-symmetrical.

G. Cernivec et al. / Thin Solid Films 511–512 (2006) 60–65 61

required. The basis for the simplex algorithm comes from

geometry as shown in Fig. 1. In two dimensional parameter

space simplex is a triangle, determined by three points

(vertices) and their interconnecting line segments [7]. At every

point the objective function is evaluated. The point with the

highest numerical value of all three points is perpendicularly

mirrored against the opposite line segment. This is called a

reflection. The reflection can be accompanied with an

expansion to take larger steps or with a contraction to shrink

the simplex where an optimization valley floor is reached. The

optimization procedure continues until the termination criteria

are met. The termination criterion is usually the maximum

number of reflections with contractions or the volume (area) of

the simplex. The algorithm is generally implemented in N

dimensions, where simplex is a hypercube with N +1 vertex

points.

The automatic optimization loop consists of the Nelder–

Mead optimization algorithm, implemented in Mathematica 5.1

[8] and the objective functions. The objective function is

optimized by varying the input parameters. In order to

converge to the physically reasonable optimal solutions of

the parameter space, the objective function is weighted by the

penalty function. If the objective function or the input

parameters exhibit non-physical values, the penalty function

is set to the value, which turns the optimization path in the

opposite direction.

In the optimization process the input parameters are the

coefficients of the 2nd order polynomial, which describes the

band-gap profile of the CIGS absorber and the conduction

band offset (DEcn/p) at the n/p hetero-interface, according to

Eq. (1):

Eg xV;Eg0; k1; k2� �

¼ qI Eg0 þ k1IxV

Wabs

þ k2IxV

Wabs

� �2" #

: ð1Þ

Variable xV is a spatial variable, relative to the n/p hetero-

interface position, Eg0, k1, k2 are the constant, linear and

quadratic polynomial coefficient, respectively, and Wabs is the

absorber’s thickness. The band-gap grading parameters k1and k2 determine the shape of the band-gap profile, while

parameter Eg0 determines the band-gap at the n/p hetero-

interface (xV=0) and the DEcn/p; since the band-gap is

changed on the account of the electron affinity (Dvn/p=�DEc

n/p) [9]. The values of the band-gap and the electron

affinity of the n-type window layer are kept fixed in our

study.

Optical simulator SunShine [10] is used to calculate

optical objective functions. Two optical objective functions

are defined; the absorbed power and the thermalization

power loss:

Pabs ¼Xk

Z Wcell

0

GL x; kð ÞI hIck

dx; ð2Þ

Pth ¼Xk

Z Wcell

0

GL x; kð ÞI hIc

k� Eg xð Þ

� �dx; ð3Þ

where h is the Planck’s constant, c is the speed of light, Wcell

is the solar cell’s thickness, Eg is spatially dependent solar

cell’s band-gap profile and GL is wavelength dependent

generation rate profile. The simulator uses one dimensional

optical model for multilayer structures with flat and/or rough

interfaces. Optical properties of the layers are defined with

wavelength dependent refractive index—n(k) and extinction

coefficient—k(k). In our optimizations both parameters are

determined based on the experimental results for the different

Cu(In1�xGax)Se2 alloys [11]. The extinction coefficients of

the sub-layers are modified during the optimization process

by shifting their energy (wavelength) axis, according to the

differences in the band-gaps [12]. The refractive index is

kept unchanged. Grading of optical band-gap is accomplished

by dividing the absorber into several discrete sub-layers (20

to 30), with constant but different optical band-gaps,

determined from the band-gap profile. In the optimization

procedure the AM1.5g spectrum is used.

Semiconductor simulator ASPIN [4] is used to calculate the

electrical objective functions. From the drift-diffusion semi-

conductor model [13] the following objective functions are

defined; the transport power (Ptsp) [14], the heat power loss

(Pq) [15] and the recombination power loss (Prec) [16] as

given by Eqs. (4)–(6):

Ptsp ¼1

q

Z Wcell

0

lEfn xð ÞJn xð Þdxþ 1

q

Z Wcell

0

lEfp xð ÞJp xð Þdx;

ð4Þ

where Efn(x), Efp(x) are the electron and hole quasi-Fermi

energies, respectively, Jn(x), Jp(x) are the electron and hole

electrical current densities, respectively, q is the electron charge

and Wcell is the thickness of the solar cell.

