Modelling of Perovskite Solar Cells · 2018-06-18 · 1.2 Solar Cells The solar cell is an important energy conversion device harnessing solar energy. The first solar cell was fabricated

Modelling of Perovskite Solar Cells

RONGSHENG WEI Master Degree

Submitted in fulfilment of the requirement for the degree of Master of

Engineering (Research)

FINAL THESIS

SCHOOL OF CHEMISTRY, PHYSICS AND MECHANICAL ENGINEERING

SCIENCE AND ENGINEERING FACULTY

Queensland University of Technology

2018


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CONTENTS

STATEMENT OF ORIGINAL AUTHORSHIP ............................................................................ 3

ACKNOWLEDGEMENT ............................................................................................................... 4

ABSTRACT .................................................................................................................................... 5

THE SUPERVISORS AND THEIR CREDENTIALS .................................................................... 7

CHAPTER 1. INTRODUCTION .................................................................................................. 8

1.1 Energy Issues ................................................................................................................................... 8

1.2 Solar Cells ....................................................................................................................................... 10

1.2.1 Working Mechanism in Solar Cells .............................................................................................. 10

1.2.2 Main Parameters in Solar Cells ..................................................................................................... 12

1.3 Perovskite Solar Cells ................................................................................................................... 13

1.3.1 Perovskite Materials ......................................................................................................................... 15

1.3.2 Development of Perovskite Solar Cells ...................................................................................... 17

1.4 SCAPS-1D Based Numerical Simulation .................................................................................. 18

1.4.1 Basic Semiconductor Physics in SCAPS-‐1D .............................................................................. 20

1.4.2 Grading Models in SCAPS-‐1D ...................................................................................................... 23

1.4.3 Operation Theory ............................................................................................................................. 24

1.4.4 Strengths and Limitations .............................................................................................................. 25

1.5 Research Objective and Outline ................................................................................................. 26

CHAPTER 2. NUMERICAL MODEL FOR PSCs ..................................................................... 28

2.1 Introduction ................................................................................................................................... 28

2.2 Physics Model in PSCs ................................................................................................................. 30

2.2.1 Basic Equations .................................................................................................................................. 30

2.2.2 Generation and Recombination Mechanism ........................................................................... 32

2.3 Numerical Simulation Method .................................................................................................... 34

2.3.1 Discretization of Equations ............................................................................................................ 34

2.3.2 Linearization of Equations .............................................................................................................. 36

2.4 Results and Discussion ................................................................................................................. 39

2.4.1 The Effect of Effective Density of State ...................................................................................... 39

2.4.2 The Effect of Relative Dielectric Permittivity ............................................................................. 43


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2.4.3 The Effect of Band Gap Energy .................................................................................................... 46

2.5 Conclusion ...................................................................................................................................... 49

CHAPTER 3. SCAPS SIMULATION FOR HTM LAYER IN PSCs ......................................... 50

3.1 Introduction ................................................................................................................................... 50

3.2 Device Model .................................................................................................................................. 52

3.3 Results and Discussion ................................................................................................................. 56

3.3.1 The Influence of HTM Layer Characteristics ............................................................................. 56

3.3.2 Comparison for Different Hole Transporting Materials ........................................................ 59

3.4 Conclusion ...................................................................................................................................... 62

CHAPTER 4. SUMMARY AND FUTURE WORKS ................................................................. 63

4.1 Summary ........................................................................................................................................ 63

4.2 Future Works ................................................................................................................................. 63

REFERENCES ............................................................................................................................... 65

APPENDIX ................................................................................................................................... 72


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STATEMENT OF ORIGINAL AUTHORSHIP

The work contained in this report has not been previously submitted to meet requirements for

an award at this or any other higher education institution. To the best of my knowledge and

belief, the report contains no material previously published or written by another person except

where due reference is made.

Signature:

Date:

QUT Verified Signature


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ACKNOWLEDGEMENT

First of all, I would like to thank my supervisor, Associate Professor Hongxia Wang. She is

very professional, and she has given me a considerable amount of help during my study at QUT.

I really appreciate her sharing her knowledge of perovskite solar cells and discussing questions

about physical models with me. She also gave me many valuable suggestions regarding my

project and thesis. I also would like to thank Professor Aijun Du and Fawang Liu for giving

me the guidance regarding my thesis and numerical modelling. As well, I would like to thank

my research group members, including Shengli Zhang, Teng Wang, Nima and Disheng Yao.

My sincere thanks also go to Libo Feng, who is a member of another group. All of them gave

me lots of assistance during my project and I have really appreciated their generosity. I

acknowledge the services of professional editor, Diane Kolomeitz, who provided copyediting

and proofreading services, according to the guidelines laid out in the university-endorsed

national ‘Guidelines for editing research theses’. Finally, I want to thank my family members

for their ongoing support, both financially and emotionally. Without them, I would not have

achieved what I have got today.


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ABSTRACT

Perovskite solar cells have become a hot topic in the solar energy device area. With 10 years

of development, the energy conversion efficiency has seen a great improvement from 2.2% to

more than 22%, and it still has great potential for further enhancement. Numerical simulation

is a crucial technique in deeply understanding the operational mechanisms of solar cells and in

predicting the maximum value of a solar cell with controlled design. This technique can also

give guidance on the structure optimisation for different devices. In this project, two main areas

of research have been discussed. The first part of this thesis illustrates a numerical model of

perovskite solar cells (PSCs) by using the Matlab program. It also introduces a specific

computation process for this model. This model is used to study semiconductor physics in PSCs

and investigate the effect of three parameters on device performance: density of state, relative

dielectric permittivity and band gap energy. The simulation results reveal that a large value of

effective density state can decrease both short-circuit current (Isc) and open-circuit voltage (Voc)

in particular on Voc. A large value of relative dielectric permittivity can cause a large Voc and

Isc, but the effect is not significant. Larger band gap energy is beneficial for the Voc and Isc

without considering reduction of light absorption. The second part shows a model of a

perovskite solar cell with the structure of glass/ FTO/ TiO2/ CH3NH3PbI3/ HTM/ Au by using

SCAPS-1D software. The influence of hole transport material layer characteristics, including

hole mobility and band gap offset, on the performance of PSCs are investigated by using this

model with Spiro-MOeTAD. Besides Spiro-OMeTAD, two-hole transport materials based on

inorganic materials including CuO and Cu2O are also discussed. The simulation results show

that with the increase of hole mobility in the HTM layer, the value of PCE is enhanced. When


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hole mobility reaches 5e-3 cm2/Vs, PCE closes to the maximum value and the change reaches

a plateau. The device can obtain high performance when the value of the valance band gap

offset is between -0.2 eV and 0.2 eV. Furthermore, the device shows a better performance by

using inorganic materials (CuO, Cu2O) as the HTM layer, than by using spiro-MOeTAD,

especially to Cu2O with a PCE of 21.87%.

Key Words: Perovskite solar cell, Matlab, SCAPS-1D, Modelling


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THE SUPERVISORS AND THEIR CREDENTIALS

Principal Supervisor: Assoc. Prof. Hongxia Wang

Hongxia Wang is an Associate Professor of Energy and Process Engineering in the Faculty of

Science and Engineering of Queensland University of Technology (QUT). She is a member of

the Royal Australian Chemical Institute, MRACI, and a member of the Australian Photovoltaic

Institute. Her research areas focus on energy conversion and storage devices, semiconductor

materials, and the charge transport process.

Associate Supervisor: Prof. Aijun Du

Aijun Du is a Professor of Energy and Process Engineering in the Faculty of Science and

Engineering of Queensland University of Technology (QUT). He is a member of the Australian

Research Council Nanotechnology Network, American Nano Society and American Chemical

Society. His research areas are related to clean energy, nanoelectronics, and environment,

through the use of advanced theoretical modelling.


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CHAPTER 1. INTRODUCTION

1.1 Energy Issues

With the rapid development of the economy, the demand for energy is increasing. An

investigation from the World Energy Resources has predicted that global population will reach

to 8.1 billion in 2020, and total primary energy supply is expected to reach 17208 Mtoe in 2020

[1]. Shafiee and Topal have reported that the world’s reserves of oil, gas and coal will be nearly

used up after 35 years, 37 years and 107 years respectively [2]. In addition, a growing number

of research works have shown that the issues, such as pollution and global warming, are the

consequence of burning of fossil fuels [3]. Along with the issue of depletion of fossil fuels,

environmental pollution and global warming caused by burning of fossil fuels have raised

significant concerns as well. Thus, an urgent mission is to find an alternative source, which is

clean, renewable and sustainable, to replace fossil fuels. Solar energy is an infinite, clean and

flexible energy, and it can be converted into many other categories of energy for different

demands. Compared with other renewable energy resources such as hydropower, wind energy,

bioenergy, geothermal and nuclear energy, solar energy is more accessible and rich [4]. Solar

technology has developed rapidly in the past decades. It successfully overcomes the technical

difficulties for improving efficiency of energy conversion, meanwhile, it reduces the cost that

meets the requirements of commercialisation [5]. Fig.1.01 illustrates different energy

consumptions from 2005 to 2015; from this, it is clear that the proportion of solar energy

consumption has increased more than 40 times from 2005 to 2015. Furthermore, the cost of

electricity generated from the solar PV and solar thermal has reduced significantly compared

with 2010 and 2015, as shown in Fig.1.02 [6]. It is clear that solar energy as a renewable energy


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is promising to change the world’s energy pattern. It also has a lot of room to progress with the

continuous improvement of technology.

Fig.1.01The comparative energy consumption in 2005, 2010, and 2015 [1]

Fig.1.02 The tendency of global renewable energy cost of electricity in 2010 and 2015 [1]


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1.2 Solar Cells

The solar cell is an important energy conversion device harnessing solar energy. The first solar

cell was fabricated by using single silicon crystal in Bell labs, which showed an energy

conversion efficiency of 6% in 1954 [7]. The first commercial silicon solar cell was reported

in 1955, with a 2% efficiency. Because of the high cost and price, very few people actually

used the solar cell for home application, but it gradually began to be widely used in daily life

after 1973 [8]. A new type of solar cell with low fabrication cost and 10% efficiency was made

at the Institute of Energy Conversion of University of Delaware in 1980; it was called thin film

solar cell [9]. In 1982, a polysilicon solar cell was widely produced using a casting method, by

the Kyocera Corporation [10]. In 1991, Michael Gratzel and Brian O’Regan reported a new

type of solar cell called dye-sensitized solar cells (DSSCs), and further reducd the cost of

fabrication by 50% compared to silicon solar cells, by using solution processes in 1988 [11].

Since then, research on solar cells has entered the third generation. Perovskite solar cells (PSCs)

are derived from the research concept of dye-sensitized solar cells.

1.2.1 Working Mechanism in Solar Cells

Solar cells are made by semiconductor materials, which can generate electricity from sunlight

directly by using a photovoltaic effect. When a solar cell is exposed to light, a portion of the

photon with the energy larger than the bandgap is absorbed by the semiconductor. The

absorbed photons with sufficient excitation energy (E>Egap) can cause the transport of electrons

and holes; electrons in the conduction band and holes in the valence band move in different

directions, as shown in Fig.1.03 [12].


