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OPTICAL MODELING OF ORGANIC PHOTOVOLTAIC SOLAR CELLS

A Thesis

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

Maithili Ghamande

August, 2011 OPTICAL MODELING OF ORGANIC PHOTOVOLTAIC SOLAR CELLS

Maithili Ghamande

Thesis

Approved: Accepted:

Advisor Dean of the College Dr. Jutta Luettmer-Strathmann Dr. Chand Midha

Faculty Reader Dean of the Graduate School Dr. Robert R. Mallik Dr. George R. Newkome

Faculty Reader Date Dr. Yu-Kuang Hu

Department Chair Dr. Robert R. Mallik

ii ABSTRACT

Organic photovoltaic devices consist of several thin layers of material with different electro-optical properties. Since the conversion of incident to charge carriers occurs only in the active layers, the intensity distribution of within the device has an important effect on the efficiency of a cell. The intensity in turn depends upon properties of the layers, such as , absorption coefficient, and thickness, as well as on properties of the incident light, such as of incidence and spectral distribution. In this work, we investigate the absorption of light in thin-film organic solar cells with computational methods. Since interference effects play an important role in thin-films, we implement a transfer matrix method to calculate the complex amplitude of the electric field at the interfaces and propagate the electromagnetic wave within the layers. We apply the method to conjugated / bilayer solar cells and investigate devices of two planar geometries for the relevant part of the solar spectrum and a range of of incidence. Our results show that the angle of incidence has a small effect on the distribution of the electric field in the active layers for a wide range of angles. For normal incidence, we confirm that the thickness of one of the layers, the adjacent to the metal , has a large effect on the electric field distribution and find that the of light in the

iii active layers depends strongly on the of the incident light. A reweighting of the absorbance with the solar illustrates that optimizing the design of solar cells requires a compromise between materials properties and device geometry.

iv ACKNOWLEDGEMENTS

I would like to express my deepest gratitude to my advisor, Dr. Jutta Luettmer-

Strathmann, for her constant support, patience and especially her guidance through- out this work. I would also like to thank Dr. Hu and Dr. Mallik for being on my committee and giving valuable advice and encouragement. Special thanks to my re- search group members and my friend Kiran Khanal for valuable assistance. I am grateful to the faculty and staff in the Department of Physics for their direct or in- direct help during my study and research. Finally, I would like to thank my parents and sister, for their support during this work.

v TABLE OF CONTENTS

Page

LIST OF TABLES ...... viii

LIST OF FIGURES ...... ix

CHAPTER

I. INTRODUCTION ...... 1

1.1 Organic Solar Cells ...... 1

1.2 ...... 3

1.3 Thesis Outline ...... 6

II. MAXWELL’S EQUATIONS ...... 8

2.1 Maxwell’s Equation and Boundary Conditions ...... 8

2.2 Time Harmonic Maxwell’s Equations ...... 9

2.3 Electromagnetic Waves in Media ...... 10

III. LIGHT PROPAGATING THROUGH STRATIFIED MEDIA ...... 15

3.1 Boundary Conditions at a Plane Interface ...... 15

3.2 Transfer Matrix for a Single Interface ...... 19

3.3 Multilayer System ...... 24

IV. ORGANIC MODEL AND CHARACTERISTICS . . . . . 34

vi 4.1 Model ...... 34

4.2 Modeling Parameters ...... 36

V. RESULTS ...... 41

VI. SUMMARY AND CONCLUSION ...... 55

BIBLIOGRAPHY ...... 57

vii LIST OF TABLES

Table Page

3.1 Parameters for the example of an air/glass interface with light inci- dent from air with θi = 0...... 22 4.1 Thickness of layers in the solar cell for two different geometries. . . . . 39

viii LIST OF FIGURES

Figure Page

1.1 Schematic drawing of the working principle of an organic photo- voltaic cell. Illumination of the donor material through the trans- parent ITO electrode has created an , which diffuses to the interface between the donor material and the acceptor material. The curved arrow indicates the path of the exciton. At the interface, the exciton dissociates, and the is transferred to the acceptor material, leaving a hole in the donor material. These charged car- riers are then transported to and collected at their respective . 2

1.2 Standard AM 1.5 G solar irradiation spectrum [10]...... 5

1.3 Absorption coefficients of active materials (left axis) and the AM 1.5 standard solar spectrum (right axis). MDMO-PPV and P3HT are conjugated that act as donor materials, PCBM is a soluble derivative of the fullerene C60 and an acceptor material. Absorption of sunlight occurs only for of about 300 nm to 700 nm [11]. 6

