A semi-analytical model for semiconductor solar cells
D. Ding, S. R. Johnson, S.-Q. Yu, S.-N. Wu, and Y.-H. Zhang
Citation: Journal of Applied Physics 110, 123104 (2011); doi: 10.1063/1.3671061
View online: http://dx.doi.org/10.1063/1.3671061
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JOURNAL OF APPLIED PHYSICS 110, 123104 (2011)

A semi-analytical model for semiconductor solar cells
D. Ding, S. R. Johnson, S.-Q. Yu,a) S.-N. Wu,b) and Y.-H. Zhangc)
Center for Photonics Innovation and School of Electrical, Computer, and Energy Engineering,
Arizona State University, Tempe, Arizona 85287, USA

(Received 4 August 2011; accepted 15 November 2011; published online 23 December 2011)
A semi-analytical model is constructed for single- and multi-junction solar cells. This model
incorporates the key performance aspects of practical devices, including nonradiative recombination,
photon recycling within a given junction, spontaneous emission coupling between junctions, and
non-step-like absorptance and emittance with below-bandgap tail absorption. Four typical planar
structures with the combinations of a smooth/textured top surface and an absorbing/reflecting
substrate (or backside surface) are investigated, through which the extracted power and four types
of fundamental loss mechanisms, transmission, thermalization, spatial-relaxation, and recombination
loss are analyzed for both single- and multi-junction solar cells. The below-bandgap tail absorption
increases the short-circuit current but decreases the output and open-circuit voltage. Using a
straightforward formulism this model provides the initial design parameters and the achievable
efficiencies for both single- and multiple-junction solar cells over a wide range of material quality.
The achievable efficiency limits calculated using the best reported materials and AM1.5 G one sun
for GaAs and Si single-junction solar cells are, respectively, 27.4 and 21.1% for semiconductor
slabs with a flat surface and a non-reflecting index-matched absorbing substrate, and 30.8 and
26.4% for semiconductor slabs with a textured surface and an ideal 100% reflecting backside
surface. Two important design rules for both single- and multi-junction solar cells are established:
i) the optimal junction thickness decreases and the optimal bandgap energy increases when
nonradiative recombination increases; and ii) the optimal junction thickness increases and the
C 2011 American Institute of
optimal bandgap energy decreases for higher solar concentrations. V
Physics. [doi:10.1063/1.3671061]

I. INTRODUCTION

Photovoltaic solar cell modeling is often approached in
two different directions: i) numerical simulations based on
drift and diffusion models,1 and ii) analytical analysis based
on detailed balance models.2,3 Although commercial simulation software can predict a reasonable performance for a
given design, complex simulations consume a large amount
of computation time and it is not straightforward to extract
clear device physics from complicated numerical calculations, especially for multi-junction solar cells. Moreover,
most commercial software packages consider both radiative
and non-radiative recombination4–6 however, they typically
do not include some important aspects related to radiative
recombination, such as photon recycling7,8 within a given
junction and spontaneous emission coupling between adjacent junctions.9–12 On the contrary, detailed balance models
are only capable of clarifying the fundamental limitations of
ideal solar cells by neglecting many important mechanisms
that occur in real devices. It is, therefore, necessary to
develop a semi-analytical model that not only takes into

a)

Present address: Department of Electrical Engineering, University of
Arkansas, Fayetteville, Arkansas 72701, USA.
b)
Present address: Philips Lumileds Lighting Company, San Jose, California
95131, USA.
c)
Author to whom correspondence should be addressed. Electronic mail:
yhzhang@asu.edu.
0021-8979/2011/110(12)/123104/17/$30.00

account all of the important properties of materials and
device structures to provide a more thorough understanding
of photovoltaic solar cells, but also enables fast computation
to offer basic guidance for practical device design. The
development of such a model has been carried out and some
of the key results were briefly reported in Ref. 13.
This paper reports the detailed work that examines the
detailed balance theories and extends them in the construction of a semi-analytical model for single- and multijunction solar cells, which explicitly elucidates the impact of
real material and device properties, including nonradiative
recombination, photon recycling, spontaneous emission coupling, and the non-step-like absorptance and emittance of
junctions and materials with absorption tails below the
bandgap. As a result, this model provides: i) an in-depth
analysis of radiative losses, ii) a straightforward formulism
compared to drift and diffusion models, and iii) important
optimal design parameters, such as bandgap energy, junction
thickness, and junction number, over a wide range of material quality and solar concentration.
II. FUNDAMENTAL LOSS MECHANISMS IN
SOLAR CELLS

It is imperative to identify the loss mechanisms in conventional p-n junction solar cells and to establish which of
these limit efficiency. For example, the loss mechanisms for
incident solar photons include reflection, contact shadowing,

110, 123104-1

C 2011 American Institute of Physics
V

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absorption in barrier layers and by free-carriers, and transmission through the junction. Of these losses, reflection can
be significantly reduced using anti-refection coatings14 and
textured surfaces,4,15 contact shadowing can be minimized
or eliminated by using transparent16 or backside contacts,17
and parasitic absorption can be minimized using widebandgap barriers and optimal doping profiles in the junction.
Conversely, transmission losses are inherent to photovoltaic
solar cells and strongly depend on the optical properties of
the junction materials.
The loss mechanisms related to photogenerated carriers
include: i) thermalization of carriers from their initial excited
states to the band edges, ii) spatial relaxation of carriers as
they are swept along the band-edges to the contacts, iii) radiative recombination of electron-hole pairs, iv) ShockleyRead-Hall (SRH), Auger, and surface recombination,6 and v)
series and shunt resistance. Of these losses, surface recombination is structure dependent and can be significantly
reduced using wide-bandgap barriers, the series resistance
can be minimized using well designed junctions and contacts, and shunt resistance can be minimized using high quality single-crystal materials and optimized fabrication
techniques.18 Therefore, without losing generality, this study
focuses on all of the fundamental loss mechanisms and
neglects surface recombination and series and shunt
resistance.
The band edge diagram of the single-junction heterostructure solar cell shown in Fig. 1 schematically illustrates
the key loss mechanisms. The solar radiation transmitted
through the junction at energies below the optical bandgap is
the major component of the transmission loss. The solar radiation absorbed in the junction generates excess carriers that
thermally relax to the band-edges (almost instantaneously) in
a loss process called thermalization. As the carriers are swept
to the junction contacts, they lose potential energy as they
move along the band edge in a loss mechanism referred to as
spatial relaxation in this work. Before the electrons and holes
are extracted from the junction contacts, some are lost to
radiative and nonradiative recombination.
To clearly demonstrate the impact of these loss mechanisms on the performance of single-junction solar cells, two
characteristic current-voltage curves are shown in Fig. 2;
Fig. 2(a) is the ideal case where only radiative recombination
is present, and Fig. 2(b) is the non-ideal case where both

FIG. 1. Schematic band-edge diagram showing the key loss mechanisms for
a typical solar cell.

J. Appl. Phys. 110, 123104 (2011)

FIG. 2. Current vs voltage graph for (a) an ideal GaAs solar cell without
nonradiative recombination losses, and (b) a non-ideal GaAs solar cell with
nonradiative recombination losses. The area of the various shaded regions
represents the power extracted and lost. Also shown are the short-circuit current, Jsc , operating current, Jm , average absorbed solar photon energy, h,
bandgap plus thermal energy, Eg þ kT, open-circuit voltage, Voc , and operating voltage, Vm .

radiative and nonradiative recombination are present. Here,
the material properties of GaAs and the AM1.5 G solar spectrum19 are used. The equations utilized in the calculation are
presented in the following sections. A relatively large nonradiative recombination rate (representing poor quality material) is used in Fig. 2(b) to clearly illustrate the differences
between the two cases. Furthermore, the amount of incident
solar power that is extracted and lost is quantified by the
areas of the current-voltage rectangles that are labeled
accordingly. This diagram shows both the relative contribution of each loss mechanism and the relative interplay
between the loss mechanisms. Although the power lost
directly to carrier recombination is not significantly
increased, the impact on spatial relaxation is substantial; this
is a result of the large internal electrical field required to rapidly sweep carriers out of the junction in order to limit the
otherwise large and ever-present carrier recombination
losses. Note that the trade-off between recombination and