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Fig.1.03 Excitation and charge separation [12] Fig.1.04 Working principle of solar cell [13]

The carriers, which are generated around the P-N junction, reach the space charge region

without recombination. Due to the effect of an internal electric field, holes diffuse into the P-

type region, and electrons flow into the N-type region. As a result, there are excess holes in the

P-type region and electrons, which are stored in the N-type region, have the same situation.

Excess electrons and holes form an electrostatic field near the P-N junction, which has the

opposite direction to the potential energy barrier. The electrostatic field can not only offset

parts of the potential energy barrier effects, but also make the P-type region have positive

electricity and N-type region have negative electricity. As a consequence, the electromotive

force is created between these two regions [13]. If there is a circuit loop, current can be

produced. The complete process is shown in Fig.1.04.


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1.2.2 Main Parameters in Solar Cells

The performance parameters of a solar cell mainly refer to the output characteristics of the

device (I-V) including short-circuit current, open-circuit voltage, fill factor and power

conversion efficiency [14], as shown in Fig.1.05.

Fig.1.05 Output characteristics of the device (I-V) [14]

l Open-circuit voltage: The open-circuit voltage is the maximum output voltage of solar

cells. It can be obtained when the value of the output current is zero. Open-circuit voltage

can be expressed by Eq.1.1

𝑉"# =%&'(𝐿𝑛[ ,

,-+ 1] (1.1)

where, 𝑘2 is the Boltzmann constant, T is temperature, q is elementary charge, I is the

light-generated current density, I0 is the saturation current density.

l Short circuit-current: Set a solar cell under a standard light source, when the output is in

short circuit state, which means the voltage value is 0, the current is the maximum output

current called short-circuit current. It is given as:

𝐼4# = 𝐼 − 𝐼6(𝑒9:;&< − 1) (1.2)


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l Fill factor: This is the ratio of maximum output power to the open circuit voltage and short

circuit current. Fill factor is associated with maximum output power. A higher fill factor

leads to a greater output power. The value of fill factor depends on series resistance and

voltage. Fill factor is defined by using the equation below:

𝐹𝐹 = ?@ABC,DC

(1.3)

Where, Pm is the maximum output power. Voc is the open-circuit voltage. J is the short-

circuit current.

l Power conversion efficiency: Power conversion efficiency (PCE) is a significant

parameter. It is defined as the ratio of the maximum output power to the incident light

power, as shown in Eq.14.

𝜂 = ,DC∗GG∗ABC?HI

(1.4)

where, 𝑃KL is the incident light power. It can be easily seen from this equation that the

value of power conversion efficiency is determined by Isc, FF, Voc and Pin.

1.3 Perovskite Solar Cells

In 2012, Nature published an article of a perovskite solar cell with an efficiency over 10%.

Since then, perovskite solar cells have drawn a lot of attentions due to their high performance,

low cost of fabrication and great potential for commercialisation [15]. Generally, the structure

of perovskite solar cells consists of five layers, which are a metal-based cathode layer, hole

transporting material (HTM) layer, perovskite layer, electron transporting material (ETM)

layer and anode (FTO/ITO). A typical perovskite solar cell with planar structure is shown in


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Fig.1.06. The perovskite layer is used as a light absorber in the device, where photon excitation

occurs. The generated electrons and holes are separated and transferred to the ETM layer and

HTM layer respectively. Due to its ambipolar characteristic, perovskite has a high mobility of

both electrons and holes. The function of the ETM layer is to extract and transfer electrons and

block holes. For the HTM layer, it is used to extract and transfer holes and block electrons. It

means high mobility for electrons in the ETM layer, high mobility for holes in the HTM layer,

and appropriate band offsets between ETM layer/perovskite layer/HTM layer that are

necessary for high efficiency [16-20]. The main processes of charge transport in perovskite

solar cells are shown in Fig.1.07. The green arrows represent the favourable processes for

energy conversion including photon excitation, transportation of electrons from perovskite

layer to ETM layer, and transportation of holes from perovskite layer to HTM layer. The red

arrows indicate the undesirable processes, which cause energy loss, in perovskite solar cells

involving the recombination of charge carriers, back electron flow from the ETM layer and

hole flow from the HTM layer to perovskite layer [21].

Fig.1.06. The configuration of perovskite solar cell with planar structure [40]. Fig.1.07

Schematics showing of charge transport in perovskite solar cells [40]


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1.3.1 Perovskite Materials

Perovskite material is a type of material that shares similar crystal structure with calcium

titanate (CaTiO3) and can be expressed as ABX3 [22]. As shown in Fig.1.08, A is organic

ammonium, such as CH3- NH+3, B is metal cation, (e.g. Pb2+), X is halide anion (e.g. Cl, Br, I.

In the crystal structure, A ion is surrounded by eight three-dimensional structure of a corner

sharing octahedral BX6 units [23].

Fig.1.08 The Perovskite crystal structure [40]

The probable crystallographic structure of a perovskite material can be predicted by analysing

a tolerance factor t and an octahedral factor u. t can be expressed as Eq.1.01.

𝑡 = (𝑟O + 𝑟P)/( 2[𝑟2 + 𝑟P]) (1.01)

where, rA, rB and rX are the ionic radii of A, B and X respectively. The report shows that the

tolerance factors of most perovskite materials lie in the range from 0.75 to 1 [24]. However, it

is not enough to deduce the probable crystallographic structure of perovskite materials by only


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considering tolerance factors. Therefore, the octahedral factor u is used as an additional

indicator to predict the formation of perovskite structure. U can be described by Eq.1.02.

𝑢 = 𝑟2/𝑟P (1.02)

Generally, the ABX3 perovskite structure can be formed under the conditions of 0.813< t

<1.107 and 0.442< u <0.895 [25]. The structure map by considering t and u for ABX3

perovskite materials is shown in Fig.1.09.

Fig.1.09 The structure map for ABX3 perovskite materials [25]

Compared with other materials, perovskite materials have their own unique characteristics such

as appropriate band gap energy (around 1.55eV), long carrier lifetime and diffusion length, and

high extinction coefficient [26-39]. Due to these favourable characteristics, perovskite

materials are widely applied as a light absorber in solar cell device. However, they also have

some limitations including low stability under moisture and ultraviolet radiation environments,

which are easily to result in degradation of performance, and the effect of toxicity problems,

caused from the toxicity ion of Pb2+ on both the environment and the human body during the

fabrication and disposal processes [30-31].


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1.3.2 Development of Perovskite Solar Cells

The first perovskite solar cell was reported by Miyasaka and colleagues in 2006. They used

CH3NH3PbBr3 as the material and obtained a solar cell with an efficiency of 2.2% [32]. Three

years later in 2009, they used iodine instead of bromine, and the efficiency achieved by the

solar cell rose to 3.8% [33]. Park and co-workers furthered the increase in efficiency to 6.5%

through the TiO2 surface treatment in 2011. The perovskite quantum dot (QD) sensitized solar

cell has a better light absorption compared with the dye-sensitized solar cell. However, the

stability of the perovskite QD-sensitized solar cell is low, because QD dissolves into electrolyte

solution [34]. In order to avoid the effects from electrolytes, Park and teammates attempted to

use solid state organic molecules or polymers as HTM to replace the liquid electrolytes. They

used Spiro-MeOTAD based HTM with mesoscopic TiO2 as ETM to improve the device

stability. They achieved a solar cell with an efficiency of 9.7% in 2012 [35]. In the same year,

Snaith and colleagues reported perovskite solar cells using Spiro-MeOTAD based HTM and

Al2O3 as scaffold. The device presented an efficiency of 10.9%. In their report, they showed

that a better performance can be obtained by using mixed-halide (CH3NH3PbI3-xClx) due to

improved ability of charge transport. They also showed that perovskites had a bipolar charge

carrier transport of electron and hole [36]. In 2013, Seok, Grätzel and colleagues reported an

efficiency of 12.3% by using the structure of nanoporous TiO2 infiltrated by mixed-halide

perovskite [37]. Simultaneously, an efficiency over 15% was reported from Burschka and

group members by using TiO2 scaffold and iodide deposition [38]. Snaith removed scaffolding

from the device and applied planar structure, obtaining similar conversion efficiency compared

with the report by Burschka et al [39]. Subsequently, efficiencies of 16.2% and 17.9% were


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reported by Seok et al. in early 2014 by using CH3NH3PbI3-xClx and a poly-triarylamine HTM

[40-41]. In 2016, Saliba introduced a perovskite solar cell with an efficiency of 21.1% by using

a mixture of triple cation (Cs/MA/FA). It showed high stability and reproducibility [42]. Seok

and co-workers introduced an approach to reduce defects in the perovskite layer by using an

intramolecular exchanging process in 2017, which is favorable in reducing the defects

concentration, and they obtained an efficiency of over 22% [43].

1.4 SCAPS-1D Based Numerical Simulation

Numerical simulation is a crucial and efficient way to investigate the physical mechanism in a

solar cell device without actually making the device. It can save both time and money in device

development. Much simulation software has been developed and applied in the research of

solar cell devices such as AMPS-1D, SCAPS-1D, PC1D, AFORS-HET and so on. SCAPS-1D

is a one-dimensional simulation software developed by the University of Gent, Belgium. It has

been applied to the study of different types of solar cells such as CZTS, CdTe, CIGS, etc. [43-

47]. Compared with other software, SCAPS has a very intuitive operation window and

diversified models for grading, defects and recombination. The main features of SCAPS

including [48] materials and defects properties can be defined in 7 semiconductor layers, as

shown in Fig.1.10 and Fig.1.11, where plentiful grading laws are provided for almost all

parameters of materials and defects, and the defect definition can be set in both bulk and

interface. There are five defect types and distributions available in the software and a variety

of properties related with solar cells, such as energy bands, concentrations, currents, I-V

characteristics, C-V, C-f, and QE can be determined by SCAPS. SCAPS can also provide


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flexible calculation and record functions including single shot, batch calculation, curve fitting,

data and diagram recording.

Fig.1.10 Layer definition panel in SCAPS-1D

Fig.1.11 Material and defect definition panel in SCAPS-1D


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1.4.1 Basic Semiconductor Physics in SCAPS-1D

l Basic Equations

Poisson’s equation is used to describe the relationship between potential and space charges, as

shown in Eq.1.03

TU

TVU𝜑 𝑥 = (

Y[𝑛 𝑥 − 𝑝 𝑥 − 𝑁\](𝑥) + 𝑁O^(𝑥) − 𝑝_ 𝑥 + 𝑛_(𝑥)] (1.03)

where, 𝜑 is the potential, q is the elementary charge, 𝜀 is the permittivity, n is the density of

free electron, p is the density of free hole, 𝑁\] is the ionised donor-like doping density, 𝑁O^

is the ionised acceptor-like doping density, 𝑝_ is the trapped hole density, 𝑛_ is the trapped

electron density.