Pq ¼Z Wcell

0

GL xð ÞI Eg xð Þ � leh xð Þ� �

dx; ð5Þ

Prec ¼Z Wcell

0

R xð ÞIleh xð Þdx: ð6Þ

The symbols GL(x) and R(x) correspond to the volume

generation and recombination rate profile, respectively, and

JSC [A/m2]230 250 270 290 310 330 350

P [

W/m

2]

80

120

140

160

180

220

600

100

200

500

-Voc

[V]

0.50

0.55

0.60

0.65

0.70

0.75

0.80

Ptsp+PrecPqPthPabs

- Voc

Pout

Fig. 3. Power-balance of the CIGS solar cell as a generator under STC. The

absorbed power ( Pabs), reduced for the thermalization power loss ( Pth) and the

heat power loss ( Pq) produces the electro-chemical power. The electro-

chemical power is in the process of carrier transport and recombination reduced

for the internal transport power ( Ptsp) and the recombination power loss ( Prec),

which results in output power ( Pout). Since the Jsc is targeted with the penalty

function, the band-gap grading profile is optimized in a way to produce the

overall maximum Voc and maximum Pout. The absorber’s thickness is Wabs=

1 Am; sbulk=1 ns; S int =10�2 v th, where v th is thermal velocity.

G. Cernivec et al. / Thin Solid Films 511–512 (2006) 60–6562

leh(x) is a difference of the quasi-Fermi energies, i.e. the

electro-chemical energy per electron-hole pair. The Pout

electrical objective function is defined as maximum power

point on the current– voltage (J–V) characteristic of a solar

cell:

Pout ¼ f J�V Eg0; k1; k2� �� �

: ð7Þ

The quality of the CIGS absorber layer and the hetero-

interface is modeled with the recombination lifetime for the

minority carriers (sbulk) [17] and surface recombination

velocity for the majority carriers (Sint) [18], respectively.

The back contact is modeled with surface recombination

velocity equal for majority and minority carriers (Sback). The

sbulk, Sint, Sback and the effective density of states (Nc, Nv)

are in our study modeled as independent of Eg(xV).

3. Band-gap optimization of the Cu(In1�xGax)Se2 absorber

In our optimization study a ZnO:Al/ZnO/CdS/CIGS/Mo

structure is chosen. The semiconductor properties of the

layers are taken from the literature [19]. No electrostatic

fluctuations [1] are assumed; the conduction and the valence

band are smooth and continuous. In the optimization process

the minority carrier lifetime (sbulk) in the CIGS absorber and

the interface recombination velocity (Sint) at the CdS/CIGS

hetero-interface are set as independent parameters. The sbulkvalues in the range from 1 to 10 ns are assumed as typical for

the CIGS absorber [17], while larger sbulk values are used in

order to explore the efficiency limits and improvement

possibilities. The Sint values in the range from 5 I10�4 vthto 5 I10�2 vth are assumed as physically reasonable [2,18],

where vth is a thermal velocity. Sint below this range is used

with the same purpose as an overestimated sbulk. A constant

recombination velocity (Sback) of 105 m s�1 is assumed at the

back contact. When the CIGS absorber is optimized as non-

graded, the optimization parameter space consists of Eg0,

while k1 and k2 are set to zero. When the optimization is

x [μm]0.4 0.6 0.8 1.0 1.2 1.4

Eg

[eV

]

1.0

1.2

1.4

1.6

1.8

2.0

2.2J sc = 230 A/m2

J sc = 250 A/m2

J sc = 270 A/m2

J sc = 290 A/m2

J sc = 310 A/m2

J sc = 330 A/m2

J sc = 350 A/m2

min(Eg)

CdS/CIGSCIGS/Mo

hetero-interfacebackcontact

X'

Fig. 2. Band-gap profiles which produce minimum sum of the thermalization

( Pth) and heat ( Pq) power losses in the absorption process. The Jsc is targeted

with the penalty function. The absorber’s thickness is Wabs=1 Am; sbulk=1 ns;

S int=10�2 v th, where v th is a thermal velocity.

carried out for a graded absorber, the optimization parameter

space consists of Eg0, k1 and k2.