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J. Appl. Phys. 110, 123104 (2011)

spatial-relaxation losses also occurs during the presence of
other loss mechanisms such as surface recombination and series and shunt resistance.
Also shown in Fig. 2 are the short-circuit current, Jsc ,
the open-circuit voltage, Voc , the current, Jm , and voltage,
Vm , at maximum power output,1,6,18 the average energy of
the absorbed solar photons, hv, the average energy separation
of the thermalized electron and hole populations ðEg þ kTÞ,
and the electron charge, q. The recombination current loss is
ðJsc  Jm ÞðEg þ kTÞ=q, the spatial-relaxation loss is
Jm ðEg þ kT  qVm Þ=q, and the thermalization loss is
Jsc ½h  ðEg þ kTÞ=q, which is the same for both cases in
Fig. 2,
 and the total amount of solar power absorbed is
Jsc h q, assuming 100% internal quantum efficiency under
the short-circuit condition.18 These examples are considered
under the optimal working conditions; however, the principle
previously discussed and shown in the figures can also be
applied to other operation points.
III. A SEMI-ANALYTICAL MODEL FOR SOLAR CELLS
A. Assumptions and basic formula

As pointed out by Henry,3 in addition to nonradiative
recombination, detailed balance models ignore many of the
extrinsic losses in practical solar cells such as reflection, contact shadowing, series and shunt resistance, incomplete collection of photogenerated carriers, absorption in window
layers, and a rise in junction temperature. Other assumptions
commonly made that are not explicitly stated are absorbing
substrates, step-like absorbance and emittance (i.e., absorbing layers are assumed to be opaque at energies above the
bandgap and transparent at energies below the bandgap), the
Boltzmann approximation for photon and carrier statistics,
constant refractive index, and unity internal quantum efficiency.18 These assumptions are carefully examined and
clarified in this work.
The current density of a single-junction solar cell is,


(1)
J ¼ Jsc  Jrad þ JSRH þ JAuger ;
where J is the total current density, Jsc is the short-circuit
current density, Jrad is the radiative recombination current
density, JSRH is the SRH recombination current density, and
JAuger is the Auger recombination current density. Equation
(1) is the basic formula for the model and the analysis of
each term is provided in the following sections.
B. Short-circuit current density

The short-circuit current density is the rate at which
photogenerated carriers are collected under the short-circuit
condition. Taking into account the absorption within the
width of the Urbach tail,20 the short-circuit current density
per unit area is,
ð1
ð1
Aa gi nsun  dh ¼ qAa
nsun  dh; (2)
Jsc ¼ q
0

Eg Eu

where q is the electron charge, Aa is the energy-dependent
absorptance of the junction, gi is the energy-dependent inter-

nal quantum efficiency (i.e., capture efficiency),16 Eg is
the bandgap energy, Eu is the width of the Urbach tail, and
nsun is the solar photon flux density per unit area. For simplicity, gi ¼ 1 is assumed, which is valid since, typically, a
very large fraction of the photogenerated carriers are collected under the short-circuit condition. The Eg  Eu term
represents an effective bandgap cut-off that is Eu smaller
than Eg when the influence of localized tail states is
considered.
The absorptance term is moved outside the integral in
Eq. (2) by defining the energy independent effective absorptance as,
Ð1
Aa nsun  dh
Aa  Ð 01
:
(3)
Eg Eu nsun  dh
Here, the effective absorptance represents a convenient measure of the “optical thickness” of a junction.
The concept of below-bandgap absorption with a characteristic width can be extended to include other extrinsic
mechanisms such as impurity bands, quantum wells, quantum wires, and quantum dots which add localized energy
states below the bandgap. The details can be slightly different but the basic principle remains the same in that the effective bandgap shifts to a lower energy as the width of the tail
increases which results in an increase in the short-circuit current and a reduction in the open-circuit voltage.
C. Radiative recombination current density

Radiative recombination is one of the intrinsic loss
mechanisms in solar cells. Photons generated through radiative recombination typically undergo many absorption/emission cycles before escaping. This process is commonly
referred as photon recycling7,8 and is not negligible in
optically-thick semiconductor devices such as solar cells.
Taking photon recycling into account, the radiative recombination current density is,


Jrad ¼ qd ð1  cr ÞRsp ¼ qd ce;upper þ ce;lower Rsp


(4)
¼ q Nsp;upper þ Nsp;lower ;
where d is the junction thickness, cr is the photon recycling
factor,8 and Rsp is the radiative recombination rate per unit
area per unit length. Furthermore, assuming that the parasitic
absorption of spontaneous emission due to impurities and
free carriers is negligible, then ð1  cr Þ ¼ ce , where ce is the
photon extraction factor.8 Moreover, photon extraction factors for both the upper and the lower surfaces are specified,
since the upper and lower surface configurations of a given
junction are different. In the right-most equation, the radiative recombination current is given in terms of Nsp;upper and
Nsp;lower , which provide the spontaneous emission extracted
through the respective upper and lower surfaces of the
junction.
Taking ambient blackbody radiation into account, the
net spontaneous emission fluxes from the upper surface into
free space and from the lower surface into a similar semiconductor material are, respectively,

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Nsp;upper ¼

Nsp;lower

ð1

nsp;upper  dh
0
ð


2p 1
eupper n;sp  n;bb ðh Þ2 dh
¼ 3 2
h c 0
ð1


2p
n;sp  n;bb ðh Þ2 dh
¼ 3 2 eupper
h c
Eg Eu
2p
ffi 3 2 eupper kTðEg  Eu þ kTÞ2 eðEg Eu Þ=kT
h c


 eqV=kT  1 ;
(5a)
ð1
¼
nsp;lower dh
0
ð


2p 1 2
nr elower n;sp  n;bb ðh Þ2 dh
¼ 3 2
h c 0
ð1
2pn2g


n;sp  n;bb ðh Þ2 dh
¼ 3 2 elower
h c
Eg Eu

ffi

2pn2g

elower kTðEg  Eu þ kTÞ2 eðEg Eu Þ=kT
h3 c2


 eqV=kT  1 ;
(5b)

where nsp;upper and nsp;lower are the spontaneous emission
fluxes per unit energy interval from the upper and lower
surfaces, respectively, h is Planck’s constant, c is the speed
of light in vacuum, k is Boltzmann’s constant, nr is the refractive index, h is the photon energy, and the emittance is
eupper for the upper surface and elower for the lower surface.
The respective photon occupation numbers for the spontaneous
emission
and
blackbody
background
are
n;sp ¼ ½eðhqV Þ=kT  11 and n;bb ¼ ½eh=kT  11 , where
V is the output voltage, assuming qV is equal to the separation between the electron and hole quasi-Fermi levels in the
junction (which implies no series or contact resistance).
Equation (5) implies that the absorptance of the junction, in terms of ambient blackbody radiation, is equal to the
emittance of the junction, in terms of spontaneous emission,
which is a reasonable assumption when the internal optical
loss is negligible.21 Note that the spectral spontaneous emission from the upper surface, nsp;upper , is experimentally measurable, however, the information it provides must be
carefully interpreted. From Eq. (5a), nsp;upper is clearly a
function of temperature, bandgap, tail width, emittance of
upper surface, and injection level, which is qV in this study,
however, more precisely, it is the quasi Fermi-level separation of electrons and holes. Parameters such as capture efficiency influence the injection level, however, nsp;upper is not
a direct measure of these.
The refractive index is a slowly varying function compared to either the sharp cutoff of the tail states or the occupation number and is therefore moved out from the integral
of Eq. (5b), by defining an average value as,


Ð1 2
2
2
0 nr elower n;sp  n;bb ðh Þ dh
ffi n2r ðh
sp Þ; (6)
ng  Ð 1


2
0 elower n;sp  n;bb ðh Þ dh
which is approximately equal to the refractive index at the
average energy of the spontaneous emission spectrum above
the effective bandgap, Eg  Eu , with

J. Appl. Phys. 110, 123104 (2011)


Ð1 
a n;sp  n;bb ðh Þ3 dh
h
sp ¼ Ð01 

a n;sp  n;bb ðh Þ2 dh
0

Ð1 
3
Eg Eu n;sp  n;bb ðh Þ dh
ffi Eg  Eu þ kT; (7)
ffi Ð1 

2
Eg Eu n;sp  n;bb ðh Þ dh
where a is the absorption coefficient. Note that right hand
side of Eq. (7) also gives the average energy separation of
the photoexcited electron and hole populations.
As with effective absorptance, it is useful to move the
emittance term out of the integral by defining two different
energy independent effective emittances for the respective
upper and lower surfaces as,