Continuity equations Eq1.04 -1.05 define the transportation of carriers

𝑞 TLT_= TbI

TV+ 𝑞𝐺 − 𝑞𝑅

𝑞 TeT_= − Tbf


(1.04) 𝐽L = 𝑞𝑛𝜇L

TjTV+ 𝑞𝐷L

TLTV

𝐽e = −𝑞𝑝𝜇eTjTV+ 𝑞𝐷e

TeTV

(1.05)

where, G is the optical generation rate, R is the recombination rate, 𝐷L is the electron diffusion

coefficient, 𝐷e is the hole diffusion coefficient, 𝜇L is the electron mobility, and 𝜇e is the

hole mobility.

l Concentration for Electrons and Holes

In thermal equilibrium, the free carrier concentrations are expressed by Eq.1.06-1.07

𝑛 = 𝑁#exp (opôC%&'

) (1.6)

𝑝 = 𝑁qexp (orôp%&'

) (1.7)

where, Ef is Fermi level, 𝑘2 is Boltzmann constant, T is temperature, Ec and Ev are energy

levels under a steady state.


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l Diffusion Length

Diffusion length describes the transport ability of carriers in a solar cell device. It depends on

diffusion coefficient and carrier lifetime, which is shown in Eq.1.09

𝐿 = 𝐷𝜏 (1.09)

where, L is diffusion length, 𝜏 is carrier lifetime.

l Recombination Mechanism

There are three recombination models used in SCAPS.

Band-to-band recombination is a reverse process of photon absorption. The electrons in the

conduction band drop back down to the empty valence band and recombine with holes. The

band-to-band recombination rate can be expressed as:

𝑅 = 𝛾(𝑛𝑝 − 𝑛Ku) (1.10)

The Shockley-Read-Hall (SRH) recombination is also called trap-assisted recombination. It

occurs due to the defects or impurities in the materials. The SRH recombination rate can be

given by:

𝑅 = Le^LHU

vf L]L- ]vI(e]e-) (1.11)

Auger recombination is a process when a pair of electron and hole recombination occurs during

the transition from high energy level to low energy level, with the resulting energy being given

to the third carrier. It can be described by:

𝑅 = (𝑐LO + 𝑐eO)(𝑛𝑝 − 𝑛Ku) (1.12)

where, R is recombination rate, 𝛾 is recombination coefficient, 𝜏L and 𝜏e are lifetimes for

electron and hole. 𝑛6 and 𝑝6 are equilibrium electron concentration and equilibrium hole

concentration, 𝑐LO and 𝑐eO are constants, which can be set in SCAPS.


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l Work Function

Work function is the minimum energy required to move an electron from a solid to vacuum.

The value of work function is used to describe the strength of binding energy of an electron in

materials. In SCAPS, work function can be set by the user, or it can be calculated by using the

model in SCAPS, as shown in Eq.1.13-1.15

n-type contact:

𝜙y = 𝜒 + 𝑘2𝑇𝑙𝑛(}~

}�^}�) (1.13)

p-type contact:

𝜙y = 𝜒 + 𝐸��e − 𝑘2𝑇𝑙𝑛(}~

}�^}�) (1.14)

intrinsic contact:

𝜙y = 𝜒 + 𝑘2𝑇𝑙𝑛(}~LH) (1.15)

where, 𝜙y is work function, 𝜒 is electron affinity, Nc is effective density of state for

conduction band, NA and ND are acceptor and donor dopant densities. Ni is intrinsic carriers’

densities.

l Absorption Coefficient

The absorption coefficient can be defined as the extent to which a material absorbs energy. It

is determined by the nature of material. In SCAPS, the absorption coefficient is given by

Eq.1.16

𝛼 𝜆 = 𝐴 + 2��

ℎ𝜈 − 𝐸��e (1.16)


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where, A and B are the absorption constants, h is Planck’s constant, v is light speed [49].

1.4.2 Grading Models in SCAPS-1D

In SCAPS, parameter grading is an important function which can make a model more flexible.

Most of the parameters can be graded in the material and defect definition panel with various

grading models provided in SCAPS. Each layer is assumed to have composition 𝐴�^�𝐵� . If

the layer is only made of pure compound A, it means y = 0, and y = 1when the layer is only

made of pure compound B. If the layer contains both compounds A and B, the value of y is

between 0 and 1. The grading y(x) is through the whole layer; y is required to be defined at

both sides of the layer. We can describe the distribution of composition though the layer by

using grading laws. The material properties P are composition dependence, which can be

expressed as P[y(x)]. SCAPS provides several grading laws, which are listed below [48], and

these laws can be defined in the grading selection.

Uniform:

𝑃O = 𝑃2

Linear:

𝑃O 𝑦2 − 𝑦 + 𝑃2 𝑦 − 𝑦O𝑦O − 𝑦2

Parabolic:

𝑃O 𝑦2 − 𝑦 + 𝑃2 𝑦 − 𝑦O𝑦2 − 𝑦O

− 𝑏(𝑦2 − 𝑦)(𝑦 − 𝑦O)

(𝑦2 − 𝑦O)u

Parabolic2:

𝑦 < 𝑦6: 𝑃6 + (𝑃O − 𝑃6)(𝑦 − 𝑦6𝑦O − 𝑦6

)u

𝑦 > 𝑦6: 𝑃6 + (𝑃2 − 𝑃6)(𝑦 − 𝑦6𝑦2 − 𝑦6

)u


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Logarithmic:

𝑃O�&^��&^�O×𝑃2

�^��&^��

Exponential:

𝑃6 + 𝑃O − 𝑃6sinh 𝑦2 − 𝑦

𝐿Osinh 𝑦2 − 𝑦O

𝐿O

+ 𝑃2 − 𝑃6sinh 𝑦 − 𝑦O

𝐿2sinh 𝑦2 − 𝑦O

𝐿2

Beta function:

𝑃O + (𝑃O − 𝑃2)𝛽�,�(𝑦 − 𝑦O𝑦2 − 𝑦O

)

Power law:

(𝑃O�y×

𝑦2 − 𝑦𝑦2 − 𝑦𝐴

+ 𝑃2�/y×

𝑦 − 𝑦O𝑦2 − 𝑦O

)y

where, b is bowing factor, P0 is background value, LA,B is characteristic lengths, 𝛽�,� is the

incomplete beta-function, m is power.

1.4.3 Operation Theory

The main theory of SCAPS-1D is to solve Poisson’s equation and continuity equations. Fig.1.2

shows the working strategy of SCAPS-1D. Each calculation begins from the start point, and

uses the initial assumption, which is expressed by using quasi-Fermi levels, to obtain the

equilibrium situation. In this situation, no illumination and voltage are applied. When the work

point is set in a dark condition, the equilibrium condition is applied to calculate the solution.

When it is set under a light condition, the short circuit situation will be calculated, and this new

value will be used as the initial value of the next step [48].


25

Fig. 1.12. Working Strategy of SCAPS-1D [48]

The convergence of the Gummel type iteration scheme with Newton-Raphson algorithm is

used in SCAPS for numerical calculations. After a calculation point is set, SCAPS will follow

Newton-Raphson algorithm to undertake iteration calculation until obtaining the optimisation

value.

1.4.4 Strengths and Limitations

The strengths of SCAPS-1D include the following features [50]:

l It has a clear user interface, and it is a kind of interactive software. Almost all parameter

settings can be saved as external ASCI code files and read from external files.

l It allows users to write scripting languages and run them.

l In the aspect of the device parameter setting, almost all parameters can be graded in

SCAPS-1D, which is beneficial to the study of the composite layer formed by the

interdiffusion between different layers.


26

l It considers the defect in bulk and interfaces and provides a variety in types of selection

of defect charge. The energy distribution caused by defects can be single level, uniform

distribution, Gauss distribution, and Band tail.

However, there are three limitations in the SCAPS-1D. Firstly, the device model can only add

seven semiconductor layers. Due to this reason, SCAPS-1D is not suitable for a device with

graded junctions. Another limitation is that SCAPS-1D is developed for thin film solar cells,

so it cannot be applied for other device structures, such as a perovskite solar cell with

mesoporous structure. The last limitation is about the accuracy of calculation. Although the

resolution is already a small number (<10-15), it is usually insufficient for the simulation of

semiconductors.

1.5 Research Objective and Outline

There are numerous studies related to silicon solar cell, thin film solar cell and organic solar

cell. However, few studies and models have been understood in perovskite solar cells.

Numerical analysis approach can give an assistance to understand PSCs deeply, and it is a

significant way to improve the research efficiency in PSCs. This project aims to 1) understand

the operation mechanism and physical model of perovskite solar cells, and 2) study the

mathematical approaches related to numerical simulation. Furthermore, it seeks to 3)

investigate the effect of HTM layer characteristics on the performance of perovskite solar cells,

and 4) study the perovskite device performance by using the inorganic materials (CuO and

Cu2O) as the hole transport materials. This project gives two simplified models of PSCs in

order to study PSCs more clearly and easily. Numerical simulations by using Matlab and


27

SCAPS are employed to study the absorber layer and HTM layer of perovskite solar cells.

Chapter 1 gives an introduction regarding the current world energy issues, solar cell device,

perovskite solar cells and SCAPS simulation program. Chapter 2 presents a numerical

simulation with specific computation processes accordingly for perovskite solar cells by using

Matlab to investigate the influence of effective density state, relative dielectric permittivity and

band gap energy on the device performance. Chapter 3 shows a model of perovskite solar cells

with the structure of glass/ FTO/ TiO2/ CH3NH3PbI3/ HTM/ Au by using SCAPS-1D software.

The influence of hole transport material layer characteristics on the performance of PSCs are

investigated. Furthermore, three different hole transporting materials, including CuO, Cu2O

and Spiro-OMETAD are also discussed. As for Chapter 4, it is about the works which are

required to extend in the future research.


28

CHAPTER 2. NUMERICAL MODEL FOR PSCs

2.1 Introduction

Numerical simulations have been applied to study the performance of solar cells over a period

of 40 years [51]. Jerry provided an effective analysis approach to discuss the carrier transport

problem of silicon solar cells by solving Poisson’s equation and continuity equations for

electrons and holes in 1975 [52]. Several software programs are widely used for thin film solar

cells, such as SCAPS-1D, PC1D and AMPS-1D, to understand the mechanism of solar cells in

recent years [53]. The numerical analysis for dye sensitized solar cells (DSCs) is also studied

and developed by many researchers [54-56]. With the development of these numerical models,

an enhanced understanding of the mechanisms in solar cells has been gained, which in turn

contributes to the development of the device towards improved efficiency.

However, the operation mechanisms of silicon solar cells, DSCs and perovskite solar cells are

different. The model of silicon solar cells can be seen as a p-n junction with doping, and it uses

the dopant density and distribution to describe the performance of solar cells [57]. For DSCs,

the redox level in electrolytes affects the output voltage of a device, so the boundary conditions

and initial value are based on it [58]. In the model of perovskite, the output voltage is

determined by the difference between quasi-Fermi energy level for electrons and holes at the

two electrodes of device [59]. Therefore, in order to obtain accurate results, the density of

carriers and their potential need to be expressed by using the model of quasi-Fermi energy level.