3.1. Minimization of the thermalization and heat power losses

Two objective functions are used in this optimization

process; optical—the thermalization power loss (Pth) and

electrical—the heat power loss (Pq). Sum of both objective

functions represents a coupled objective function to be

minimized. Since (Pth+Pq) minimum would be obtained at

no absorption and to consider the collection of carriers, a

short-circuit current ( Jsc) is included into the optimization

as the penalty function. A simple penalty function is used,

which abruptly increases the sum of both objective

functions, if the desired Jsc is not reached. Fig. 2 shows

the optimized band-gap profiles for targeted Jsc in 1 Amthick CIGS absorber. In a desire to improve the collection

efficiency all profiles exhibit the band-gap increment

towards the back contact. When increasing the targeted

Jsc, the band-gap lowering can be observed, mainly at the

hetero-interface. All of these profiles produce minimums of

the (Pth+Pq), but only one ( Jsc=290 A/m2) produces

maximum of Pout as can be seen in Fig. 3. With this type

of the optimization local Voc or Pout maximums are found—

determined by the targeted Jsc. In order to find the overall

maximum Voc and/or Pout maximum, range of the attainable

Jsc values has to be searched. For the analyzed solar cell

with Wabs=1 Am, the overall maximum Voc is obtained with

the band-gap grading profile that produces 250 A/m2 of Jsc.

Since the minimization of (Pth+Pq) tries to maximize Eg

and leh, as seen from the Eqs. (3) and (5), this type of

optimization is more appropriate when searching for the

EC

CdS

/CIG

S[e

V]

0.06

0.08

0.10

0.12

0.14

0.16

bulk recombinationback contactrecombination

(a)

(b)

Wabs [μm]

0.0 0.5 1.0 1.5 2.0 2 3.0

P out

[W/m

2]

80

100

120

140

160

180

200

220

τbulk = 200 ns

= 50 ns

τbulk

τbulk

τbulk

= 10 ns

= 1 ns

.5

Δ

Fig. 5. Conduction band offset at the CdS/CIGS hetero-interface (a) and output

power of structures with optimized non-graded CIGS absorber (b). Positive

values of the conduction band offset indicate the ‘‘spike-like’’ offset [2]. Wabs is

the thickness of the CIGS absorber layer; S int =10�2 v th, where v th is thermal

G. Cernivec et al. / Thin Solid Films 511–512 (2006) 60–65 63

band-gap profile which will produce the overall maximum

Voc.

3.2. Maximization of output power

The minimization of the (Pth+Pq) does not consider the

transport power. By maximizing the Pout electrical objective

function the transport power loss is implicitly considered and

band-gap profile with the overall maximum output power is

found.

Following optimizations were done: (1) Optimization of

1 Am thick, non-graded absorber. To obtain an overall

Eg0� (sbulk,, Sint) dependence, the sbulk and Sint were varied

from 1 ns to 50 As and from 10�2 vth to 10�12 vth, respectively.

(2) Optimization of non-graded absorbers of different thick-

nesses with Sint set to 10�2 vth and sbulk varied from 1 to 200

ns. (3) Optimization of 1 and 3 Am thick graded absorbers for

two Sint values; 10�2 vth and 10�4 vth. The sbulk was varied

from 1 to 200 ns.

Contour plot of optimized Eg0 as a function of Sint and

sbulk is shown in Fig. 4a. When lowering sbulk, the

optimized Eg0 increases, reducing the bulk recombination.