Ð1
2
0 eupper n;sp  n;bb ðh Þ dh
eupper  Ð 1 

2
Eg Eu n;sp  n;bb ðh Þ dh
Ð1
eupper eh=kT ðh Þ2 dh
;
(8a)
ffi 0
2
kT E  E þ kT eðEg Eu Þ=kT
g

elower

u




elower n;sp  n;bb ðh Þ2 dh
 Ð1 

2
Eg Eu n;sp  n;bb ðh Þ dh
Ð1
elower eh=kT ðh Þ2 dh
:
ffi 0
2
kT E  E þ kT eðEg Eu Þ=kT
Ð1
0

g

(8b)

u

Note that in a manner similar to absorptance, detailed balance models typically utilize a step-function emittance that
is zero below the bandgap and unity above the bandgap,
which presupposes that the absorptance and emittance are
the same. However, for a typical solar cell they are not,
because the energy range of incident solar radiation is much
broader than the emission spectrum that typically occurs
over energies where the absorption coefficient is small.
Using the Boltzmann approximation for low injection
(i.e., Eg  qV > 3kT), the photon occupation number is
given by a simple exponential, which is used to obtain the
right-hand terms in Eqs. (5), (7), and (8).
It is convenient to define the radiative recombination
saturation-current density as,
JB 


2
q2pkT 
2


e
þ
e
n
Eg  Eu þ kT ;
upper
lower
g
3
2
h c

(9)

with the radiative recombination current written as,


Jrad ffi JB  eðEg Eu Þ=kT eqV=kT  1 :

(10)

Here, the radiative recombination saturation-current density
is not simply dependent on bandgap energy as in the detailed
balance model, however, it is also dependent on the material
properties and structure geometry, including the refractive
index, tail width, and junction thickness.
The preceding equations provide an important insight
into the impact of tail states on solar cells: For a given voltage, the presence of tail states substantially increases the
radiative recombination current by roughly a factor, eEu =kT ,
which is 1.29 and 1.37 for crystalline GaAs and Si, respectively, using published values of the Urbach parameter,
Eu .22,23 The enhancement of spontaneous emission by states

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below the bandgap effectively shifts the optical absorption/
emission cutoff to lower energies, and the operating and
open-circuit voltages decrease in a response to limit the
spontaneous emission losses. For a given radiative recombination current, the voltage output is reduced by the width of
the tail as quantified by comparing the exponents in Eq. (10).
The question as to whether states below the band edge
improve solar cell performance depends on the trade-off
between a higher short-circuit current and lower open-circuit
voltage. For single-junction solar cells with a bandgap
greater than optimal (Eg > 1.37 eV), there is an increase in
the energy conversion efficiency when absorption below the
bandgap is present. On the contrary, when the bandgap is
optimal or less than optimal the efficiency decreases when
states below the band edge are present. Consequently, the existence or deliberate insertion of states below the band edge
provides a method to realize an optimal absorption cutoff in
the case when an ideal bandgap material is not available;
however, they do not outperform an optimal bandgap material without the presence of these states. In addition, the preceding discussion ignores any increases in non-radiative
recombination for a material with a large number of tail
states.
It is interesting to point out that the radiative recombination saturation-current can also be written in terms of the
radiative recombination coefficient, B, that is often used to
describe the performance of many other optoelectronic devices,6 with JB  qdBce ðNc Nv Þ, which is written in the same
form as in the nonradiative recombination terms described in
Eqs. (19) and (20), with the junction thickness, d, explicitly
shown. Upon substituting the analytical expression for B,24
and
using
ce;upper ¼ eupper =ð4ag d  n2g Þ
and
ce;lower
¼ elower =ð4ag dÞ, Eq. (9) is obtained, where ag is the absorption coefficient at the average photon energy of the spontaneous emission spectrum. In Eq. (9) the junction thickness is
implicit in the emittance term and disappears in the
optically-thick limit due to photon recycling.

J. Appl. Phys. 110, 123104 (2011)

FIG. 3. Schematic diagrams of four types of planar solar cell structures, labeled Structures A, B, C, and D. Structure A is a semiconductor slab with an
absorbing substrate, Structure B is a semiconductor slab with a reflecting
substrate, Structure C is a textured semiconductor slab with an absorbing
substrate, and Structure D is a textured semiconductor slab with a reflecting
substrate.

Structure A has a smooth upper surface and an absorbing substrate which typifies a junction (i.e., a subcell) inside
a multi-junction solar cell where the adjacent lower junction
functions as an absorbing substrate. Structure D has a textured upper surface and a reflecting substrate which typifies
a high performance single-junction solar cell, especially for
a Si single-junction cell.25 Structure B has a smooth top surface and a reflecting substrate and Structure C has a textured
top surface and an absorbing substrate,which are investigated
for comparison.
The absorptance of a given junction is a function of the
absorption coefficient, junction thickness, and surface/interface properties. Using ray optics, the absorptance is given
by,

D. Four planar structures

This work categorizes solar cell structures into four different planar configurations. Schematic diagrams of these
structures are shown in Fig. 3, where Structure A (Fig. 3(a))
is a semiconductor slab with a non-reflecting index-matched
absorbing substrate, Structure B (Fig. 3(b)) is a semiconductor slab with an ideal 100% reflecting substrate, Structure C
(Fig. 3(c)) is a textured semiconductor slab with a nonreflecting index-matched absorbing substrate, and Structure
D (Fig. 3(d)) is a textured semiconductor slab with an ideal
100% reflecting substrate. Here, the term “reflecting substrate” is used to describe a reflecting layer on the backside
of the device, which can be, for example, a highly-reflective
metal layer. The terms “upper surface” and “lower surface”
in this study are used to describe the surface facing the sun
light and the bottom interface between the slab and the substrate, respectively. Moreover, it is assumed that there is no
reflection at the upper surface (i.e., Ta ¼ 1  Aa ) since the
reflection can be substantially reduced using an antireflection coating and/or surface roughening.

Aa ¼

Aa ffi 1  ead

for Structure A;

(11a)

Aa ffi 1  ea2d

for Structure B;

(11b)

A a ¼ 1  ta

for Structure C;

(11c)

for Structure D;

(11d)

1  ta


1
1  1  2 ta
nr

where a is the absorption coefficient, d is the slab thickness,
nr is the refractive index of the semiconductor slab, and ta is
the fraction of incident photons not absorbed on a single pass
through the slab, with
ta ¼

ð p=2

ead=cos h 2 cos h sin hdh for Structure C;

(12a)

e2ad=cos h 2 cos h sin hdh for Structure D;

(12b)

0

ta ¼

ð p=2
0

where h is the angle between the scattered ray and the surface normal. Maximal (i.e., Lambertian) scattering is
assumed, where 2 cos h is the probability that a ray is scattered into the solid angle, sin hdh. Here, 2 cos h can be

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123104-6

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replaced by other scattering functions for non-Lambertian
scattering. For Structures A and B, the optical path length is
slightly longer when the incident angle of the solar radiation
is not normal, however, the difference is negligible for typical semiconductors due to their large indices of refraction.
Since much of the scattered light is trapped in Structure
D, the absorptance is greatly enhanced through the multiple
reflection/scattering events. For randomized scattering from
a textured top surface, Tiedje et al.4 provide an analysis for
the absorptance where the optical thickness of semitransparent layers is increased by a factor of 4n2r on average,
and their
equation for absorptance in Structure D is
.
½1 þ 1 ð4n2r adÞ1 , which is a good approximation to Eq.
(11d) with a difference of less than 2% when the refractive
index is greater than 3.
Using crystalline Si and GaAs as the prototypical indirect and direct bandgap semiconductors, the effective absorptance given by Eq. (3) as a function of slab thickness is
plotted in Fig. 4 for the four slab structures. Here the AM1.5
G solar spectrum is used in the calculation; there is only a
slight difference in the results if the AM0 spectrum26 is used.
Published values for the refractive index, nr , absorption coefficient, a,24 and the tail widths, Eu ¼ 8.5 meV (Ref. 22) and
6.7 meV (Ref. 23) are used for Si and GaAs, respectively.
The slab thicknesses for Aa ¼ 0:90, 0.99, and 1.0 are also
summarized in Table I, where, for a given Aa , Structure A is
the thickest, Structure C is the second thickest, Structure B is
the third thickest, and Structure D is the thinnest. Moreover,
direct bandgap junctions are much thinner than indirect
bandgap junctions.
Most reported GaAs single-junction solar cells that utilize the Structure A configuration28 satisfy Aa  1. On the
contrary, most reported high-performance Si single-junction
solar cells25 that utilize the Structure D configuration to further increase the optical-thickness of the junction also satisfy
Aa  1. From a practical device design point of view, further
increases of the effective absorptance beyond Aa  1 typically result in excessive non-radiative recombination and
parasitic absorption losses. Note that the absorptance calculation assumes relatively low injection levels (i.e.,
Eg  qV > 3kT), as the absorption coefficient at equilibrium
is used. Low injection is a valid assumption for most solar
cells and is used throughout this study.