Furthermore, the boundary conditions and initial value need to be created by using the model

of quasi-Fermi energy level.


29

In this chapter, a numerical model based on Poisson’s equation and continuity equations for

perovskite solar cells has been developed. This model has been simplified as much as possible

under some appropriate assumptions in order to reduce the complication for computation. In

addition, the details of computation approaches and procedures are also introduced in this

chapter. The simulation results are obtained by using Matlab, and these results are used to

understand how the effective density of state, relative dielectric permittivity and band gap

energy affect the performance of perovskite solar cells.


30

2.2 Physics Model in PSCs

2.2.1 Basic Equations

In this section, a one-dimensional model is established by using Poisson’s equation and

continuity equations to describe the performance of perovskite solar cells. The effects of defect

density and ionised doping density are not considered in this model, so the Poisson’s equation

can be given by:

TU

TVU𝜑 𝑥 = (

Y[𝑛 𝑥 − 𝑝(𝑥)] (2.01)

where 𝜑 is the potential, q is the elementary charge, 𝜀 is the permittivity, n and p are the

density of free electrons and the density of free holes respectively.

The continuity equations of electrons and holes can be described by:

𝑞 TLT_= TbI

TV+ 𝑞𝐺 − 𝑞𝑅 (2.02)

𝑞 TeT_= − Tbf

TV+ 𝑞𝐺 − 𝑞𝑅 (2.03)

where 𝐽L is the electron current density, 𝐽e is the hole current density, G is the optical

generation rate, R is the recombination rate.

Under a stead state condition, electron and hole densities do not change with time, so the

continuity equations can be simplified as: TbITV= −𝑞𝐺 + 𝑞𝑅 (2.04)

TbfTV= 𝑞𝐺 − 𝑞𝑅 (2.05)


31

In a perovskite solar cell, the output current includes diffusion current and drift current caused

by electrons and holes. It can be obtained from the following equations:

𝐽L = 𝐽�K��4K"L + 𝐽��K�_ = 𝑞𝑛𝜇LTjTV+ 𝑞𝐷L

TLTV

(2.06)

𝐽e = 𝐽�K��4K"L + 𝐽��K�_ = −𝑞𝑝𝜇eTjTV+ 𝑞𝐷e

TeTV

(2.07)

where 𝐷L is the electron diffusion coefficient, 𝐷e is the hole diffusion coefficient, 𝜇L is the

electron mobility, and 𝜇e is the hole mobility. In this model, it is assumed that 𝐷L,e and 𝜇L,e

meet the Einstein relation, which can be expressed as

𝐷L,e = 𝜇L,e%&'(= 𝑉_ (2.08)

where 𝑘2 is the Boltzmann constant, and T is the temperature.

There are three unknowns to be solved, which are n, p and 𝜑. In order to obtain these values,

appropriate boundary conditions are required. At the cathode and anode, density of electrons

and holes obey the Boltzmann distribution, and they can be expressed as:

𝑛 0 = 𝑁#𝑒 C¡ pI¡;&< (2.09)

𝑝 𝐿 = 𝑁#𝑒 r¡ pf;&< (2.10)

in which, 𝑁# is the density of state, 𝐸�L and 𝐸�e are the electron and hole Quasi-Fermi level.

In perovskite solar cells, the output voltage is equal to the difference between 𝐸�L and 𝐸�e

without considering the work function of the metal electrode, and it can be expressed as:

𝜑 = 𝐸�L − 𝐸�e = 𝐸��e + 𝑘2𝑇𝑙𝑛L 6}C

+ 𝑘2𝑇𝑙𝑛(e(¢)}C) (2.11)

Here, in order to simplify the device model, it assumes that the output voltage value is the

bandgap energy of the perovskite, and the corresponding boundary conditions can be given by:


32

𝜑(¢) − 𝜑 6 = 𝑉� − 𝐸��e (2.12) 𝑛 6 = 𝑁#𝑛 ¢ = 𝑁# exp − o£¤f

A¥

(2.13)

𝑝 ¢ = 𝑁#𝑝 6 = 𝑁# exp − o£¤f

A¥

(2.14)

where, 𝑉� is the applied voltage.

2.2.2 Generation and Recombination Mechanism

In 2004, Hoppe showed that if the thickness of a device is very thin (under 100nm), the

exponential relationship of light absorption and distance do not affect the performance of

device significantly, so the generation rate can be set as a constant [60]. Here, the generation

constant used for perovskite is difficult to find but the absorption coefficient can be easily

found. Therefore, we still use the exponential model, and the charge generation rate can be

given by:

𝐺(V) = 𝐺 𝜆, 𝑥 𝑑𝜆 = 𝐼𝑃𝐶𝐸 𝜆 ,(¨)×©(¨)×ª¡«(¬)

�#/¨¨-6

¨-6 𝑑𝜆 (2.15)

where, IPCE is incident photon to current efficiency, 𝐼(¨) is the incident light density, 𝛼(¨) is

the absorption coefficient, h is the Plank constant, c is the light speed, 𝜆 is the photon’s

wavelength. Here, it assumes that IPCE is equal to 100%, which means all of the excitons can

be separated to pairs of hole and electron accordingly. So, the charge generation rate can be

rewritten as:

𝐺(V) = 𝐺 𝜆, 𝑥 𝑑𝜆 = ,(¨)×©(¨)×ª¡«(¬)

�#/¨¨-6

¨-6 𝑑𝜆 = 𝛼𝑁6𝑒^©V (2.16)


33

where, 𝐼(¨) is the AM1.5 standard solar spectra, which is calculated by NREL, 𝑁6 =

,(¨)�#/¨

¨-6 𝑑𝜆. Generally, the bandgap energy of perovskite used for theoretical study is 1.55ev.

So, using the linear fitting result of incident photon density and bandgap energy, 𝑁6 becomes:

𝑁6 ≈ −2.2×10�°×1.55 + 5.12×10�° = 1.71×10�° [61].

In the device, all of the carriers will move to their corresponding electrode, and when they

move, electrons and holes will try to recombine. There are two kinds of common recombination

mechanism, which are direct (band to band) and indirect (RSH) recombination. The direct

recombination is caused by the electrons’ direct transition between the conduction band and

valence band. Generally, it is used to describe the device with low defect concentration. It can

be expressed as [62]:

𝑅 = 𝛾(𝑛𝑝 − 𝑛KL_u ) (2.17)

𝑛KL_ = 𝑁#exp (−o£¤fuA¥

) (2.18)

𝛾 = (³Y

(2.19)

where, 𝛾 is the recombination coefficient, and 𝑛KL_ is the intrinsic carrier concentration. The

indirect recombination is used to describe the device with high defect concentration, and the

recombination occurs in the traps and defect centers. It can be given by [62]:

𝑅 = Le^LHI¥U

vf L]L´ ]vI(e]e´) (2.20)

where, 𝜏L is the lifetime of the electron, 𝜏e is the lifetime of the hole. In this model, it ignores

the trap and defect concentrations, so only considers the direct recombination mechanism.


34

2.3 Numerical Simulation Method

Poisson’s equation and continuity equations are partial differential equations, which are

difficult to solve directly by using conventional methods. Therefore, numerical computation

approaches are used in the simulation in order to obtain the numerical solutions. In this

simulation, the device works under small current and voltage conditions; the coupling between

the equations is not significant, so finite difference method and Gummel iteration method are

chosen to process these equations.

2.3.1 Discretization of Equations

Finite difference method is a discrete approximation computation approach for differential

equations; its calculation results are not a continuous function in the domain but the

approximate value of the functions at each mesh point. The basic idea of the difference method

is to use different coefficients to replace the derivative part in the equations. In this simulation,

it assumes the thickness of perovskite is L nm, and discretises the domain [0, L] with the

distance of h. Here, in order to obtain convenient calculation and accurate results, nested grid

method is used to discretise the device. The meshing diagram of the device can be shown in

Fig.2.01.

Fig.2.01 Meshing diagram of device


35

Based on the discretisation model of the device, the Poisson’s equation and continuity

equations with the discrete form can be described by:

j K]� ^j K ^[j K ^j(K^�)]�U

= 𝑛 𝑖 − 𝑝(𝑖) (2.21)

bI K ^bI(K^�)�

+ 𝑞𝐺 − 𝑞𝑅 = 0 (2.22)

bf K ^bf(K^�)�

− 𝑞𝐺 + 𝑞𝑅 = 0 (2.23)

where, k=1,2, 3, …, N-1, i=1, 2, …, N, k is the middle point between i-1 and I, N is the number

of the mesh point. The Scharfetter Gummel approach assumes the change of electric field,

carrier mobility and current density can be ignored between i and i+1, so that Eq.2.06 and

Eq.2.07 become:

𝐽L 𝑖 = ³I(�

𝜑 𝑖 − 𝜑 𝑖 + 1 L K]� ª¶ H ¡¶ H·´

:¥ ^L(K)

[ª¶ H ¡¶ H·´

:¥ ^�]

(2.24)

𝐽e 𝑖 = ³f(�

𝜑 𝑖 − 𝜑 𝑖 + 1 e K ª¶ H ¡¶ H·´

:¥ ê(K]�)

[ª¶ H ¡¶ H·´

:¥ ^�]

(2.25)

There are 3×(𝑁 − 2) unknowns (𝑛, 𝑝, 𝜑) after substituting Eq.2.24 and Eq.2.25 into the

Poison’s equation and continuity equations.

𝐹L 𝑛 𝑖 − 1 , 𝑛 𝑖 , 𝑛 𝑖 + 1 , 𝑝 𝑖 , 𝜑 𝑖 − 1 , 𝜑 𝑖 , 𝜑 𝑖 + 1 = 0 (2.26)𝐹e 𝑝 𝑖 − 1 , 𝑝 𝑖 , 𝑝 𝑖 + 1 , 𝑛 𝑖 , 𝜑 𝑖 − 1 , 𝜑 𝑖 , 𝜑 𝑖 + 1 = 0 (2.27)𝐹j 𝑛 𝑖 , 𝑝 𝑖 , 𝜑 𝑖 − 1 , 𝜑 𝑖 , 𝜑 𝑖 + 1 = 0 (2.28)

Here, we have 3×(𝑁 − 2) equations with the boundary conditions, so the numerical solutions

can be obtained.