1.4 Orientation of a solar cell array. A ray from the (solid line) at midday is normal to the panel when the panel is tilted by an angle β to the horizontal. For the state of Ohio, the angle is approximately 400. 7

3.1 Two media with indices of refraction n1 and n2 are separated by a plane interface perpendicular to the z-axis. An electromagnetic with wave vector k, electric field E, and magnetic field H is incident at point P of the interface with angle of incidence θi. The wave is partially reflected in the same medium with angle θr = θi and fields E0 and H0 and partially transmitted with angle of refraction 00 00 θt and fields E and H . The symbol n denotes the unit vector perpendicular to the interface, and k, k0, and k00 are the wave vectors of the incident, reflected, and refracted rays, respectively...... 16

ix 3.2 A planar interface I0ba at z = 0 separates media 0 and 1 with indices of refraction n0 and n1. A TE wave is incident on I0ba with ampli- tude E0bp and angle of incident θ0, and is reflected with amplitude E0bn. The transmitted wave has amplitude E0ap at the interface and angle of refraction θ1. E0i, E0r and E1t are the z-position dependent electric fields of the incident, reflected, and transmitted waves...... 20

0 3.3 A wave of wavelength λ = 480 nm traveling from z0 = 0 is incident on a planar interface at z = 0 between medium 0 (air) and medium 1 (glass) with angle of incidence θ0 = 0 and amplitude E0bp = 1. The lines in the top and bottom panel show the real and imaginary contributions of the total electric field amplitudes, respectively...... 23

0 3.4 A wave of wavelength λ = 480 nm traveling from z0 = 0 is incident on a planar interface at z = 0 between medium 0 (air) and medium 1 (glass) with angle of incidence θ0 = 0 and amplitude E0bp = 1. The incident (blue), reflected (red) and transmitted (black) intensities in medium 0 and 1 are shown...... 24 3.5 A TE wave is incident on the planar interface between medium 0 and medium 1 with complex amplitude E0bp. The wave is partially reflected in the same medium with amplitude E0bn and partially transmitted into medium 1 with amplitude E1ap. The wave is par- tially reflected from interface I1ba into medium 1 with amplitude E1bn and partially transmitted into medium 2 with amplitude E2ap. . . 25 3.6 Layer system consisting of ambient (0) and substrate (m) semi- infinite layers and m − 1 finite layers. Layer j has thickness dj, refractive index nj, and forward and backward traveling waves with amplitudes Ejbp and Ejbn, respectively, at the interface Ij+1. Ejap and Ejan are the corresponding amplitudes at the interface Ij...... 27

3.7 The absolute square value |E|2 of the electric field inside an air- glass-air system for normal incidence and light of wavelength λ = 480 nm...... 29

3.8 Reflected amplitudes E0r, E01r and E2r in the air (40nm)-glass (160nm)-air (50nm) system for a wavelength of λ = 480 nm and normal incidence. The top and bottom panel show the real and imaginary contributions, respectively...... 31

3.9 The absolute square value |E|2 of the electric field inside the air- glass-air system for θ0 = π/3 and light of wavelength λ = 480 nm. . . . 32

x 3.10 The absolute square value |E|2 of the electric field as a function of position z for a range of angles of incidence θ0 for the air-glass-air system. (a) The contour lines tagged by values represent lines of constant |E|2; the vertical lines show the interfaces. (b) In the col- ormap of |E|2, the shades represent values indicated in the colorbar on the right...... 33 4.1 The arrangement of layers in the solar cell. At the top is the ITO electrode, in the middle are the PEOPT and C60 active layers, and at the bottom is the aluminum electrode. A PEDOT layer is placed between ITO and PEOPT; light is incident from the ITO side...... 35

4.2 Chemical composition of the active material poly(3-(4’- (1”,4”,7”-trioxaoctyl)phenyl)thiophene) (PEOPT), of poly(3,4- ethylenedioxythiophne)-poly(styrenesulfonate) (PEDOT), and of the anion (PSS) [3] ...... 36 4.3 The absorption coefficient χ as function of wavelength λ for PEOPT (solid line) and C60 (dashed line) calculated from index of refraction data of Ref. [3]...... 37 4.4 Complex refractive indices of PEOPT and C60 as a function of wavelength λ. The nr (dashed lines) and ni (solid lines) represent the real and imaginary part of the refractive index n, respectively. [3] . 38 4.5 Complex refractive indices of ITO, PEDOT and Al as a function of wavelength. [1] The top and bottom panel show the real and imaginary parts, respectively...... 40