J. Appl. Phys. 110, 123104 (2011)
TABLE I. Slab thicknesses for the effective absorptance, Aa ¼ 0:90, 0.99,
and 1.00, for Si and GaAs.
Aa

Structure A

Structure B

Slab thickness for Si (mm)
0.90
0.26
0.99
2.95
1.00
5.60
Slab thickness for GaAs (lm)
0.90
0.99
0.99
2.59
1.00
3.44

Structure C

Structure D

0.13
1.48
2.80

0.16
1.94
3.76

0.01
0.16
0.36

0.50
1.30
1.72

0.66
1.81
2.40

0.05
0.20
0.28

Using ray optics,21 it can be derived that the upper surface emittance for the four structures is,
eupper ¼ 1  n2r

ð hc

ead=cos h 2 cos h sin hdh for Structure A;

0

(13a)
eupper ¼ 1  n2r

ð hc

e2ad=cos h 2 cos h sin hdh

for Structure B;

0

(13b)
eupper ¼ Aa

for Structures C and D;

(13c)

where in Structures A and B, h is the angle between the surface normal and the emitted ray inside the junction and
hc ¼ sin1 ð1=nr Þ is the critical angle beyond which the emitted rays do not escape from the upper surface. Since the
angular integration for Structures A and B is done over the
escape cone inside the slab, the refractive index squared
appears in front of the integral. If the integration is done outside the slab overall angles, the refractive index does not
appear, however, the expression for the path length in terms
of the outside angle is more complicated due to refraction at
the surface. For Structures C and D, the emittance equals the
absorptance given in Eq. (13c), since the maximal scattering
path length is the same during either absorption or emission
in textured structures.
On the contrary, for Structures A and B, the emittance is
slightly larger than the absorptance given in Eqs. (13a) and
(13b), because the absorption path length under normal incidence solar radiation is slightly shorter when compared to

FIG. 4. Effective absorptance, Aa , vs
physical thickness, d, for the four types
of semiconductor slabs investigated: (a)
GaAs, and (b) Si.

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FIG. 5. Effective emittance, eupper and
elower , vs physical thickness, d, for the
four types of semiconductor slabs investigated: (a) GaAs, and (b) Si.

the emission which occurs at all angles within the escape
cone. Under maximum concentration ( 47 000 suns) where
the solar radiation is incident at all angles, the absorptance is
identical to the emittance as the solar radiation is absorbed at
all angles within the escape cone.
In Structures B and D, the reflecting substrate results in
zero emission escaping from the lower surface. However, in
Structures A and C a large amount of spontaneous emission
is coupled into the absorbing substrate as it is index matched.
Furthermore,
the upper surface in both structures reflects

1  1 n2r of the internal spontaneous emission incident on it,
which is substantial when the refractive index is large. The
flux is n2r times larger inside the semiconductor compared to
free space, with
elower ¼ 1 


ð hc

ead=cos h 2 cos h sin hdh

0
ð p=2

e2ad=cos h 2 cos h sin hdh

for Structure A;

hc

(14a)

elower

The effective emittance versus effective absorptance for
GaAs and Si is shown in Figs. 5(a) and 5(b). For the same
effective absorptance, the effective emittance of the lower
surface for Structure A is the largest, the lower surfaces for
Structure C are the second largest, the upper surface of
Structure D is the third largest, the upper surface of Structure
C is the fourth largest, and the upper surfaces for Structures
A and B is the smallest (they almost overlap in the plot).
Moreover, in order to achieve the same effective emittance,
indirect bandgap junctions are much thicker than direct
bandgap junctions. In addition, it is interesting to point out
that the preceding calculations are useful for other optoelectronic devices such as light emitting diodes, for which Structure D is preferred because it has a much higher emittance
than the other three structures for the same slab thickness.
The effective emittance for GaAs and Si are also summarized in Table II for the cases where the effective absorptance
is Aa ¼ 0:90, 0.99, and 1.00.
Figure 6 shows the average refractive index, ng , versus
bandgap energy for some common group-IV, III-V, and

(14b)
elower ¼ 0 for Structure B;


1
for Structure C; (14c)
¼ ð1  ta Þ 1 þ ta 1  2
nr
elower ¼ 0 for Structure D:

(14d)

TABLE II. Effective emittance when effective absorptance is Aa ¼ 0:90,
0.99, and 1.00, for Si and GaAs.
Structure A
Aa

eupper

elower

Structure B
eupper

Effective emittance for Si
0.90
0.12
0.33
0.12
0.99
0.72
1.17
0.72
1.00
0.97
1.44
0.97
Effective emittance for GaAs
0.90
0.36
0.79
0.36
0.99
0.72
1.18
0.72
1.00
0.85
1.30
0.85

Structure C

Structure D

elower

eupper

elower

eupper

elower

0
0
0

0.13
0.75
1.01

0.24
1.02
1.29

0.19
0.92
1.24

0
0
0

0
0
0

0.40
0.77
0.90

0.64
1.06
1.18

0.58
1.08
1.20

0
0
0

FIG. 6. Refractive index at the average spontaneous emission energy,
Eg þ kT, vs bandgap energy, Eg , for the common II-VI, III-V, and group IV
semiconductors shown. The solid line is a least squares fit to the data that
provides an average refractive index as a function of the bandgap energy for
common semiconductor materials.

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FIG. 7. Radiative recombination saturation current density, JB , vs bandgap
energy, Eg , for solar cells with absorbing substrates (Structure A) and textured surfaces and reflecting substrates (Structure D). The solid circles are
calculations for Structure A using the discrete values of the refractive index
for the common II-VI, III-V, and group IV semiconductors. The solid curves
(Structures A and C) are calculated using the average refractive index as a
function of the bandgap energy and the dashed curves (Structures B and D)
do not depend on the refractive index.

FIG. 8. Lowest reported SRH recombination saturation current density per
unit length, JA =d, vs bandgap energy, Eg , for the various semiconductors
shown.



JSRH  JA eðEg Eu Þ=2kT eqV=2kT eqV=kT  1 ;


JAuger  JC e3ðEg Eu Þ=2kT eqV=2kT eqV=kT  1 ;

(17)
(18)

27

II-VI semiconductors based on published data. The solid
gray line is a linear fit to ng with ng ¼ 4:12  0:46  Eg
(where Eg is in units of eV).
The radiative recombination saturation current density
as a function of bandgap energy for the four structures is
evaluated and plotted in Fig. 7, using the effective emittances for GaAs given by Aa ¼ 1 (see Table II). The solid
circles are the calculations for Structure A using published
values of ng .27 The solid curves for Structures A and C give
the results using the linear fit for ng ; here, the curves nearly
coincide because the radiative loss for both is dominated by
the absorbing substrate (see Eq. (9)). On the contrary, the
curves for Structures B and D with the reflecting substrates
are much lower and do not depend on the refractive index
since the emittance of the lower surface is zero. As can be
seen from Fig. 7, the radiative recombination saturationcurrent density, JB , increases with the bandgap energy, Eg ,
and is substantially larger for Structures A and C as spontaneous emission is coupled into the substrate.

where JA and JC are SRH and Auger recombination saturation current densities, and Eg  Eu is the effective bandgap
described in Eq. (10). In the limiting case when the semiconductor material in question is intrinsic, JA and JC are given
by,
pffiffiffiffiffiffiffiffiffiffi
(19)
JA ¼ qdA Nc Nv ;
JC ¼ qdCðNc Nv Þ3=2 ;

(20)

JSRH ¼ qdRSRH ;

(15)

where A is the SRH recombination coefficient, C is the Auger recombination coefficient, and Nc and Nv are the conduction- and valence-band effective density of states,
respectively.6
It is informative to look at the extent of the SRH and
Auger recombination for some common group IV and III-V
semiconductors. Following Eq. (19), the SRH recombination
saturation current density per unit length versus bandgap
energy is calculated and shown in Fig. 8, where the published longest minority carrier lifetime for each material is
used.29 The results vary from 0.1 to 300 A cm2 lm1 with
no comprehensive trend. Similarly, following Eq. (20), the
Auger recombination saturation current density per unit
length versus bandgap energy is shown in Fig. 9, where published values of the Auger recombination coefficients are
used.29 The results vary from 100 to 300 000 A cm2 lm1;
Si has the largest value, which is mainly due to its large
effective densities of states.