36

2.3.2 Linearization of Equations

It can be easily seen that Jn, Jp and R are non-linear functions, so that Eq.2.26, Eq.2.27 and

Eq.2.28 are non-linear equations too. It is necessary to linearise these equations by using Taylor

expansion in order to simplify the computation. According to the theory of the Gummel

iteration method, when one variable is calculated we are assuming that other variables are

constants. So, when we calculate 𝜑, it needs to keep n and p as constants. Using the first order

Taylor expansion at 𝜑K^�6 , 𝜑K6, 𝜑K]�6 the Eq.2.28 can be rewritten as

𝐹j =TG¶-

Tj K^�∆𝜑 𝑖 − 1 + TG¶-

Tj K∆𝜑 𝑖 + TG¶-

Tj K]�∆𝜑 𝑖 + 1 + 𝐹j6 = 0 (2.29)

In order to ensure the convergence and the accuracy of results, the relationships between 𝜑

and n, p need to be considered:

𝑛 𝑖 = 𝑛KL_ exp𝜑 𝑖 − 𝐸�L 𝑖

𝑉_ (2.30)

𝑝 𝑖 = 𝑛KL_ exp𝐸�e 𝑖 − 𝜑 𝑖

𝑉_ (2.31)

Each coefficient in Eq.2.29 is shown below:

TG¶-

Tj K^�= �

�U

TG¶-

Tj K= − u

�U− (

Ye- K ]L- K

A¥TG¶-

Tj K]�= �

�U

𝐹j6 =��U𝜑6 𝑖 − 1 − u

�U𝜑6 𝑖 + �

�U𝜑6 𝑖 + 1 + (

Y𝑝6 𝑖 − 𝑛6 𝑖

(2.32)

Here, the chasing method is chosen to solve the equations. This is because the elements of

coefficient matrixes are very small, and some of them are approximately zero, where it is

possible to make the coefficient matrixes become a tri-diagonal matrix. In addition, these

matrixes meet the limitation conditions of the chasing method.


37

Solve the Poisson’s equation in an independent loop until 𝜑 meets the accuracy condition.

After getting new potential values, we substitute them into the continuity equations to calculate

n and p respectively. When n is calculated, maintain 𝜑 and p as constants. Using the first

order Taylor expansion at 𝑛K^�6 , 𝑛K6, 𝑛K]�6 the Eq.2.26 becomes:

𝐹L =TGI-

TL K^�∆𝑛 𝑖 − 1 + TGI-

TL K∆𝑛 𝑖 + TGI-

TL K]�∆𝑛 𝑖 + 1 + 𝐹L6 = 0 (2.33)

TGI-

TL K^�= ³I

�U[j K^� ^j K ]

¼½¾ ¶ H¡´ ¡¶ H:¥

^�

TGI-

TL K= − ³I

�U𝜑 𝑖 − 1 − 𝜑 𝑖

¼½¾ ¶ H¡´ ¡¶ H:¥

¼½¾ ¶ H¡´ ¡¶ H:¥

^�− ³I

�Uj K ^j K]�

¼½¾ ¶ H ¡¶ H·´:¥

^�− 𝛾𝑝6(𝑖)

TGI-

TL K]�= ³I

�U[𝜑 𝑖 − 𝜑(𝑖 + 1)]

¼½¾ [¶ H ¡¶(H·´):¥

]

¼½¾ ¶ H ¡¶ H·´:¥

^�

𝐹L6 =TGI-

TL K^�𝑛6 𝑖 − 1 + [ TGI

-

TL K+ 𝛾𝑝6(𝑖)]𝑛6 𝑖 + TGI-

TL K]�𝑛6 𝑖 + 1 + 𝐺 𝑖 − 𝑅(𝑖)

(2.34)

The first order Taylor expansion form of Eq.2.28 can be obtained by using the same approach,

as shown previously. The results are shown in Eq.2.35 and Eq. 2.36.

𝐹e =TGf-

Te K^�∆𝑝 𝑖 − 1 + TGf-

Te K∆𝑝 𝑖 + TGf-

Te K]�∆𝑝 𝑖 + 1 + 𝐹e6 = 0 (2.35)

TGf-

Te K^�= −³f

�U[𝜑 𝑖 − 1 − 𝜑(𝑖)]

¼½¾ [¶ H¡´ ¡¶(H):¥

]

¼½¾ ¶ H¡´ ¡¶ H:¥

^�

TGf-

Te K= ³f

�U𝜑 𝑖 − 𝜑 𝑖 + 1

¼½¾ ¶ H ¡¶ H·´:¥

¼½¾ ¶ H ¡¶ H·´:¥

^�− ³f

�Uj K^� ^j K

¼½¾ ¶ H¡´ ¡¶ H:¥

^�+ 𝛾𝑛6(𝑖)

TGf-

Te K]�= −³f

�U[j K ^j(K]�)]

¼½¾ ¶ H ¡¶ H·´:¥

^�

𝐹e6 =TGf-

Te K^�𝑝6 𝑖 − 1 + TGf-

Te K− 𝛾𝑛6 𝑖 𝑝6 𝑖 + TGf-

Te K]�𝑝6 𝑖 + 1 − 𝐺 𝑖 + 𝑅(𝑖)

(2.36)


38

After solving the continuity equations for electrons and holes, it is needed to be estimated

whether n and p meet the accuracy condition. If these solutions meet the accuracy condition,

the iteration loop is stopped and Jn and Jp are calculated. On the contrary, the new carrier

densities (n and p) are substituted into the Poisson’s equation and a new iteration is started until

all the solutions meet the accuracy condition. The Gummel iteration method flow diagram is

shown in Fig.2.02.

Fig.2.02 Gummel iteration method flow diagram


39

2.4 Results and Discussion

The parameters used in the simulation are summarised in Table.2.1. The parameters are

extracted from literature [63-66].

Table.2.1 Parameters used in the simulation

Parameters Symbol Value

Perovskite thickness L 100 nm

Band gap 𝐸��e 1.55 eV

Temperature T 300 K

Electron mobility 𝜇L 1 cm2/Vs

Hole mobility 𝜇e 2 cm2/Vs

Density of state Nc 2.2e18 cm-3

Relative dielectric permittivity 𝜀 30

Absorption coefficient 𝛼 5.7e4 cm-1

2.4.1 The Effect of Effective Density of State

Effective density of state is the number of states, which are available to be occupied, per energy

interval around the bottom of the conduction band and the top of the valance band. It assumes

that the effective density of states for conduction band and valance are the same in this

simulation. According to Eq.2.37, it is clear that the effective density of state depends on

effective mass, which is determined by the material, and the value of Nc can affect the density

of electrons and holes directly. So, in order to investigate the influence of effective density of

state for the device performance, 𝑁# is varied from 1018 cm-3 to 1021 cm-3, and other


40

parameters remain the same. The simulation result of I-V characteristics is shown in Fig.2.03,

and the values of Isc, Voc, and FF are summarised in Table.2.2

𝑁# = 2 (u¿y%&')ÀU

�-À (2.37)

where, 𝑁# is effective density of state, m is effective mass, 𝑘2 is the Boltzmann constant, T

is temperature, ℎ6 is the Planck constant.

Fig.2.03 The I-V curves for different value of Nc

Table.2.2 The values of Isc, Voc, and FF for different Nc

Nc/cm-3 Isc/mA/cm-2 Voc/V FF

1018 11.4 1.072 0.778

1019 11.2 0.939 0.761

1020 10.9 0.833 0.738

1021 10.7 0.701 0.747

It can be seen that Nc has a significant influence on the device performance in particular on Voc,

when it is changed. With the increase of Nc, the short-circuit current density slowly decreases,


41

the open-circuit voltage increases rapidly, and the fill factor changes irregularly. In order to see

the difference clearly and explain these simulation phenomena, the following discussions focus

on the simulation results of Nc=1018 cm-3 and Nc=1020 cm-3.

Since the carrier and potential distribution can reflect the electrical properties of a device,

therefore, the simulation based on these two aspects are carried out. Fig.2.04 and Fig.2.05 show

the carriers distribution in the device for Nc=1018 cm-3 and Nc=1020 cm-3 respectively. It can be

seen that the densities of minority carriers almost remain the same when Nc is changed from

Nc=1018 cm-3 to Nc=1020 cm-3 but the majority carriers increase significantly. It means that the

space charges go up with the increase of Nc, and the space charge effect becomes more

significant. This can be explained by the Poisson’s equation with the electric field form

TopHÁÂÃTV

= (Y(𝑛 − 𝑝).

Fig.2.04 Charge carriers’ distribution for Nc=1018 cm-3


42

Fig.2.05 Charge carriers’ distribution for Nc=1020 cm-3

Fig.2.06 illustrates the potential distribution for Nc=1018 cm-3 and Nc=1020 cm-3 respectively.

When Nc is small, the potential distribution is nearly linear. When Nc is large, the potential

changes rapidly at both sides of the device but it changes minimal in the middle of the device.

Because the electric field intensity is the derivative of the potential, so it can be easily seen that

Nc=1020 cm-3 has smaller electric field intensity than Nc=1018 cm-3 in the greater part of the

device. The voltage corresponding to the value of current is zero, and called open circuit

voltage. In this case, the value of the electric field intensity is zero. Therefore, carriers cannot

move in a controlled way in the same direction without the effect of the electric field.

Consequently, there is no current output. When the applied voltage is changed from 0 V to1.2

V with the same interval value, the smaller the electric field intensity, the easier it reduces to

zero. This is the reason why the open-circuit voltage goes down with the increase of Nc.


43

Fig.2.06 Potential distribution for Nc=1018 cm-3 and Nc=1020 cm-3

Although the electric field intensities are strong at both sides of device, it exists at a short

distance. The carriers move under a small electric field intensities in a long distance. It means

that the carriers cannot easily move through the device, and this is the reason why the short-

circuit current decreases with the increase of Nc.

2.4.2 The Effect of Relative Dielectric Permittivity

According to Poisson’s equation and continuity equations, it can be found that relative

dielectric permittivity (𝜀) is one of the crucial parameters affecting the potential and carrier

densities. Therefore, the simulation results based on relative dielectric permittivity dependent

performance are discussed in this section in order to understand the effect of relative dielectric

permittivity on the device performance. These results are obtained by varying 𝜀 from 5 to 9


44

without changing other parameters. Fig.2.07 shows the short-circuit current density increases

from 11.226 mA/cm2 to 11.289 mA/cm2 with the increase of 𝜀.

Fig.2.07 The effect of relative dielectric permittivity on current

This has the same tendency for open-circuit voltage and fill factor. Open-circuit voltage

increases gradually from 1.012 V to 1.0358 V, and fill factor increases from 0.7416 to 0.750,

as shown in Fig.2.08. But overall, the effect of relative dielectric permittivity on Isc, Voc and FF

is small.

Fig.2.08 The effect of relative dielectric permittivity on voltage and fill factor

Fig.2.09 depicts net generation rate for 𝜀=5 and 𝜀=9. It is clear that high relative dielectric

permittivity has a relatively large net generation rate. In this simulation, if the absorption

coefficient is fixed, the generation rate is the function of position. Therefore, a large net


45

generation rate can be obtained due to the reduction of recombination loss. This proves that

high relative dielectric permittivity can reduce the bimolecular recombination. It is also one

factor why short-circuit current goes up slightly with the increase of 𝜀.