On the other hand the increased Sint (Sint >10�6 vth) starts

limiting the Voc [2,18]. The optimized Eg0 is lowered to

increase the absorption and maintain the efficiency. The

lowest output power (Fig. 4b) is determined by finite

minority carrier mobility, while the highest output power is

limited by the absorption properties of the CIGS absorber

[1].

1.55

1.551.55

1.55

1.50

1.501.50

1.50

1.45

1.451.45

1.45

1.601.60

1.60

1.40

1.40

1.40

1.40

1.35

1.35

1.35

1.30

τ bulk

[s]

τ bulk

[s]

10-8

10-7

10-6

10-5(a) Eg0 [eV]

210 210 210

210

210

200 200 200200

200

200

190 190 190

190

190

190

180

180 170

Sint [vth]

10-210-410-610-810-1010-1210-9

10-8

10-7

10-6

10-5 Pout [W/m2](b)

Fig. 4. Optimized Eg0 (a) and Pout contours (b) of 1 Am thick, non-graded CIGS

absorber under STC. Parameter optimization space: Eg0.

velocity. Parameter optimization space: Eg0.

When optimizing non-graded absorbers of different thick-

nesses, the conduction band offset (DEcCdS/CIGS) gives the

information about the CdS/CIGS hetero-interface:

DECdS=CIGSc ¼ ECdS

g � DECdS=CIGSv � Eg0; ð8Þ

where EgCdS is the band-gap of the CdS window layer,

DEvCdS/CIGS is the valence band offset at the CdS/CIGS

hetero-interface and Eg0 is the absorber’s band-gap at the

hetero-interface. DEcCdS/CIGS changes only on the account of

the Eg0, since the valence band edge is not affected by gallium

(Ga) addition [9]. At sbulk>1 ns the DEcCdS/CIGS exhibits the

absorber thickness dependence, shown in Fig. 5a. With

decreasing CIGS absorber’s thickness (Wabs<0.7 Am), the

DEcCdS/CIGS decreases, i.e. optimized Eg0 increases. The

minority carrier (electron) concentration at the back contact

is lowered and the back-contact recombination is reduced [3].

In thick (Wabs>0.7 Am) CIGS absorbers this effect is less

pronounced. Major efficiency improvement is observed when

changing the absorber’s quality to sbulk=10 ns (Fig. 5b). At

higher sbulk the efficiency improvement saturates. The

thickness dependent Pout saturates due the absorption proper-

ties of the CIGS absorber.

Optimized band-gap profiles of 1 Am thick CIGS absorber

exhibit double grading (Fig. 6a). With improved hetero-

interface quality (Sint=10�4 vth) the Voc increase from 60 to

Eg

[eV

]E

g [e

V]

1.2

1.4

1.6

1.8

= 1ns

τbulk

τbulk

τbulk

τbulk

τbulk

= 10ns

= 50ns

= 200ns

τbulk

τbulk

bulk

τbulk

τ> 1μs

CIGS/Mo

x [μm]0.4 0.6 0.8 1.0 1.2 1.4

1.2

1.4

1.6

= 60 mV

= 130 mV

= 140 mV

= 150 mV

< 240 mV

CdS/CIGS

Sint= 10-2 vth

Sint = 10-4 vth

(a)

(b)

x [μm]0.5 1.0 1.5 2.0 2.5 3.0 3.5

V

∇

ocV

∇

ocV

∇

ocV

∇

ocV

∇

oc

V

∇

oc

V

∇

oc

V

∇

oc

V

∇

oc

V

∇

oc

= 110 mV= 125 mV= 115 mV= 123 mV< 150 mV

CdS/CIGS

= 1ns = 10ns= 50ns= 200ns > 1μs

CIGS/Mo

Sint = 10-2 vth

Sint = 10-4 vth

(c)

(d)

Fig. 6. Optimized graded 1 Am thick CIGS absorber with S int=10�2 v th (a), S int=10

�4 v th (b) and optimized gradedWabs=3 Am thick CIGS absorber with S int=10�2

v th (c) and S int =10�4 v th (d), where v th is thermal velocity. Band-gap profiles are optimized for different absorber qualities. Parameter optimization space: Eg0, k1, k2.