JAuger ¼ qdRAuger ;

(16)

F. Loss and extracted power

E. Non-radiative recombination current densities

To further evaluate non-radiative recombination losses
in solar cells, it is necessary to establish simple models for
the SRH and Auger recombination currents in solar cells.
The SRH and Auger recombination current densities are
given by,

where RSRH and RAuger are SRH and Auger recombination
rates,6 respectively.
Under low injection, Eqs. (15) and (16) are approximated by,

The equations for single-junction solar cells including
extracted power, conversion efficiency, and losses are listed
in Table III. Where the extractable energy of the photogenerated carriers is qV, the average energy separation of the

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J. Appl. Phys. 110, 123104 (2011)

red-shifted as it undergoes photon recycling and, in general,
has an average energy less than Eg  Eu þ kT. In this study,
precise numerical calculations are used to calculate the average emitted photon energy for GaAs and Si single junction
cases.
The transmission and thermalization losses are independent of the voltage, while the spatial-relaxation and
recombination losses are dependent upon the voltage. The
trade-off between spatial-relaxation and recombination loss
determines the optimal operating condition (maximum
power output) for solar cells. The spatial-relaxation loss is
maximum (minimum) and the recombination loss is minimum (maximum) at the short-circuit (open-circuit)
condition.

G. Application to multi-junction solar cells
FIG. 9. Lowest reported Auger recombination current density per unit
length, JC =d, vs bandgap energy, Eg , for the various semiconductors shown.

electron and hole populations is Eg  Eu þ kT, and the average energy of the absorbed solar radiation is,
Ð1
Aa nsun hdh
;
(21)
h sun ¼ 0Ð 1
0 Aa nsun dh
which monotonically deceases as the junction thickness
increases, saturating at Aa  1. The Boltzmann approximation provides Eg  Eu þ kT as the average energy separation
of the electron and hole populations and consequently, the
average energy of the internal spontaneous emission (see Eq.
(7)). Note that the emitted spontaneous emission is spectrally

The preceding model is further extended to multijunction solar cells consisting of j (j 1) junctions numbered in ascending order from top to bottom (i.e., from the
largest bandgap to the smallest bandgap). The ith junction
has thickness, d, bandgap energy, Eg , Urbach tail width, Eu ,
absorptance, Aa , and emittance, eupper and elower . In the following, the superscript, i, is used as necessary for clarity and
left out where the context makes it apparent. The current
density for the ith junction is,


i
i
i
i
 Jrad
þ JSRH
þ JAuger
for 1 i j;
(22)
J i ¼ Jsc
which shares the same form as that of single-junction solar
cells. The radiative, SRH, and Auger recombination current

TABLE III. Definitions and equations for losses and extracted power in single-junction solar cells.
Parameter
Extracted power

Conversion efficiency

Transmission loss

Thermalization loss

Spatial relaxation loss

Radiative loss

SRH loss

Auger loss

Definition and equation
Power extracted at maximum output condition
pout ¼ JmVm
Extracted power
Incident solar power
Pout
Jm Vm
g¼
¼ Ð1
Pin
0 nsun  hv  dhv
Solar power not absorbed by solar cell
Ð1
Jsc
Ltr ¼ 0 ð1  Aa Þ  nsun  hv  dhv ¼ Pin 
 hvsun
q
Energy lost as photoexcited carriers equilibrate to their respective populations




Ð1
Jsc
Lth ¼ 0 Aa nsun  hv  Eg  Eu þ kT dhv ¼
 hvsun  Eg  Eu þ kT
q
Potential energy lost as the electron-hole population is separated and swept to each respective contact

J 
Lsr ¼  Eg  Eu þ kT  qV
q
Energy lost to radiative recombination

Jrad 
Lrad ¼
 Eg  Eu þ kT
q
Energy lost to SRH recombination

JSRH 
LSRH ¼
 Eg  Eu þ kT
q
Energy lost to Auger recombination

JAuger 
LAuger ¼
 Eg  Eu þ kT
q

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J. Appl. Phys. 110, 123104 (2011)

densities for the ith junction are also the same as these for
single-junctions solar cells (see Eqs. (10), (17), and (18)).
The short-circuit current density becomes complicated
when more than one junction is present. Carriers are generated by the absorption of both solar radiation, nisun , and
coupled spontaneous emission from the adjacent larger9–12
i1
ni1
where Nsp;lower
bandgap
sp;lower ,
Ð 1 i1 i  1 junction,
0
¼ 0 nsp;lower dh (with nsp;lower ¼ 0) is the total spontaneous
emission received, which is typically completely absorbed,
since it has energies well above the bandgap of the ith junction where the absorption coefficient is sufficiently large.
Taking into consideration the coupled spontaneous emission
from the lower surface of the preceding larger-bandgap junction, the short-circuit current density per unit area of the ith
junction is,
ð1 

i
Aa nisun þ ni1
Jsc ¼ q
sp;lower dh
0
!
ð1
i1
for 1 i j; (23)
ffi q Aa
nisun dh þ Nsp;lower
Eg Eu

u

assuming the bandgap separation between the junctions are
much greater than the absorption tail width. Note that the
condition, Aa ¼ 1, provides an opportunity to generally
investigate multi-junction solar cells without knowing the
absorption coefficient and thickness for each junction.
Moreover, to study the best performance for devices, it
is assumed that the multi-junction solar cell in question is
current-matched at the maximum power output condition for
each junction. The total power extracted is determined by
summing overall junctions, with
Pout ¼ Jm

j
X

Vmi :

(24)

i¼1

Ejg Eju

0

which is approximately the transmission loss for the bottom
junction, since typically the solar photons with energy above
the bandgap of the bottom junction are completely absorbed.
1, the transmission loss is simplified to
WhenÐ Aj a ¼
E Ej
Ltr ¼ 0 g u njsun h  dh.
The thermalization loss includes the thermalization of
both solar radiation and coupled spontaneous emission, and
is written as,
Lth ¼

j ð 1
X
i¼1

ffi

i j is the solar radiation
where nisun ¼ Tai1 ni1
sun for 2
incident on the ith junction and Tai is the transmittance of ith
junction, with Ta0 ¼ 1 and n1sun ¼ nsun as the solar radiation
incident on the top junction. When Aa ¼ 1 is satisfied, each
junction is equivalent to a long-pass “filter” with a cut-off at
Eig  Eiu for the solar radiation spectrum which leads to
Ð Ei1 Ei1

i
i1
¼ q EigEi u nsun þ Nsp;lower
(with E0g  E0u ¼ 1),
Jsc
g

The transmission loss is given by
!
ð1 Y
j
Ltr ¼
Tri nsun h  dh
0
1
ð1
ð1
j
j
nsun h  dh  Aa
njsun h  dh; (29)
ffi

0

Aia



j  i h
X
Jsc

q

i¼1

i1
þNsp;lower

nisun þni1
sp;lower



i
h
i
i
h  Eg Eu þkT dh


i
i
h sun  Eig Eiu þkT

h

 
io
i1
i
i
Ei1
E
E

E
;
g
u
g
u

with n0sp;lower ¼ 0. When Aa ¼ 1, the average energy of the
absorbed solar photons for the ith junction is,
,ð i1 i1
ð i1 i1
i