Fig.2.09 Net generation rate for 𝜀=5 and 𝜀=9

The potential distribution for 𝜀=5 and 𝜀=9 is shown in Fig.2.10. The large value of 𝜀 has a

low rate of change for potential, compared with a small value of 𝜀 at both sides of the device,

which means that the electric field intensities are weak at these parts, but electric field

intensities are strong in the middle part of device. Therefore, the open-circuit voltage would

increase with the increase of 𝜀. Although the electric field is weak near the electrodes, it is

strong in the greater part of device, which is beneficial for the carriers moving though the

device, and this is another important factor to explain why short-circuit current goes up with

the increase of 𝜀.


46

Fig.2.10 Potential distribution for 𝜀=5 and 𝜀=9

2.4.3 The Effect of Band Gap Energy

Many reports show that the band gap energy of MAPbI3 is between 1.45 eV and 1.7 eV

[64,65,66]. As the material used as the absorber layer, the band gap energy of MAPbI3 has a

significant influence on device performance. Here, the effect of band gap energy on device

performance is investigated. The value of band gap energy is changed from 1.45 eV to 1.6 eV.

Fig.2.11 shows the I-V characteristic for different values of band gap energy. It is clear that

both short-circuit current and open-circuit voltage increases with the increment of band gap

energy, and the change of open-circuit voltage is more significant to compare with short-circuit

current. Open-circuit voltage increases from 0.97 V to 1.106 V, and the value of short-circuit

current rises from 11.34 mA/cm2 to 11.4 mA/cm2. The influence of band gap energy on fill


47

factor is displaced in Fig.2.12. With the increase of the value of band gap energy, the fill factor

decreases from 0.773 to 0.68.

Fig.2.11 The I-V curves for different value of band gap energy

Fig.2.12 The effect of band gap energy on fill factor


48

In the actual situation, short-circuit current should decrease slightly due to the reduction of

light absorption caused by the increase of the band gap energy. In this simulation, G is set as

constant following the exponential distribution, so it does not consider the reduction of light

absorption. Therefore, the enhancement of short-circuit current can be explained as the result

of the increase of the net generation rate. Broadening of the band gap energy can strengthen

the electric field, as shown in Fig2.13 It can be seen that the different potentials between the

electrodes increase with the increment of band gap energy, and the change rate of potential in

the device has the same tendency. This is the reason why open-circuit voltage goes up.

Fig.2.13 Potential distribution for different band gap energy


49

2.5 Conclusion

This chapter mainly introduces a numerical model for perovskite solar cell and a specific

computation process for Poisson’s equation and continuity equations. The simulation results

are obtained by using the Matlab program. The influence of effective density of state, relative

dielectric permittivity, and band gap energy on device performance is investigated. Through

the analysis of the simulation results, it is found that a large value of effective density state can

decrease open-circuit voltage significantly, meanwhile it can also decrease the short-circuit

current slightly. Thus, in order to obtain good device performance, it is necessary to choose a

small value of effective density state material or control the density of state of the material used

in the absorber layer. The device model with large value of relative dielectric permittivity

shows a high open-circuit voltage and short-circuit current performance, and the simulation

results also prove that high relative dielectric permittivity can reduce the bimolecular

recombination. In addition, with the increment of relative dielectric permittivity, the value of

fill factor increases slightly. Broader band gap energy is benefit for the open-circuit voltage

and short-circuit current, while it is not good for the fill factor.


50

CHAPTER 3. SCAPS SIMULATION FOR HTM LAYER IN PSCs

3.1 Introduction

The emergence of perovskite solar cells has attracted a lot of attention due to their high

efficiency. However, the unstable performance and expensive hole transport materials limit its

development [67]. Generally, organic materials (Spiro-OMETAD, PEDOT:PPS, MEH-PPV),

which have relatively high cost and low hole mobility, are used as HTM in perovskite solar

cells [68-69]. Some reports also showed that the HTM layer affects the device performance

and the degradation of device performance was observed by using organic material as HTM

[70-72]. Therefore, in order to commercialise the perovskite solar cells and dominate the

photovoltaic industry, it is necessary to find an alternative HTM layer for PSCs. Inorganic

materials are considered as promising materials for the HTM layer because of their favourable

characteristics, such as high mobility of hole, low cost of fabrication, better chemical stability,

and appropriate valance band position [73-75].

Numerical simulation can give assistance to studying solar cell devices. SCAPS-1D has been

developed by the University of Gent and has been widely used for thin film solar cells.

Perovskite solar cells with a planar structure have similar structure and excitation type

comparing with thin film solar cells. Therefore, many simulation-based PSCs have been also

built by using SCAPS. Hossain and co-workers built a simulation based on PSCs with ZnO as

the ETM layer by using SCAPS. They showed that the device with ZnO as the ETM layer can

obtain a high efficiency of 22.83% compared with TiO2 as the ETM layer [69]. The comparison

of different organic hole transport materials including NPB, MEH-PPV, PEDOTPSS, Spiro-


51

OMeTAD and P3HT was reported by Karimi and colleagues by using SCAPS. They showed

that the device with Spiro-OMeTAD as the HTM layer achieves the best performance [66]. A

simulation of lead-free CH3NH3SnI3 perovskite solar cells was provided by Du and colleagues.

They showed that the device based on lead-free CH3NH3SnI3 as the absorber layer obtained an

efficiency of 23.36% [76]. Minemoto and Murata investigated the effect of work function on

device performance by using SCAPS. Their simulation results illustrated that work function

matching is necessary to improve the device performance especially to Voc [77].

In this chapter, a numerical model based on a perovskite solar cell with the structure of

glass/FTO/TiO2/CH3NH3PbI3/HTM/Au is built by using SCAPS-1D software. The influences

of HTM layer characteristics on perovskite solar cells based on this model are discussed here.

Furthermore, the simulation results of two inorganic materials, Cu2O and CuO, as candidates

for the HTM layer, are also shown here to compare with Spiro-OMETAD, a standard HTM

used in PSCs.


52

3.2 Device Model

The device model is built by SCAPS-1D software. The configuration of the perovskite solar

cell used in this simulation is showed in Fig.3.1. It is a planar heterojunction structure

consisting of four layers. Following the direction of incident light, the device configurations

are front contact(TCO), SnO2:F(FTO), n type electron transportation layer(TiO2), perovskite

material layer(CH3NH3PbI3), p type hole transportation layer(Spiro-OMETAD,CuO,Cu2O),

and back contact (Au).

Fig.3.1 The structure of perovskite solar cell in the simulation

The main idea of SCAPS-1D is to solve Poisson’s equation and continuity equations for

electrons and holes:

𝜕u

𝜕𝑥u 𝜑 𝑥 =𝑞𝜀 [𝑛 𝑥 − 𝑝 𝑥 − 𝑁\](𝑥) + 𝑁O^(𝑥) − 𝑝_ 𝑥 + 𝑛_(𝑥)]

𝑞 TLT_= TbI


𝑞 TeT_= − Tbf



53

𝐽L = 𝑞𝑛𝜇L𝜕𝜑𝜕𝑥 + 𝑞𝐷L

𝜕𝑛𝜕𝑥

𝐽e = −𝑞𝑝𝜇e𝜕𝜑𝜕𝑥 + 𝑞𝐷e

𝜕𝑝𝜕𝑥

where, 𝜑 is the potential, q is the elementary charge, 𝜀 is the permittivity, n is the density of

free electron, p is the density of free hole, 𝑁\] is the ionised donor-like doping density, 𝑁O^

is the ionized acceptor-like doping density, 𝑝_ is the trapped hole density, 𝑛_ is the trapped

electron density, G is the optical generation rate, R is the recombination rate, 𝐷L is the electron

diffusion coefficient, 𝐷e is the hole diffusion coefficient, 𝜇L is the electron mobility, and 𝜇e

is the hole mobility.

Here, it needs to set characteristics’ parameters for each layer and device parameters in SCAPS.

The parameters used in the simulation are adopted from literatures, experimental work and

theoretical study. The main parameters of materials and defects are summarised in Table.3.1(a)

and Table.3.1(b) [63,65,78,79,80]:

Table.3.1(a) Materials parameters used in the numerical analysis

Parameters FTO TiO2 CH3NH3PbI3

Thickness(nm) 400 50 450

Acceptor density(𝑐𝑚^Æ) 0 0 0

Donor density(𝑐𝑚^Æ) 1019 1017 1013

Bandgap energy(eV) 3.5 3.26 1.55

Electron affinity(eV) 4 4.2 3.9

Relative dielectric permittivity 9 10 6.5

Mobility of electron 20 20 2

Mobility of hole 10 10 2

Defect density(𝑐𝑚^Æ) 1015 1015 2.5*1013


54

Table.3.1(b) Materials parameters used in the numerical analysis

Parameters Spiro-OMETAD CuO Cu2O

Thickness(nm) 150 150 150

Acceptor density(𝑐𝑚^Æ) 1018 1018 1018

Donor density(𝑐𝑚^Æ) 0 0 0

Bandgap energy(eV) 2.9 1.3 2.17

Electron affinity(eV) 2.2 4.07 3.2

Relative dielectric permittivity 3 18.1 7.11

Mobility of electron 10-4 0.1 80

Mobility of hole 10-4 0.1 80

Defect density(𝑐𝑚^Æ) 1015 1015 1015

Other parameters of materials such as effective density of state of conduction band and

effective density of state of valance band are set as 2.2*1018 cm-3, electron thermal velocity and

hole thermal velocity are set as 1.8*1018 cm-3, 107cm/s, respectively. In order to simplify the

device model, the absorption coefficient for each layer is set as 105 cm-1, and the defect type is

set as neutral, which means that the defect can lead to the Shockley-Read-Hall recombination

but cannot contribute to the space charge. The energetic distribution uses the Gaussian model,

and its characteristic energy uses the default value.

The main device parameters are shown in Table.3.2. The standard AM1.5G spectrum is used

in this simulation. The values of working point and numerical setting use the default value. The

scanning voltage is set from 0 V to 1.2 V. All the simulations will operate under these

conditions.


55

Table.3.2 Device parameters used in the numerical analysis

Back contact electrical properties

Thermionic emission/surface recombination

velocity of electron(cm/s)

105


velocity of hole(cm/s)

107

Metal(Au) work function(eV) 5.1

Front contact electrical properties


velocity of electron(cm/s)

107


velocity of hole(cm/s)

105

Work function of TCO(eV) 4.4

Operation temperature(k) 300


56

3.3 Results and Discussion

3.3.1 The Influence of HTM Layer Characteristics

The investigation of the influence of HTM layer characteristics on the device performance is

based on Spiro-MeOTAD, which is widely used in current perovskite solar cells as a standard

material. Hole mobility and valance band gap offset are two crucial parameters, which need to

be considered as criteria in the selection of hole transport materials. Here, these two parameters

are investigated in order to understand how they affect device performance.

3.3.1.1 Hole mobility

The influence of defect density in the HTM layer on device performance is not significant, so

a HTM layer without defect is applied here in order to see the effect of performance only caused

by hole mobility.