5.1 Distribution of the absolute square value of the electric field |E|2 normalized by the absolute square value of the incident electric field 2 |E0| as a function of position inside the solar cell for geometry (a) and geometry (b). The wavelength is λ = 460 nm and the angle of incidence is θ0 = 0 (normal incidence). The layers are, from left to right, ITO, PEDOT, PEOPT, C60, and Al...... 42 5.2 Distribution of the normalized absolute square value of the electric 2 2 field |E| / |E0| for a range of wavelengths from λ = 300 nm to λ = 700 nm and angle of incidence θ0 = 0 (normal incidence). In the colormap for geometry (a) and geometry (b), different shades corre- 2 2 spond to different values of |E| / |E0| as indicated in the colorbar on the side...... 44

xi 5.3 Distribution of the normalized absolute square value of the electric 2 2 field |E| / |E0| as a function of position inside the solar cell for 0 angle of incidence θ0 = 0 and the wavelengths λ = 340 nm (solid line), λ = 460 nm (dashed line), λ = 540 nm (dash-dotted line) for geometry (a) and geometry (b)...... 45 5.4 Distribution of the normalized absolute square value of the electric 2 2 field |E| / |E0| for geometry (a) and wavelength λ = 460 nm. The tagged lines in the contour plot (1) represent lines of constant values 2 2 of the normalized electric field |E| / |E0| for a range of angles of 0 0 incidence from θ0 = 0 to θ0 = 80 . In the colormap (2) different 2 2 shades correspond to different values of |E| / |E0| as indicated in 0 0 the colorbar on the side; the range of angles is θ0 = 0 to θ0 = 40 . . . 46 5.5 Distribution of the normalized absolute square value of the electric 2 2 field |E| / |E0| for geometry (b) and wavelength λ = 460 nm. The tagged lines in the contour plot (1) represent lines of constant values 2 2 of the normalized electric field |E| / |E0| for a range of angles of 0 0 incidence from θ0 = 0 to θ0 = 80 . In the colormap (2) different 2 2 shades correspond to different values of |E| / |E0| as indicated in 0 0 the colorbar on the side; the range of angles is θ0 = 0 to θ0 = 40 . . . 47 5.6 Distribution of the normalized absolute square value of the electric 2 2 field |E| / |E0| as a function of position inside the solar cell for wavelength λ = 340 nm and three angles of incidence for geometry 0 0 (a) and geometry (b); the angles are θ0 = 0 (solid line), θ0 = 30 0 (dashed line), θ0 = 60 (dash-dotted line)...... 49 5.7 Distribution of the normalized absolute square value of the electric 2 2 field |E| / |E0| as a function of position inside the solar cell for wavelength λ = 460 nm and three angles of incidence for geometry 0 0 (a) and geometry (b); the angles are θ0 = 0 (solid line), θ0 = 30 0 (dashed line), θ0 = 60 (dash-dotted line)...... 50 5.8 Distribution of the normalized absolute square value of the electric 2 2 field |E| / |E0| as a function of position inside the solar cell for wavelength λ = 540 nm and three angles of incidence for geometry 0 0 (a) and geometry (b); the angles are θ0 = 0 (solid line), θ0 = 30 0 (dashed line), θ0 = 60 (dash-dotted line)...... 51

xii 5.9 Scaled absorption function As defined in equation (5.1) as a function of position inside the PEOPT (230 nm - 270 nm) and C60 layers for 0 geometry (a) and geometry (b); the angle of incidence is θ0 = 0 , and the wavelengths are λ = 340 nm (solid line), λ = 460 nm (dashed line), and λ = 540 nm (dash dotted line)...... 53

∗ 5.10 Reweighted absorption function As defined in equation (5.2) as a function of position inside the PEOPT (230 nm - 270 nm) and C60 layers for geometry (a) and geometry (b); the angle of incidence is 0 θ0 = 0 , and the wavelengths are λ = 340 nm (solid line), λ = 460 nm (dashed line), and λ = 540 nm (dash dotted line)...... 54

xiii CHAPTER I

INTRODUCTION

1.1 Organic Solar Cells

Fossil fuels are non- sources and therefore depleted with time. To address the decrease in fossil fuels it is important to research and develop efficient renewable energy sources such as solar cells. Organic solar cells can be produced with low cost techniques in contrast to inorganic, -based solar cells

[1]. In addition, they may be fabricated on top of flexible substrates, which imparts mechanical flexibility to the devices [2].