Eg Eu

h sun ffi

Eg Eu

nsun hdh
Eig Eiu

nsun dh;
Eig Eiu

where E0g  E0u ¼ 1. Note that for a given bandgap, the average energy of the absorbed solar photons, and hence the
thermalization losses, decrease as the junction number
increases.
The total radiative loss is obtained by summing the
spontaneous emission lost from the upper surfaces of all
junctions and the lower surface of the bottom junction, and
is written as,
Lsp ¼

j1
X
i¼1


 Jj 

i
Nsp;upper
Eig  Eiu þ kT þ rad Ejg  Eju þ kT :
q
(31)

Additionally, the energy conversion efficiency is,
Jm

g¼

j
X

Vmi

Pout
i¼1
:
¼ Ð1
Pin
n
h
 dh
sun
0

(25)

The total SRH, Auger, and spatial-relaxation losses are also
obtained by summing each of them overall junctions, with
LSRH ¼

j
X
Ji

SRH

i¼1

LAuger ¼

q




Eig  Eiu þ kT ;

j
i

X
JAuger
i¼1

q


Eig  Eiu þ kT ;


i
Jm X h  i
Eg  Eiu þ kT  qVmi :
q i¼1

Here, the spontaneous emission coupled into each lower
junction is excluded as a loss (except for the bottom junction), since it is reasonably assumed to be completely
absorbed in the adjacent smaller bandgap junctions.
IV. MODELING RESULTS AND DISCUSSION
A. Single-junction solar cells

(26)

(27)

j

Lsr ¼

ð30Þ

(28)

In this section, important design parameters such as
junction thickness, solar concentration, bandgap, junction
number, and material quality in terms of the SRH recombination current density per unit length are studied using the
previously established equations.
Junction thickness is a critical design parameter for
single-junction solar cells. To make the discussion more

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J. Appl. Phys. 110, 123104 (2011)

FIG. 10. Single-junction GaAs solar cell
under one sun AM1.5 G with typical Auger recombination current density,
JC =d ¼ 120 A cm2 lm1 . (a) Efficiency
vs junction absorptance (i.e., optical
thickness) for various SRH recombination current densities, JA =d. (b) Solar
power extracted and lost vs absorptance,
assuming SRH recombination current
density, JA =d ¼ 10 A cm2 lm1 .

general and useful for practical devices, the effective absorptance (i.e., the “optical thickness” of a solar cell junction) is
investigated, rather than the physical thickness of the junction, which is strongly material dependent. Using a GaAs
single-junction solar cell with Structure A as an example, the
energy conversion efficiency versus the effective absorptance for the AM1.5 G solar spectrum is plotted in Fig. 10(a)
for various SRH recombination saturation current densities
per unit length, JA =d. In the calculation, a typical Auger
recombination saturation current density per unit length of
JC =d ¼ 120 A cm2 lm1 is used, which is negligible compared to the other recombination losses. The gray curve with
JA ¼ 0 A=cm2 shows the efficiency limit with no SRH
recombination. This efficiency limit monotonically increases
as the short-circuit current monotonically increases with
effective absorptance.
When SRH recombination is present, there is an optimal
effective absorptance (i.e., junction thickness) given by a
peak value in the efficiency. To further analyze the factors
involved in this optimal junction thickness, the relative
fractions of input power that are extracted and lost are shown
by the areas under each curve in Fig. 10(b) versus the
effective absorptance, where the best reported value of
JA =d ¼ 10 A cm2 lm1 for GaAs is used in the calculations. From Fig. 10(b) it can be seen that the transmission
loss decreases linearly with the effective absorptance, while

the SRH recombination and spatial-relaxation losses increase
super-linearly with effective absorptance. The combination
of these trends results in an optimal thickness for a singlejunction solar cell. Note that even for the best reported GaAs
material quality, the Auger and radiative recombination
losses are too small to be clearly seen in Fig. 10(b) and SRH
recombination dominates the total recombination loss.
The optimal effective absorptance and energy conversion efficiency are plotted in Fig. 11(a) as a function of the
SRH saturation current density per unit length, JA =d, for the
four single-junction structures previously discussed, using
GaAs as the material. The general trend is that the optimal
effective absorptance decreases with material quality. For a
given material quality with JA =d < 15 A cm2 lm1 , the
optimal effective absorptance for Structure D is the largest,
Structure C is the second largest, Structure A is the third
largest, and Structure B is the smallest. While for a given
material quality with JA =d > 15 A cm2 lm1 , the optimal
effective absorptance for Structure B is the largest, Structure
A is the second largest, Structure C is the third largest, and
Structure D is the smallest. Here, JA =d  15 A cm2 lm1 is
a transition point where all four structures share the same
optimal effective absorptance. As will be further analyzed,
there is a transition from a radiative recombination dominated region to a SRH recombination dominated region
between 1 < JA =d < 15 A cm2 lm1 , which results in a

FIG. 11. Single-junction GaAs solar
cell under one sun AM1.5 G with typical
Auger recombination current density,
JC =d ¼ 120 A cm2 lm1 . (a) Optimal
effective absorptance (left-hand y-axis)
and the corresponding conversion efficiency (right-hand y-axis) vs SRH
recombination current density, JA =d, for
the four solar cell structures investigated
(A, B, C, and D). (b) Solar power
extracted and lost for Structure A at the
optimal junction thickness vs SRH
recombination current density, JA =d.

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J. Appl. Phys. 110, 123104 (2011)

FIG. 12. Single-junction Si solar cell
under one sun AM1.5 G with typical Auger recombination current density
JC =d ¼ 3  105 A cm2 lm1 . (a) Efficiency vs junction absorptance (i.e., optical thickness) for various SRH
recombination current densities, JA =d.
(b) Solar power extracted and lost vs absorptance for Structure D, assuming
SRH recombination current density,
JA =d ¼ 0:1 A cm2 lm1 .

slope change in the optimal junction thickness curves shown
in Fig. 11(a). Moreover, for a given material quality, the efficiency for Structure D is always the best; Structure B is next,
Structure C is third, and Structure A is the lowest.
For very high quality materials (JA =d < 102
A cm2 lm1 ) Structures D and B with reflecting back surfaces offer similar efficiencies at about 33.9%, while Structures C and A on absorbing substrates offer similar and
slightly lower efficiencies at about 31.7%. In this case, the
much larger radiative losses associated with an absorbing
substrate results in a lower efficiency for Structures C and A.
Moreover, the efficiency of the non-textured structures (A
and B) approaches that of the textured structures (C and D)
because the average optical path lengths related to the
absorption of solar radiation and the emission loss are the
same. On the contrary, as the material quality degrades, the
textured top surface of Structures D and C provides a clear
enhancement in performance as the radiative losses become
insignificant and the nonradiative SRH losses take over. In
this case, it is the thickness of the junction that becomes important, where thinner is better as long as the junction is
thick enough to absorb most of the solar radiation; where, in
order of performance, Structure D is the thinnest, Structure
B is next, Structure C is next, and Structure A is the thickest.
To further understand these results, the extracted power
and losses are plotted against the SRH saturation current
density per unit length, JA =d, in Fig. 11(b), for a GaAs based
solar cell with an absorbing substrate (Structure A).
Although the nonradiative recombination loss increases with
the SRH recombination saturation current, the main contributor to the decrease in conversion efficiency is the large
increase in the spatial-relaxation loss.
It is necessary to point out that state-of-the-art GaAs
single-junction solar cells still operate in the SRH recombination dominated region, where JA =d varies from tens to
hundreds of A cm2 lm1. For Structure A, the optimal
junction thickness remains close to one that is optically-thick
for 103 < JA =d < 105 A cm2 lm1 , thus the transmission
loss remains almost unchanged. On the contrary, for Structure D the optimal junction thickness deviates from one that
is optically-thick and the transmission loss increases as the
SRH recombination losses increase. For the best reported
GaAs material quality (JA =d  10 A cm2 lm1 ), the