57

Fig.3.2 The device performance for different hole mobility

Fig.3.2 shows the simulation results by changing the value of hole mobility from 1e-4 cm2/Vs

to 1e-2 cm2/Vs. The value of Isc, FF and PCE gradually rises with the increase of hole mobility,

while the value of Voc decreases slightly. When the hole mobility reaches to 5e-3 cm2/Vs, the

change of Isc, Voc, FF and PCE tends to become stable, at this time, Isc is nearly 23.5 mA/cm2,

Voc is approximately to1.3141 V, FF closes to 65% and PCE is around 17%. A high hole

mobility can lead to the improvement of the conductivity and performance of the device. The

hole mobility of Spiro-MeOTAD is low, which is not good for the device performance, and

this is the reason why p-type doping to enhance hole mobility in the HTM layer is necessary.

Therefore, materials with large hole mobility should be considered for the selection of hole

transport material.

3.3.1.2 Band gap offset

As we know, the HTM layer is used to transfer holes from the perovskite layer to the HTM

layer while blocking the back charges flow from HTM layer to perovskite layer, so an

appropriate valance band position is important in the device design. The influences of valance

band offset on the device performance are investigated in this section. It is assumed that if

Eabsorber_vc-EHTM_vc>0, the band offset is called a positive band gap offset. Otherwise, the band

offset is called a negative band gap offset. Different values of band gap offset can be obtained

by changing electron affinity. Both positive and negative valance band offsets are varied from

0.1 eV to 0.5 eV by every 0.1 eV. The simulation results are showed in Fig.3.3 and Fig.3.4


58

Fig.3.3 Device performance for different values of positive valance band offset

Fig.3.3 shows the simulation results from different positive valance band gap offsets. In this

case, the valance band position of the HTM layer is lower than the perovskite layer. With the

increase of band gap offset, the higher potential barrier is formed between these two layers.

The high potential barrier means that the holes generated in the perovskite layer are difficult to

move to the HTM layer. From Fig.3.4, it can be seen that when the value of the positive valance

band gap offset goes up, the values of Isc and Voc are almost unchanged, while the value of FF

decreases, which causes the reduction of PCE. Furthermore, the reduction of PCE becomes

significant when the positive band gap offset is larger than 0.2 eV.

Fig.3.4 Device performance for different values of negative valance band offset


59

Fig.3.4 illustrates the simulation results from different negative valance band gap offsets. In

this situation, the valance band position of the HTM layer is higher than the perovskite layer.

It seems that the device can achieve a good performance with the enhancement of negative

valance band gap offset, because of the contribution from the electric field to the holes.

However, the results present an opposite phenomenon. It can be seen that the Isc is nearly kept

the same, but PCE, FF and Voc of the device go down with the increase of negative valance

band gap offsets. Moreover, the effect of negative valance band gap offset on Voc and PCE are

significant when the negative valance band gap offset is larger than 0.2eV.

3.3.2 Comparison for Different Hole Transporting Materials

Copper oxide (CuO) and Cuprous oxide (Cu2O) are P-type semiconductors. Both of them have

a relatively wide band gap. The band gap energy of CuO is 1.3 eV [81]. The band gap energy

of Cu2O is 2.17 eV [82]. The energy level diagram of CuO and Cu2O in the device is shown in

Fig.3.5. From Chapter 3.3.1, we know that in order to reduce energy loss and obtain high device

performance, it is expected to have a small valance band offset (-0.2 eV to 0.2 eV) between the

HTM layer and the perovskite layer. Thus, it can be seen from Fig.3.5, that both CuO and Cu2O

have an appropriate valance band position, and the values of valance band offset of CuO and

Cu2O are less than 0.1 eV. In addition, Cu2O exhibits a high hole mobility up to 80 cm2V-1s-1,

and CuO also has a relative high hole mobility 0.1 cm2V-1s-1 compared with organic materials

(Spiro-MeOTAD 0.0001 cm2V-1s-1). Therefore, for all these reasons that have been mentioned,

it is believed that CuO and Cu2O are the probable alternative materials that can be utilised in


60

HTM layers. The simulations are carried out by using three types of hole transporting materials

(CuO, Cu2O, and Spiro-MeOTAD) here.

Fig.3.5 Energy level diagram of CuO and Cu2O in the device

The simulation results are shown in Fig.3.6 and Table.3.3. Fig.3.6 illustrates the I-V

characteristics for the perovskite solar cell with different HTM layers including CuO, Cu2O

and Spiro-MeOTAD. Table.3.3 presents the results of device performance including Isc, Voc,

FF and PCE for different HTM layers. From the simulation results, it can be easily seen that

the device with CuO or Cu2O as HTM shows a better performance than Spiro-MeOTAD.

Especially using Cu2O as HTM, the PCE of the device reaches 21.87%. Furthermore, relative

high Isc 24.01 mA/cm3 and Voc 1.128 V are obtained by using Cu2O, because of its appropriate

band gap energy, valance band position and high hole mobility. These simulation results

provide evidence to prove that Cu2O and CuO are probable materials to be used as hole

transport material.

TiO2

CH3NH3PbI3 CuO

Cu2O

-4.4

-4.2

-7.46

-3.9

-5.45 -5.37

-4.07

-5.37

-3.2

-5.1

E(eV)


61

Fig.3.6 I-V characteristics for different HTMs

Table.3.3 Device performance for different HTMs

HTMs Isc mA/cm3 Voc V FF % PCE %

Spiro-MeOTAD 23.93 1.13 71.24 19.32

CuO 24.04 1.07 77.05 19.83

Cu2O 24.01 1.128 80.67 21.87


62

3.4 Conclusion

In this chapter, a device model with the structure of glass/FTO/TiO2/CH3NH3PbI3/HTM/Au

has been created by using SCAPS-1D software. The effects of HTM layer characteristics,

including hole mobility and valance band offset, have been discussed. Furthermore, the device

performance by using CuO, Cu2O and Spiro-MeOTAD as hole transport materials has also

been illustrated. The simulation results show that high value of hole mobility in the HTM layer

is good for the device performance. For Spiro-MeOTAD, when hole mobility reaches 5e-3

cm2/Vs, PCE closes to the maximum value, and the change of it tends to become stable with

the continuing increase of hole mobility. The value of valance band gap offset shows a crucial

influence on the device performance. Both positive band gap offset and negative band gap

offset can affect the device performance significantly. The results reveal that the band gap

offset between -0.2 eV and 0.2 eV can obtain a good device performance. Through the

comparison of CuO, Cu2O and Spiro-MeOTAD, it is found that the inorganic materials CuO

and Cu2O used as HTM, exhibit a better performance than Spiro-MeOTAD, and Cu2O achieves

a superior performance with a PCE of 21.87%.


63

CHAPTER 4. SUMMARY AND FUTURE WORKS

4.1 Summary

In this thesis, I illustrate the influence of effective density state, relative dielectric permittivity,

band gap energy on the device performance by using matlab model firstly. The simulation

results show that a large value of effective density state can decrease open-circuit voltage

significantly, meanwhile it can also decrease the short-circuit current slightly. A large value of

relative dielectric permittivity can lead to a large Voc and Isc, but the effect is not significant.

Broader band gap energy is benefit for the open-circuit voltage and short-circuit current, while

it decreases the fill factor. Then I show the influence of hole transport material layer

characteristics, including hole mobility and band gap offset, on the performance of PSCs and

comparing device performance with different HTM including CuO, Cu2O and Spiro-

MeOTAD by using SCAPS model. The simulation results show that high value of hole

mobility in the HTM layer is good for the device performance. The value of valance band gap

offset shows a crucial influence on the device performance. The device can obtain high

performance when the value of the valance band gap offset is between -0.2 eV and 0.2 eV. In

addition, the device shows a better performance by using inorganic materials (CuO, Cu2O) as

the HTM layer, than by using spiro-MOeTAD, especially to Cu2O with a PCE of 21.87%.

4.2 Future Works

This report gives two numerical models based on perovskite solar cells. One was built and

solved by myself, another was created by using SCAPS-1D directly. The performance effect

factors in the perovskite layer and HTM layer including effective density state, relative


64

dielectric permittivity, band gap energy, hole mobility, and valance band gap offset have been

discussed. However, there are still some problems that need to be solved. Firstly, the perovskite

solar cell model built by myself only considers perovskite layers with some reasonable

assumptions to simplify the physical equations. This model can only get a qualitative

explanation for limited effect factors. Thus, in order to investigate more factors with a

quantitative relation, the physical model needs to consider the effects from HTM layer and

ETM layer. Through the optimisation of the device model, the simulation results that can be

obtained are more accurate and valuable. Secondly, there are only two factors in the HTM layer

that have been investigated; other factors such as the thickness of device, doping density and

carriers’ lifetime are also worthy to be studied. Finally, this report only presents two kinds of

inorganic materials (CuO and Cu2O) used as HTM layer, but there are lots of inorganic

materials in the world that could be considered as likely materials to be used as the HTM layer,

such as NiO, CuI, and CuSCN. Consequently, there needs to be some further study and

researches into this area.


65

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72

APPENDIX

Matlab Code (main function) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% %%

%% Device Model Solver--Non-coupled algorithm %%

%% %%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clc;

clear;

format long e

%Fundamental and Material Constants%

E_gap=1.55; % E_gap

mu_n=1; % mu_n

mu_p=2; % mu_p

Nc=2.2e+18; % N_c

q=1.602e-19;

k=1.38e-23;

T=300;

V_t=k*T/q;

V_a=0;

n_int=Nc*exp(-E_gap/(V_t*2));

e0=30*8.85*10^-14;

L=100e-7;

delta=0.0001;

n_max=399;

%Grid Size%

N=401;

delx=L/(N-1);

%Initial Value%

ga=q*(mu_n+mu_p)/e0;

x=0:delx:L;

n0=Nc*exp(-E_gap/(V_t)*x/L);


73

p0=Nc*exp(-E_gap/(V_t)*(1-x/L));

% fei0=(V_a-E_gap)*(x/L-1/2);

fei0=(V_a-E_gap)*x/L;

G0=(9.71e21)*exp(-5.7e4*x);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% %%

%% Calculation Procedure - at Equilibrium %%

%% %%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Solve fi %

for i=1:N-2

a(i)=1/delx/delx;

b(i)=-2/delx/delx-(q/e0)*(n0(i+1)+p0(i+1))/(V_t);

c(i)=1/delx/delx;

f(i)=-(fei0(i)/delx/delx-2*fei0(i+1)/delx/delx+fei0(i+2)/delx/delx+q/e0*(p0(i+1)-n0(i+1)));

end

a(1)=0;

c(N-2)=0;

con_v=0;

while(~con_v)

fi=chase1(a,b,c,f);

fi0=zeros(1,N);

fi0(2:N-1)=fi;

X1=max(abs(fi));

if X1<delta

fei0=fei0+fi0;

con_v=1;

else

fei0=fei0+fi0;

for i=1:N-2


end

end


74

end

con_n=0;

while(~con_n)