The simplest organic solar cell consists of photocurrent generating active materials sandwiched between two layers acting as electrodes; a transparent (ITO) layer at the top and an aluminum (Al) layer at the bottom as shown in figure 1.1. ITO and Al are chosen as electrode materials because the significant difference in their work functions creates a high built-in electric field. Light enters from the transparent ITO side and travels to the active layer, which consists of a pair of donor-acceptor materials forming a p-n junction. The illumination of the active material results in the formation of so-called , which are bound electron- hole pairs. Both active layers have a characteristic band-gap energy and excitons

1 Figure 1.1: Schematic drawing of the working principle of an organic photovoltaic cell. Illumination of the donor material through the transparent ITO electrode has created an exciton, which diffuses to the interface between the donor material and the acceptor material. The curved arrow indicates the path of the exciton. At the interface, the exciton dissociates, and the electron is transferred to the acceptor material, leaving a hole in the donor material. These charged carriers are then transported to and collected at their respective electrodes.

are generated only if the photoactive material is excited by light of higher energy than the . The excitons diffuse inside the active material and dissociate into free charges, and holes, when they reach the donor-acceptor interface.

From there, the electrons enter the acceptor material and the holes enter the donor material and diffuse to their respective electrodes. Due to the built in electric field created by the difference in work functions of the electrodes, the electrons travel to

Al and the holes travel to ITO. Since excitons are short lived particles it is important that they reach the interface of the active layer in their lifetime in order to contribute to the photocurrent generation.

Light entering the solar cell leads to a non-uniform distribution of the optical electric field since interference effects play an important role in these devices. Since

2 the generation of excitons depends strongly on the local electric field [3, 4, 5] we investigate in this work how the electric field distribution is affected by the optical properties of the materials, the geometry of the device, and the wavelength and angle of incidence of the light.

The transfer matrix method was developed to analyze the propagation of electromagnetic waves in stratified media [6] and has been used recently to model the optical electric field inside the solar cell [3, 4, 7, 8]. Petterson et al. [3] applied the method to calculated the electric field distribution in thin-film devices and found that the thickness of the layer adjacent to the metal electrode has the largest effect on the efficiency of a solar cell. Since then, several groups have used the method to optimize layer dimensions for maximum absorption of light [4, 7]. In addition, Yang and

Forrest [8] modeled the electric field distribution to calculate the exciton generation rate from the optical field intensity and the wavelength-dependent material absorption coefficient and used the calculated exciton generation rate in computer simulations of solar cells. The transfer matrix method is not the only numerical method to calculate optical field distributions. Recently, Wang et al. [9] employed a finite-difference time- domain (FDTD) algorithm to investigate the system studied by Petterson et al. [3] and find very similar results.

1.2 Sunlight

The electrical current generated by photovoltaic devices is influenced by the spectral distribution of sunlight. Since the active materials in a solar cell have a characteristic 3 band gap, the portion of the solar light that organic solar cells absorb is limited.

For an efficient collection of photons, the absorption spectrum of the active organic layer should be in the high intensity part of the solar spectrum. Therefore, terrestrial solar spectral data are important for the development and design of photovoltaic devices. Since the solar spectral distribution varies with atmospheric conditions, time of day, and the angle at which sun rays reach the ground, it is useful to select a representative spectral distribution as a common basis. For this purpose, the American Society for Testing and Materials (ASTM) has developed irradiance standards. The standard “AM 1.5 Global for a 370 tilted surface” [10] is representative of the average conditions in the 48 contiguous states of the United

States. The air-mass, AM 1.5, gives an indication of the amount of which the light has passed through to reach the surface of the and a 370 tilted surface is appropriate for a geographic location of 41.80 . Figure 1.2 shows the AM

1.5 G solar irradiance spectrum used in this work, where the minima in the spectrum are due to absorption processes in the atmosphere.

Absorption coefficients of typical organic active materials together with the standard solar spectrum are presented in figure 1.3. The graphs show that these materials absorb light mostly in the 300 nm - 700 nm range of wavelengths [11]. The

figure illustrates an important problem of organic solar cells. The acceptor material

(PCBM, a soluble derivative of C60 [11]) absorbs predominantly at short wavelengths, when the solar irradiance is low, and neither donor nor acceptor materials can make use of the long-wavelength part of solar spectrum.

4 Figure 1.2: Standard AM 1.5 G solar irradiation spectrum [10].

Another important factor is the angle at which light from the sun reaches a device [12]. In general, the transfer of light into the device is best at normal incidence.

Therefore, a photovoltaic device needs to be mounted at an angle with respect to the horizontal to receive maximum solar radiation, where the angle is dependent on the geographical location. For example, at the equator a device should be mounted horizontally since the sunlight at midday is perpendicular to the ground. For higher , such as in the state of Ohio, the angle is around 400 as shown in figure 1.4.