achievable efficiency limit under AM1.5 G one sun is 27.4%
for Structure A and 30.8% for Structure D. In theory there
could be a 2-3% absolute efficiency improvement from the
current record if Structure D is adopted for GaAs singlejunction solar cells.
Similar results are plotted in Figs. 12(a), 12(b), 13(a),
and 13(b) for single-junction Si solar cells. In the limiting
case when SRH recombination is negligible, Si solar cells
are dominated by Auger recombination (see the large Auger
recombination saturation current in Fig. 9) due to a large
effective density of states. Unlike direct bandgap materials,
this results in a maximum in the efficiency limit curve (see
Fig. 12(a)) that is a trade-off between the transmission and
Auger recombination losses. For JA =d  0:1 A cm2 lm1
(see Fig. 12(b)), Auger recombination dominates the loss
when the absorptance is less than 0.85 and SRH recombination dominates when the absorptance is greater than 0.85.
As shown in Fig. 13(a), in the case when SRH recombination is negligible, the efficiency for non-textured front
surfaces (Structures A and B) does not approach that of textured surfaces (Structures C and D). Unlike radiative losses,
Auger losses scale with the junction thickness and, hence,
are different for each optimal thickness. As shown in
Fig. 13(b), there is a transition from an Auger dominated
region to an SRH dominated region between 0:1 < JA =d
< 1 A=cm2 , which is about one order of magnitude smaller
than that for GaAs. For the best reported Si material quality
(JA =d  0:5 A=cm2 ), the achievable efficiency limit under
AM1.5 G one sun is 21.1% for Structure A and 26.4% for
Structure D. Note that Structure D is ideal in that it assumes
maximal random scattering at the textured top surface. The
upper surface scattering for a real Si solar cell may not be
entirely random, as is assumed in this study, which leads to
thicker junction designs for optimal efficiency.25
Increasing the concentration of solar radiation is a successful approach to improve the conversion efficiency of solar cells18 and it is important to determine how the optimal
design changes with solar concentration, which is presented
in Figs. 14 and 15. In Fig. 14(a), the optimal junction absorptance (left-hand y-axis) and the corresponding conversion
efficiency (right-hand y-axis) is plotted against solar concentration for single-junction GaAs solar cells for the four
structures investigated. It can be seen that the optimal

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FIG. 13. Single-junction Si solar cell
under one sun AM1.5 G with typical Auger recombination current density,
JC =d ¼ 3  105 A cm2 lm1 . (a) Optimal effective absorptance (left-hand yaxis) and the corresponding conversion
efficiency (right-hand y-axis) vs SRH
recombination current density, JA =d, for
the four solar cell structures investigated
(A, B, C, and D). (b) Solar power
extracted and lost for Structure D at the
optimal junction thickness vs SRH current density, JA =d.

absorptance increases with solar concentration. For solar
concentrations greater than one the effective absorptance of
Structure D is the largest, Structure C is next, Structure A
next, and Structure B is the smallest. The margin by which
the textured surfaces outperform the non-textured surfaces
decreases as the solar concentration increases.
In Fig. 14(b), the extracted power and losses are plotted
against solar concentration for an optimally thick singlejunction GaAs solar cell with an absorbing substrate (Structure A). The efficiency monotonically increases with solar
concentration since series resistance and other losses that
increase with solar concentration are not included. As the solar concentration increases, the overall recombination loss
slightly decreases, the spatial-relaxation losses substantially
decreases, and the dominant recombination loss gradually
changes from SRH to radiative. The main contribution of the
concentration is that it increases the average free (extractable) energy of the photogenerated carriers.
Similar results are plotted in Figs. 15(a) and 15(b) for
optimally thick single-junction Si solar cells with a textured
front surface and reflecting back surface (Structure D). The
change in optimal absorptance (junction thickness) with solar concentration is much greater for Si cells than GaAs cells.
Moreover, the dominant recombination loss gradually
changes from SRH to Auger for the Si cells. For solar concentrations less than 500 suns, the SRH recombination still

dominates the total recombination loss for the GaAs and Si
solar cells.
The preceding calculations and analysis provide a clear
physical picture of single-junction devices and also deliver
reasonable results compared to the actual devices.25,28
B. Multi-junction solar cells

To limit the number of possible combinations to model,
the analysis of multi-junction solar cells assumes that the
properties of each junction, other than bandgap energy, are
based on prototypical GaAs, where the width of the absorption tail is Eu ¼ 7 meV, the effective absorptance is Aa ¼ 1,
the effective emittance from the upper surface is
eupper ¼ 0:85, the effective emittance of the lower surface is
elower ¼ 1:30, and the Auger recombination loss is given by
JC ¼ 380 A/cm2. Note that the nonradiative losses are now
expressed in terms of the junction property saturation current
density, rather than the material property saturation current
density per unit length used in the previous section. Nevertheless, in general, each parameter can be set independently
for each junction in the model, provided specific material parameters are available.
Using a four-junction solar cell as an example, the overall
energy conversion efficiency and the bandgaps of the junctions are calculated and plotted in Fig. 16(a), where JA is set

FIG. 14. Single-junction GaAs solar cell
under AM1.5 G solar concentration with
Auger recombination current density,
JC =d ¼ 120 A cm2 lm1 and SRH
recombination
current
density,
JA =d ¼ 10 A cm2 lm1 . (a) Optimal
effective absorptance (left-band y-axis)
and the corresponding conversion efficiency (right-hand y-axis) vs solar concentration for the four solar cell
structures investigated (A, B, C, and D).
(b) Solar power extracted and lost for
Structure A at the optimal junction
thickness vs solar concentration.

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FIG. 15. Single-junction Si solar cell
under AM1.5 G solar concentration with
Auger recombination current density,
JC =d ¼ 3  105 A cm2 lm1 and SRH
recombination
current
density,
JA =d ¼ 0:5 A cm2 lm1 . (a) Optimal
effective absorptance (left-hand y-axis)
and the corresponding conversion efficiency (right-hand y-axis) vs solar concentration for the four solar cell
structures investigated (A, B, C, and D).
(b) Solar power extracted and lost for
Structure D at the optimal junction
thickness vs solar concentration.

at 30 A/cm2 (i.e., a reasonable estimate for state-of-the-art
high-quality material) and spontaneous emission coupling is
taken into account. To further identify the trend of each loss
mechanism versus the top-junction bandgap, the extracted
power and losses versus the bandgap energy of the top junction are plotted in Fig. 16(b). In Fig. 16(a), there are two efficiency peaks at 44% when the bandgaps of the top junction
are 2.02 and 2.10 eV. This double peak characteristic is
mainly due to the atmospheric absorption features in the
AM1.5 G spectrum. Any increase (or decrease) of the top
junction bandgap apart from these maximum points will result
in a lower conversion efficiency due to an increase in the total
loss. Moreover, the energy separation between bandgaps
increases as the top junction bandgap decreases and the
bandgap energy separation is smaller for the bottom junctions.
Furthermore, the bottom junction cannot provide sufficient
current to satisfy the current matching condition when the top
junction bandgap is below 2.01 eV under one sun condition.
To further understand the impact of SRH recombination
on multi-junction solar cells, the extracted power and losses
and the maximum conversion efficiency and corresponding
bandgap of each junction as a function of the SRH recombination saturation current density, JA , are modeled. The
results for four-junction solar cells are shown in Fig. 17(a)
for the maximum conversion efficiency and bandgap and in
Fig. 17(b) for the extracted power and losses. Here, the SRH

recombination saturation current density, JA , is the same for
each junction to simplify the discussion, without a loss in
generality since the overall results remain the same if the
values differ between junctions.
When JA < 1 A/cm2 the energy conversion efficiency is
very close to the theoretical limit and the recombination loss
is dominated by the radiative recombination. As JA
increases, the SRH nonradiative recombination gradually
displaces the radiative recombination. For JA > 1 A/cm2,
there is a substantial increase of the spatial-relaxation loss,
as discussed in Sec. II and the optimal bandgaps for all the
junctions deviate from those for ideal materials, resulting in
greater transmission losses and a slightly smaller thermalization loss due to the smaller energy spacing between the
bandgaps of adjacent junctions. Clearly, the total recombination loss is dominated by the SRH recombination in state-ofthe-art multi-junction solar cells under one sun when JA
varies from tens to hundreds of A/cm2.
The maximum efficiency and the corresponding
bandgap energies versus solar concentration for fourjunction solar cells are shown in Fig. 18(a) and the extracted
power and losses versus solar concentration are shown in
Fig. 18(b). Figure 18(a) shows that the peak efficiency
increases as the optimum bandgap energies for the junctions
decrease with the solar concentration. Figure 18(b) shows
that such an efficiency increase is mainly due to the

FIG. 16. Efficiency of four-junction solar cell vs top-junction bandgap energy
with SRH recombination current density,
JA ¼ 30 A/cm2. (a) Power conversion efficiency (left-hand y-axis) and bandgap
energy of each junction (right-hand
y-axis). (b) Solar power extracted and
lost (y-axis).