G. Cernivec et al. / Thin Solid Films 511–512 (2006) 60–6564

150 mV can be observed (Fig. 6b). 3 Am thick CIGS absorber

with sbulk>1 ns is optimized with double graded band-gap

profiles (Fig. 6c). The sbulk=1 ns band-gap profile is optimized

in single grading in order to minimize the bulk recombination.

The Voc increase due the hetero-interface improvement ranges

from 110 to 120 mV (Fig. 6d). Normal grading, i.e. band-gap

increase towards the back contact, increases the collection

efficiency due to the effective field, which repels the electrons

from the back contact. Reverse grading, i.e. band gap increase

towards the CdS/CIGS hetero-interface, increases the Voc, but

0.0 0.5 1.0 1.5 2.0 2.5 3.0

P n

orm

1.05

1.10

1.15

1.20

1.25τbulk

τbulk

τbulk

τbulk

τbulk

> 1 μs

= 200 ns

= 50 ns

= 10 ns

= 1 ns

Wabs [μm]

Fig. 7. Optimized graded absorber’s Pout normalized with the optimized non-

graded absorber’s Pout for various absorber qualities. Wabs is the thickness of

the CIGS absorber layer; S int=10�2 v th, where v th is thermal velocity.

opposes the electric field in the space charge region (SCR) and

decreases the collection efficiency. In order to minimize bulk

recombination, band-gap profiles of CIGS absorbers with low

sbulk are overall optimized to higher values irrespective to the

Sint.

Optimized band-gap graded profiles improve the efficiency

for all CIGS absorber thicknesses (Fig. 7). Maximum

efficiency improvements, reaching 15% to 23 % relative,

are achieved in 0.5 Am thick band-gap graded CIGS absorber.

3.3. Discussion

From Fig. 4 we can observe that maximal output power

under STC of a CIGS solar cell with 1 Am thick, non-graded

absorber (for given realistic absorption and transport proper-

ties) is 210 W/m2, resulting in 21% conversion efficiency.

With optimal band-gap grading the output power can be

improved up to 19% relatively, as shown in Fig. 7. This

would result in band-gap graded solar cell with 25%

conversion efficiency. The fundamental mechanisms that

limit CIGS solar cell’s efficiency to 25% are the transport

power and the recombination power losses. The sum of both

(Ptsp+Prec) varies under STC from 70 to 90 W/m2— as

plotted in Fig. 3. This loss accounts for a decrease of

theoretical limit of 33% conversion efficiency ranging from

7% to 9% absolutely.

With realistic bulk and hetero-interface recombination

properties (sbulk =10 ns, Sint = 10�2 v th) the conversion

G. Cernivec et al. / Thin Solid Films 511–512 (2006) 60–65 65

efficiencies of non-graded absorber would reach 18% and

with optimizing the grading profile the efficiency could be

improved for additional 13% relatively (Fig. 7) to 20.5%

absolutely.

4. Conclusions

A procedure to automatically optimized absorber layer in

CIGS solar cells has been developed. The Nelder–Mead

simplex optimization algorithm was implemented to examine

and optimize n-type window layer/p-type CIGS absorber layer

hetero-structure. Two optimization criteria were applied; the

minimization of the thermalization power loss and heat power

loss and the maximization of the output power.

When optimizing the CIGS absorber’s band-gap profile with

the electrical function maximization criterion, the overall

maximum output power is obtained, since the transport loss

is implicitly considered in the output power. The absorbers

with sbulk<10 ns are optimized in normal graded band-gap

profiles, while optimized profiles of the absorbers with

sbulk>10 ns exhibit double grading. Optimal Eg0 depends on

the hetero-interface recombination velocity (Sint). The overall

maximum output power for physically reasonable Sint values is

obtained with Eg0 ranging from 1.3 to 1.4 eV. Band-gap

engineering renders higher output power improvements for

structures with thinner CIGS absorber layer.

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