To receive full solar radiation, the orientation of a needs to be adjusted continuously. Since most panels are not set up to do this, it is interesting to calculate the effect of the angle of incidence on the optical electric fields distribution in a solar cell.

5 Figure 1.3: Absorption coefficients of active materials (left axis) and the AM 1.5 stan- dard solar spectrum (right axis). MDMO-PPV and P3HT are conjugated polymers that act as donor materials, PCBM is a soluble derivative of the fullerene C60 and an acceptor material. Absorption of sunlight occurs only for wavelengths of about 300 nm to 700 nm [11].

1.3 Thesis Outline

The remainder of the thesis is organized in five chapters. Chapter II describes

Maxwell’s equation that form the foundation of . Chapter III gives a detailed description of light propagation in stratified media and introduces the transfer ma- trix method that we implement to model the solar cells described in Chapter IV. Our results are presented in chapter V and chapter VI contains a summary and discussion.

6 Figure 1.4: Orientation of a solar cell array. A ray from the sun (solid line) at midday is normal to the panel when the panel is tilted by an angle β to the horizontal. For the state of Ohio, the angle is approximately 400.

7 CHAPTER II

MAXWELL’S EQUATIONS

2.1 Maxwell’s Equation and Boundary Conditions

Electrodynamics in macroscopic media are described by Maxwell’s equations [13]

∂B ∇ × E = − , (2.1) ∂t ∇ · D = ρ, (2.2) ∂D ∇ × H = J + , (2.3) ∂t ∇ · B = 0, (2.4) where t is the time, E is the electric field, H is the magnetic field, J is the current density and ρ the charge density. In linear isotropic media, the fields D and B are related to the electric and magnetic fields through the relations

D = E, (2.5)

B = µH, (2.6)

Here  is the electric and µ is the magnetic permeability, which are scalar quantities for isotropic media.

8 2.2 Time Harmonic Maxwell’s Equations

To discuss electromagnetic fields in dispersive media, we focus on monochromatic light with angular frequency ω and write the real electromagnetic fields, E and H, as the real parts of complex quantities Ec and Hc

1 E(r, t) = Re[E (r, t)] = Re[E(r)e−iωt] = [E(r)e−iωt + E∗(r)eiωt], (2.7) c 2 1 H(r, t) = Re[H (r, t)] = Re[H(r)e−iωt] = [H(r)e−iωt + H∗(r)eiωt], (2.8) c 2 √ where r is the position, i = −1, Re denotes the real part of the expression in parentheses, and a * indicates complex conjugate. In the following, we assume that there are no free charges (ρ = 0) and that the current density and electric field are related by Ohm’s law

J = σE. (2.9)

Evaluating the time derivatives in Maxwell’s equation and using equations (2.5)–(2.6), we obtain

∇ × E = iωµH, (2.10)

∇ · E = 0, (2.11)

∇ × H = −iωE + σE. (2.12)

∇ · H = 0 (2.13)

9 2.3 Electromagnetic Waves in Media

To determine plane wave solutions of equation (2.10) –(2.13) we take the curl of

(2.10),

∇ × (∇ × E) = iωµ(∇ × H) (2.14) and use the identity

∇ × (∇ × E) = −∇(∇ · E) − ∇2E, (2.15) together with equation (2.11) we obtain

∇2E = −iωµ(∇ × H). (2.16)

Substituting the expression for ∇ × H from equation (2.12) we find

∇2E = −(µω2 + iωµσ)E. (2.17)

Defining

k2 = ω2µ + iσωµ, (2.18) we obtain

∇2E + k2E = 0. (2.19)

±ik·r Equation (2.19) has solutions of the form E(r) = E0e , where E0 and k are complex vectors. The plane wave solutions for Maxwell’s equation at a fixed angular frequency ω are obtained when we combine this result with equation (2.7)

−i(ωt±k·r) Ec(r, t) = E0e , (2.20)

10 where the sign of k·r depends on the direction of propagation. Similarly the magnetic

field has plane wave solution

−i(ωt±k·r) Hc(r, t) = H0e . (2.21)

From Maxwell’s equations (2.11) and (2.13) it follows that k · Ec = 0 and k · Hc = 0, which implies that Ec and Hc are perpendicular to the wave vector k. The solutions

Ec and Hc describe plane polarized light, where the direction of E0 determines the plane of .