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J. Appl. Phys. 110, 123104 (2011)

FIG. 17. Efficiency of four-junction solar cell vs SRH recombination current
density, JA . (a) Power conversion efficiency (left-hand y-axis) and optimal
bandgap energy for each junction (righthand y-axis). (b) Solar power extracted
and lost (y-axis).

reduction in transmission, spatial-relaxation, and the SRH
recombination losses. It is interesting to point out that the
thermalization loss increases with concentration as the optimal junction bandgaps decrease with concentration. Moreover, the radiative recombination loss gradually increases
in terms of the overall recombination loss. For the solar
concentration less than 500 suns, SRH recombination
still dominates in the total recombination loss. These findings are important for multi-junction solar cell designs
because they show that the bandgaps need to be carefully
optimized, taking into account material quality and solar
concentration.
It is interesting to further investigate how much
improvement in efficiency can be achieved by utilizing more
junctions. The peak efficiency versus the junction number is
shown in Fig. 19(a) for various values of JA , while the
extracted power and losses versus the junction number are
shown in Fig. 19(b) for JA ¼ 30 A/cm2. A net increase in the
conversion efficiency is achieved in solar cell designs with a
larger number of junctions because the reduction of transmission and thermalization losses more than offsets the
increase in the spatial-relaxation and SRH recombination
losses caused by the decrease in the injection level of each
junction. A careful examination of Fig. 19(a) further reveals
that the benefit of having more junctions gradually disappears as the material quality is reduced.

In all of the preceding calculations, the AM1.5 G solar
spectrum is used. If the AM0 (Ref. 26) spectrum is used, the
principles remain the same, but the results will be different.
In general, the calculated curves for AM0 are much
smoother than that for AM1.5 G.
This work extends existing detailed balance models for
solar cells by considering the impact on performance of: i)
SRH and Auger recombination, ii) optical design in terms of
light trapping, absorption, and reflection, iii) less than ideal
photon recycling, and iv) the non-ideal absorption of solar
radiation and the spontaneous emission by tail states.
Although it is possible to further incorporate more extrinsic
loss mechanisms, such as surface recombination and resistance, these extrinsic mechanisms are device structure dependent and are not compatible with this generic model that
only takes into account the fundamental parameters of bulk
materials. In general, this work provides the achievable limits for solar cells made of practical materials and provides
clear device design principles and the related underlying
physics. Although not presented here, more practical design
principles can be accessed using this model, such as optimal
junction thicknesses of all junctions in a multi-junction cell
with non-ideal bandgaps that are not current matched in the
optically-thick limit.
It is necessary to point out that it takes only a few hours
to calculate all of the preceding diagrams using a personal

FIG. 18. Efficiency of four-junction solar cell vs AM1.5 G solar concentration
with SRH recombination current density,
JA ¼ 30 A/cm2. (a) Power conversion efficiency (left-hand y-axis) and optimal
bandgap energy for each junction (righthand y-axis). (b) Solar power extracted
and lost (y-axis).

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J. Appl. Phys. 110, 123104 (2011)

FIG. 19. Solar cell efficiency vs number
of junctions. (a) Optimal efficiency for
various SRH recombination current densities, JA. (b) Solar power extracted and
lost for JA ¼ 30 A/cm2.

computer. Normally, the calculation time is much longer if
sophisticated numerical simulation tools are used to reproduce these diagrams. It is known that the complexity of the
solar cell design dramatically increases with the junction
number since junctions are made of dissimilar materials with
different material parameters. For a given practical multijunction solar cell design, the optimization processes are
time-consuming because the necessary current-matching
requirement requires a large number of iterations to solve the
drift and diffusion equations for all of the junctions to find
the optimal bandgap energies and layer thicknesses.30 In this
regard, this model offers a powerful approach to gathering
initial device design parameters such as bandgap and layer
thicknesses. In addition, the principles and equations used in
this study are significant for other optoelectronic devices.
For example, the equations used to calculate emittance are
suitable for LED device design without the use of complex
ray-tracing simulations.
V. CONCLUSIONS

A semi-analytical model for photovoltaic singlejunction and multi-junction solar cells is established that provides insight into the intricate workings of ideal and practical
solar cells by taking into account the impact of photon recycling, spontaneous emission coupling, non-radiative recombination, and non-ideal step-like absorptance and emittance
due to the presence of tail states in real materials. Four types
of fundamental losses for solar cells are discussed, including
transmission, thermalization, spatial-relaxation, and recombination. To quantitatively analyze these losses, the average
absorptance and emittance are defined and their trends versus
junction thickness are determined for single-junction GaAs
and Si solar cells for four different structures with: i) a
smooth top surface and an absorbing substrate which typifies
a subcell within a multi-junction solar cell, ii) a smooth top
surface and a reflecting substrate (i.e., a reflecting back surface), iii) a textured top surface and an absorbing substrate,
and iv) a textured top surface and a reflecting substrate which
typifies a high performance single-junction solar cell.
The contribution of the width of the Urbach tail is found
to be significant to the overall emission since it is on the
order of the width of the spontaneous emission spectrum,

while it is insignificant to the overall absorption of solar radiation since it is small compared to the width of the solar
spectrum. The SRH, radiative, and Auger recombination saturation currents (JA , JB , and JC ) are analyzed based upon
published material parameters for several common semiconductors. The SRH recombination saturation current density
per unit length, JA =d, for the best reported material quality
varies from 0.03 to 300 A cm2 lm1 for various group IV
and III-V semiconductors and has no clear trend versus the
bandgap energy. The radiative recombination saturation current density, JB , increases monotonically with the bandgap
energy and is substantially lower for structures with reflecting back surfaces compared to that for structures with
absorbing substrates. The Si has the largest Auger recombination saturation current density per unit length due to its
large effective densities of states.
The optimal thickness of single-junction solar cells
decreases as the material quality decreases due to a trade-off
between transmission and SRH recombination losses. For
both GaAs and Si solar cells using the best reported material
quality, SRH is the dominant bulk recombination loss. Moreover, whenever the SRH saturation current is greater than
10 A/cm2, SRH recombination dominates, and whenever the
SRH saturation current is less than 1 A/cm2, radiative recombination dominates for GaAs and Auger recombination dominates for Si.
The energy conversion efficiency and optimal junction
thickness increases with the solar concentration. The efficiency improvement is mainly due to a reduction in spatialrelaxation. As the solar concentration increases from 100 to
1000 suns, there is a transition in the dominant loss mechanism from the SRH to radiative in GaAs and the SRH to Auger in Si. Moreover, in multi-junction solar cells, as the solar
concentration increases the optimal bandgaps decrease and
as the SRH recombination current density increases, the optimal bandgaps increase. Furthermore, the efficiency increase
with junction number is due to a substantial decrease in the
transmission and thermalization losses. However, since the
spatial-relaxation and SRH recombination losses increase
with the junction number, the efficiency gain eventually saturates as the diminishing reduction in the transmission and
thermalization losses approaches the increase in the spatialrelaxation and SRH losses. For a four junction solar cell

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structure, the optimal top-junction bandgap is 2.02 or 2.10
eV, which results in an efficiency of around 44% under
AM1.5 G one sun.
ACKNOWLEDGMENTS

This work is supported by the Science Foundation Arizona Contract Nos. SRG 0190-07 and SRG 0339-08, the Air
Force Research Laboratories/Space Vehicles Directorate
Contract No. FA9453-08-2-0228, and the National Science
Foundation Grant No. 1002114. The authors thank Swee H.
Lim, Shi Liu, Jing-Jing Li, and Shuming Wang for valuable
discussions on some of the topics in this paper.
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