% Solve n %

R=ga*(n0.*p0-n_int^2);

for i=1:N-2

b1(i)=-mu_n/delx/delx*(fei0(i)-fei0(i+1))*exp((fei0(i)-fei0(i+1))/V_t)/(exp((fei0(i)-

fei0(i+1))/V_t)-1)-mu_n/delx/delx*(fei0(i+1)-fei0(i+2))/(exp((fei0(i+1)-fei0(i+2))/V_t)-1)-

ga*p0(i+1);

a1(i)=mu_n/delx/delx*(fei0(i)-fei0(i+1))/(exp((fei0(i)-fei0(i+1))/V_t)-1);

c1(i)=mu_n/delx/delx*(fei0(i+1)-fei0(i+2))*exp((fei0(i+1)-fei0(i+2))/V_t)/(exp((fei0(i+1)-

fei0(i+2))/V_t)-1);

n1(i)=-((mu_n/delx/delx*(fei0(i)-fei0(i+1))/(exp((fei0(i)-fei0(i+1))/V_t)-1))*n0(i)+(-

mu_n/delx/delx*(fei0(i)-fei0(i+1))*exp((fei0(i)-fei0(i+1))/V_t)/(exp((fei0(i)-fei0(i+1))/V_t)-1)-

mu_n/delx/delx*(fei0(i+1)-fei0(i+2))/(exp((fei0(i+1)-fei0(i+2))/V_t)-

1))*n0(i+1)+(mu_n/delx/delx*(fei0(i+1)-fei0(i+2))*exp((fei0(i+1)-fei0(i+2))/V_t)/(exp((fei0(i+1)-

fei0(i+2))/V_t)-1))*n0(i+2)+G0(i+1)-R(i+1));

end

a1(1)=0;

c1(N-2)=0;

n=chase1(a1,b1,c1,n1);

n10=zeros(1,N);

n10(2:N-1)=n;

% Solve p %

for i=1:N-2

b2(i)=mu_p/delx/delx*(fei0(i+1)-fei0(i+2))*exp((fei0(i+1)-fei0(i+2))/V_t)/(exp((fei0(i+1)-

fei0(i+2))/V_t)-1)+mu_p/delx/delx*(fei0(i)-fei0(i+1))/(exp((fei0(i)-fei0(i+1))/V_t)-1)+ga*n0(i+1);

p1(i)=-((-mu_p/delx/delx*(fei0(i)-fei0(i+1))*exp((fei0(i)-fei0(i+1))/V_t)/(exp((fei0(i)-

fei0(i+1))/V_t)-1))*p0(i)+(mu_p/delx/delx*(fei0(i+1)-fei0(i+2))*exp((fei0(i+1)-

fei0(i+2))/V_t)/(exp((fei0(i+1)-fei0(i+2))/V_t)-1)+mu_p/delx/delx*(fei0(i)-fei0(i+1))/(exp((fei0(i)-

fei0(i+1))/V_t)-1))*p0(i+1)+(-mu_p/delx/delx*(fei0(i+1)-fei0(i+2))/(exp((fei0(i+1)-fei0(i+2))/V_t)-

1))*p0(i+2)-(G0(i+1)-R(i+1)));


75

a2(i)=-mu_p/delx/delx*(fei0(i)-fei0(i+1))*exp((fei0(i)-fei0(i+1))/V_t)/(exp((fei0(i)-

fei0(i+1))/V_t)-1);

c2(i)=-mu_p/delx/delx*(fei0(i+1)-fei0(i+2))/(exp((fei0(i+1)-fei0(i+2))/V_t)-1);

end

a2(1)=0;

c2(N-2)=0;

p=chase1(a2,b2,c2,p1);

p10=zeros(1,N);

p10(2:N-1)=p;

X2=max((abs(n10)+abs(p10))./(n0+p0));

if X2<delta

n0=n0+n10;

p0=p0+p10;

for k=1:N-1

Jn(k)=mu_n*q/delx*(fei0(k)-fei0(k+1))*(n0(k+1)*exp((fei0(k)-fei0(k+1))/V_t)-

n0(k))/(exp((fei0(k)-fei0(k+1))/V_t)-1);

Jp(k)=mu_p*q/delx*(fei0(k)-fei0(k+1))*(p0(k)*exp((fei0(k)-fei0(k+1))/V_t)-

p0(k+1))/(exp((fei0(k)-fei0(k+1))/V_t)-1);

end

AA=G0-R;

J=mean(Jn+Jp);

con_n=1;

else

n0=n0+n10;

p0=p0+p10;

for i=1:N-2



end

con_v1=0;

while(~con_v1)

fi=chase1(a,b,c,f);

fi0=zeros(1,N);


76

fi0(2:N-1)=fi;

X3=max(abs(fi));

if X3<delta

fei0=fei0+fi0;

con_v1=1;

else

fei0=fei0+fi0;

for i=1:N-2

f(i)=-(fei0(i)/delx/delx-2*fei0(i+1)/delx/delx+fei0(i+2)/delx/delx+q/e0*(p0(i+1)-

n0(i+1)));

end

end

end

end

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% %%

%% Calculation Procedure - at Non-Equilibrium %%

%% %%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Vscan=1.1;

interv=60;

Vinterval=Vscan/interv;

for j=1:interv+1

V_a=Vinterval*(j-1);

%Initial Value%

x=0:delx:L;

n0=Nc*exp(-E_gap/(V_t)*x/L);

p0=Nc*exp(-E_gap/(V_t)*(1-x/L));

fei0=(V_a-E_gap)*(x/L-1/2);

%Coefficient Matrix%

for i=1:N-2


77

a(i)=1/delx/delx;


c(i)=1/delx/delx;


end

a(1)=0;

c(N-2)=0;

con_v=0;

while(~con_v)

% Solve fi %

fi=chase1(a,b,c,f);

fi0=zeros(1,N);

fi0(2:N-1)=fi;

X1=max(abs(fi));

if X1<delta

fei0=fei0+fi0;

con_v=1;

else

fei0=fei0+fi0;

for i=1:N-2


end

end

end

con_n=0;

while(~con_n)

% Solve n %

R=ga*(n0.*p0-n_int^2);

for i=1:N-2

b1(i)=-mu_n/delx/delx*(fei0(i)-fei0(i+1))*exp((fei0(i)-fei0(i+1))/V_t)/(exp((fei0(i)-

fei0(i+1))/V_t)-1)-mu_n/delx/delx*(fei0(i+1)-fei0(i+2))/(exp((fei0(i+1)-fei0(i+2))/V_t)-1)-

ga*p0(i+1);

a1(i)=mu_n/delx/delx*(fei0(i)-fei0(i+1))/(exp((fei0(i)-fei0(i+1))/V_t)-1);


78

c1(i)=mu_n/delx/delx*(fei0(i+1)-fei0(i+2))*exp((fei0(i+1)-fei0(i+2))/V_t)/(exp((fei0(i+1)-

fei0(i+2))/V_t)-1);

n1(i)=-((mu_n/delx/delx*(fei0(i)-fei0(i+1))/(exp((fei0(i)-fei0(i+1))/V_t)-1))*n0(i)+(-

mu_n/delx/delx*(fei0(i)-fei0(i+1))*exp((fei0(i)-fei0(i+1))/V_t)/(exp((fei0(i)-fei0(i+1))/V_t)-1)-

mu_n/delx/delx*(fei0(i+1)-fei0(i+2))/(exp((fei0(i+1)-fei0(i+2))/V_t)-

1))*n0(i+1)+(mu_n/delx/delx*(fei0(i+1)-fei0(i+2))*exp((fei0(i+1)-fei0(i+2))/V_t)/(exp((fei0(i+1)-

fei0(i+2))/V_t)-1))*n0(i+2)+G0(i+1)-R(i+1));

end

a1(1)=0;

c1(N-2)=0;

n=chase1(a1,b1,c1,n1);

n10=zeros(1,N);

n10(2:N-1)=n;

% Solve p %

for i=1:N-2

b2(i)=mu_p/delx/delx*(fei0(i+1)-fei0(i+2))*exp((fei0(i+1)-fei0(i+2))/V_t)/(exp((fei0(i+1)-

fei0(i+2))/V_t)-1)+mu_p/delx/delx*(fei0(i)-fei0(i+1))/(exp((fei0(i)-fei0(i+1))/V_t)-1)+ga*n0(i+1);

p1(i)=-((-mu_p/delx/delx*(fei0(i)-fei0(i+1))*exp((fei0(i)-fei0(i+1))/V_t)/(exp((fei0(i)-

fei0(i+1))/V_t)-1))*p0(i)+(mu_p/delx/delx*(fei0(i+1)-fei0(i+2))*exp((fei0(i+1)-

fei0(i+2))/V_t)/(exp((fei0(i+1)-fei0(i+2))/V_t)-1)+mu_p/delx/delx*(fei0(i)-fei0(i+1))/(exp((fei0(i)-

fei0(i+1))/V_t)-1))*p0(i+1)+(-mu_p/delx/delx*(fei0(i+1)-fei0(i+2))/(exp((fei0(i+1)-fei0(i+2))/V_t)-

1))*p0(i+2)-(G0(i+1)-R(i+1)));

a2(i)=-mu_p/delx/delx*(fei0(i)-fei0(i+1))*exp((fei0(i)-fei0(i+1))/V_t)/(exp((fei0(i)-

fei0(i+1))/V_t)-1);

c2(i)=-mu_p/delx/delx*(fei0(i+1)-fei0(i+2))/(exp((fei0(i+1)-fei0(i+2))/V_t)-1);

end

a2(1)=0;

c2(N-2)=0;

p=chase1(a2,b2,c2,p1);

p10=zeros(1,N);

p10(2:N-1)=p;

X2=max((abs(n10)+abs(p10))./(n0+p0));

if X2<delta


79

n0=n0+n10;

p0=p0+p10;

for k=1:N-1

Jn(k)=mu_n*q/delx*(fei0(k)-fei0(k+1))*(n0(k+1)*exp((fei0(k)-fei0(k+1))/V_t)-

n0(k))/(exp((fei0(k)-fei0(k+1))/V_t)-1);

Jp(k)=mu_p*q/delx*(fei0(k)-fei0(k+1))*(p0(k)*exp((fei0(k)-fei0(k+1))/V_t)-

p0(k+1))/(exp((fei0(k)-fei0(k+1))/V_t)-1);

end

J(j)=mean(Jn(j)+Jp(j));

con_n=1;

else

n0=n0+n10;

p0=p0+p10;

for i=1:N-2



n0(i+1)));

end

con_v1=0;

while(~con_v1)

fi=chase1(a,b,c,f);

fi0=zeros(1,N);

fi0(2:N-1)=fi;

X3=max(abs(fi));

if X3<delta

fei0=fei0+fi0;

con_v1=1;

else

fei0=fei0+fi0;

for i=1:N-2


n0(i+1)));

end


80

end

end

end

end

end

V_a=0:1.1/60:1.1;

figure(4)

plot(V_a,J*1000,'LineWidth',2);

axis([0 1.1 0 15]);

xlabel('Voltage(V)');

ylabel('Current(mA/cm2)');

title('IV Characteristics');

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