Equation (2.18) for the wave vector suggests to introduce a complex electric

2 2 permittivity c, k = ω µc with

σ  =  + i . (2.22) c ω

The phase velocity of the wave is given by

ω 1 c r 1 v = = √ = with c = , (2.23) k µc n 0µ0 where c is the speed of light in and the index of refraction is defined as n = c/v. Since c is a complex quantity, both n and k are complex as well

r µ c n = = nr + ini (2.24) µ0 0

k = ωn/c = ω(nr + ini)/c. (2.25)

where nr and ni are the real and imaginary parts of the refractive index, respectively.

11 2.3.1 Poynting Vector and

The Poynting vector is defined as the cross product of the real electromagnetic fields

S = E(r, t) × H(r, t), (2.26) and describes the flow of energy per unit time across a unit area. The direction of the Poynting vector S is the direction of propagation of the wave.

In this work we are interested in layers of materials with different optical properties arranged perpendicular to the z axis. For the case of a wave propagating in the positive direction along the z axis, the complex electric field may be written as

±ωniz/c −iω(t±nrz/c) Ec(r, t) = E0e e , (2.27) where the sign in the first exponential has to be negative to describe the decay of the amplitude of the wave due to absorption of energy by the medium. With the definitions in equations (2.7), (2.8), and (2.26), the Poynting vector becomes

1 S = e−2ωniz/c E × H e−2iω(t±nrz/c) + E∗ × H e0 4 0 0 0 0

∗ ∗ 2iω(t±nrz/c) ∗ 0 +E0 × H0e + E0 × H0e . (2.28)

When we average this expression over time, the time-dependent terms yield zero and we obtain

1 1 hSi = e−2ωniz/c [E × H∗ + E∗ × H ] = e−2ωniz/cRe [E × H∗] . (2.29) 4 0 0 0 0 2 0 0

From the first Maxwell equation, (2.10), we have

k × Ec = ωµHc, (2.30) 12 which, together with equations (2.20)–(2.21) and the relations k·Ec = 0 and k·Hc = 0 yields for the amplitudes

H0 = (k/µω)E0 = (n/µc)E0, (2.31)

∗ 2 ∗ and for the cross product, E0 ×H0 = (1/ωµ) |E0| k . Substituting in equation (2.29) we find for the magnitude of the time-averaged Poynting vector

1 S = e(−2ωniz/c)n |E |2 . (2.32) 2µc r 0

This yields for the layer at position z

1 S(z) = n |E (z)|2 . (2.33) 2µc r c

For this geometry the so-called absorption distribution function is defined as the derivative −∂S/∂z [5] and satisfies

∂S A = − = χS(z), (2.34) z ∂z where χ is the absorption coefficient

χ = 2ωni/c. (2.35)

Combining equations (2.23), (2.34), and (2.35), we find for the absorption distribution function n 2π A = r χ |E (z)|2 = c n n |E (z)|2 . (2.36) z 2µc c λ 0 i r c

When a wave traveling in the stratified medium makes an angle θ with the z axis, the attenuation is still dependent only on the distance traveled along the z 13 axis [6]. Therefore, the plane wave solutions to Maxwell’s equations have complex amplitudes whose absolute values are constant in planes perpendicular to the z-axis

[14]. In this case, the expression for the average energy flow along z becomes

1 S(z) = Re(k ) |E (z)|2 . (2.37) 2ωµ z c

14 CHAPTER III

LIGHT PROPAGATING THROUGH STRATIFIED MEDIA

3.1 Boundary Conditions at a Plane Interface

Consider two media, with complex permittivity c1 and c2, magnetic perme- p p ability µ, and indices of refraction n1 = µc1/µ00, and n2 = µc2/µ00, separated by a plane interface as shown in figure 3.1. A plane wave with wave vector k is in- cident at point P on the interface. We draw our coordinate system such that the interface is in the x − y plane at z = 0 and the plane of incidence is the x − z plane.

The reflected and refracted waves have wave vectors k0 and k00, respectively. The electric and magnetic fields are described by complex quantities, see equation (2.20)–

(2.21). To ease notation, we drop the subscripts ‘c’ from now on and write E and H for complex field variables. The electric field vector points along the y-direction and the magnetic field points along the x-direction. In this way, E is perpendicular to the plane of incidence and parallel to the interface; the polarization is transverse electric

(TE). We write the waves on either side of the interface as

−i(k·r−ωt) E = E0e , (3.1)

0 0 −i(k0·r−ωt) E = E0e , (3.2)

00 00 −i(k00·r−ωt) E = E0e . (3.3)

15 Figure 3.1: Two media with indices of refraction n1 and n2 are separated by a plane interface perpendicular to the z-axis. An electromagnetic with wave vector k, electric field E, and magnetic field H is incident at point P of the interface with angle of incidence θi. The wave is partially reflected in the same medium with angle θr = θi 0 0 and fields E and H and partially transmitted with angle of refraction θt and fields E00 and H00. The symbol n denotes the unit vector perpendicular to the interface, and k, k0, and k00 are the wave vectors of the incident, reflected, and refracted rays, respectively.

To satisfy boundary conditions at z = 0, the phase of the waves must be the same at all points on the interface and for all times, which implies

k · r = k0 · r = k00 · r at z = 0 . (3.4)

Therefore, the wave vectors of the incoming, reflected, and transmitted waves all lie in the same plane. Introducing unit vectors kˆ and kˆ00 for the incident and transmitted

16 waves, respectively, we write

k · r = kkˆ · r, (3.5)

k00 · r = k00kˆ00 · r. (3.6)

With the aid of equation (3.4) and the components of the unit vectors in the x − z

ˆ ˆ00 plane, k = (sin θi, 0, cos θi), k = (sin θt, 0, cos θt), the phase of the incident and transmitted waves in the x − z plane is written as

00 k sin θix = k sin θtx at z = 0 . (3.7)

With the definition of the index of refraction in equation (2.25) we obtain Snell’s law for complex refractive indices n1, n2 and complex angles θi, θt

n1 sin θi = n2 sin θt. (3.8)

Because this relation is complex, sin θi and sin θr can no longer be interpreted physi- cally as simple angles except for the case of normal incidence [6]. Maxwell’s equations

(2.10)–(2.13) imply that the parallel components of E and H must be continuous at the boundary, z = 0. With the relation between E and H given by equation (2.30) we find

[1/µ(k × E + k0 × E0) − 1/µ0(k00 × E00)] × n = 0 (3.9)

(E + E0 − E00) × n = 0, (3.10) where n is the unit vector perpendicular to the interface. For TE polarization, as shown in figure 3.1, the electric field is parallel to the interface. From equation (3.9) 17 and the relation between k and the index of refraction n given in equation (2.25) we obtain

0 00 n1E cos θi − n1E cos θi = n2E cos θt, (3.11)

E + E0 − E00 = 0. (3.12)

00 Eliminating E from equation (3.11) we obtain the reflection coefficient rs and trans- mission coefficient ts associated with the interface between two media

0 E n1 cos θi − n2 cos θt rs = = , (3.13) E n1 cos θi + n2 cos θt 00 E 2n1 cos θi ts = = , (3.14) E n1 cos θi + n2 cos θt where the subscript ‘s’ represents the TE polarization of light. The complex indices of refraction n1 and n2 are given by equation (2.24) and the angles θi and θt are obtained using Snell’s law in equation (3.8). In this work, we focus on TE (transverse-electric) polarized light, however, the methods described here are readily applied to transverse- magnetic (TM) polarization when appropriate Fresnel coefficients are used in place of equations (3.13) and (3.14).

For normal angle of incidence, equations (3.13)–(3.14) reduce to

n1 − n2 rs = , (3.15) n1 + n2 2n1 ts = . (3.16) n1 + n2

Let Ii, Ir and It be the incident, reflected, and transmitted intensities, given by the magnitude of the time averaged Poynting vector in equation (2.32). The reflectance R is defined as the ratio of reflected intensity to incident intensity and the 18 T as the ratio of transmitted intensity to incident intensity at the interface. Taking ratios of intensities and using the Fresnel coefficients in equations (3.15)–(3.16) we obtain,

0 2 Ir |E | 2 R = = 2 = |rs| , (3.17) Ii |E| 00 2 It |E | Re(n2) T = = 2 . (3.18) Ii |E| Re(n1)

3.2 Transfer Matrix for a Single Interface

Now consider a two layer system with the physical properties described above. To develop the transfer matrix method [6, 14, 15] for the arrangement illustrated in figure

3.2, we introduce new notation. Since the plane-wave solution to Maxwell’s equations in stratified media have complex amplitudes whose absolute value depends only on the distance to the next interface, we focus on the z-dependent part of the wave functions.

We denote the incident wave by E0i, the reflected wave by E0r, and the transmitted wave by E1t, where the subscripts ‘i’, ‘r’ and ‘t’ stand for incident, reflected and transmitted waves, and ‘0’ and ‘1’ for medium 0 and medium 1, respectively. The interface between medium 0 and medium 1 is denoted by I0ba at z = 0. A TE wave is incident on the interface from medium 0. To distinguish amplitudes of waves traveling in the positive z-direction (incident and transmitted waves) and the negative z-direction (reflected wave) we use the subscripts ‘p’ and ‘n’. The incident, reflected and transmitted electric field amplitudes adjacent to the interface I0ba are given by

E0bp, E0bn and E1bp as shown in figure (3.2). Let θ0 be the complex angle of incidence

19 in medium 0 and θ1 be the complex angle of refraction in medium 1, where θ0 and θ1 are related by Snell’s law given by equation (3.8).

Figure 3.2: A planar interface I0ba at z = 0 separates media 0 and 1 with indices of refraction n0 and n1. A TE wave is incident on I0ba with amplitude E0bp and angle of incident θ0, and is reflected with amplitude E0bn. The transmitted wave has amplitude E0ap at the interface and angle of refraction θ1. E0i, E0r and E1t are the z-position dependent electric fields of the incident, reflected, and transmitted waves.

The boundary conditions on the electric field given by equations (3.11)–(3.12) are, in the notation of this section,

n0E0bp cos(θ0) − n0E0bn cos(θ0) = n1E1ap cos(θ1) (3.19)

E0bp + E0bn = E1ap. (3.20)

Defining the Fresnel reflection and the transmission coefficients associated with the interface I0ba, r01 = E0bn/E0bp and t01 = E1ap/E0bp (compare equations (3.13)–(3.14)),

20 we obtain for the field amplitudes

E0bp = (1/t01)E1ap, (3.21)

E0bn = (r01/t01)E1ap. (3.22)

Equations (3.21)–(3.22) may be rewritten in matrix form as     E E  0bp  1ap   = M0   , (3.23)     E0bn 0 where M0 is called the interface matrix and is given by   1 r 1  01 M0 =   . (3.24) t01   r01 1

The interface matrix M0 describes the linear relationship between the electric fields immediately adjacent to the interface. Once inside the medium, the electric field propagates as a plane wave. Therefore, the transmitted wave amplitude E1t(z) at a distance z1 from the interface I0ba at z = 0 may be written as,

−ik1z(z1−z0) E1t(z) = E1ape , (3.25)

where k1z = (2π/λ)n1 cos θ1. For real index of refraction n1, k1z(z1 − z0) ≡ δ is the phase change due to the wave traveling through a layer of material of thickness z −z0.

For complex n, δ also describes the damping of the amplitude. Similarly, the incident

0 and reflected waves traveling from the distance z0 to the interface at z0 are written as,

0 −ik0z(z −z0) E0i(z) = E0bpe 0 , (3.26)

0 ik0z(z −z0) E0r(z) = E0bne 0 . (3.27) 21 The total electric field in medium 0 is a superposition of waves traveling in the positive and negative direction. Therefore, if the variables Em0t and Em1t represent the total electric fields in medium 0 and medium 1, respectively, we may write,

0 0 −ik0z(z −z0) ik0z(z −z0) Em0t(z) = E0bpe 0 + E0bne 0 , (3.28)

−ik1z(z1−z0) Em1t(z) = E1ape (3.29)

Equations (3.28)–(3.29) describes the position-dependent total electric field in the

0 media when we replace z1 or z0 with the z-coordinate of any point in a medium.

As a simple example, we discuss the case of light striking an air-glass interface at normal incidence. Table (3.1) shows the refractive indices of both media and the wavelength of the incident light. The reflection and transmission coefficients are calculated using equations (3.15)–(3.16).

Table 3.1: Parameters for the example of an air/glass interface with light incident from air with θi = 0.

λ (nm) n0 n1 r01 t01

480 1.0 1.5 -0.2000 0.8000

0 When the wave traveling from z0 = 0 is incident on the interface I0ba at z = 100 nm with electric field amplitude E0bp = 1, the interface matrix M0 becomes,   1 −0.2000 1   M0 =   . (3.30) 0.8000   −0.2000 1

22 Once the wave enters through the interface, it travels to a distance z1 = 200 nm with phase factor δ1 = (2π/λ)n1(z1 −z0). For the numerical evaluation, the distances

0 (z0 − z0) and (z1 − z0) are divided into 300 discrete intervals each. The total electric

field in the media is calculated using equations (3.28)–(3.29). To find the amplitude

0 0 at z0, the electric fields E0bp and E0bn are propagated backwards to z0 = 0 using the

0 phase factor δ0 = +(2π/λ)n0(z0 − z0).

0 Figure 3.3: A wave of wavelength λ = 480 nm traveling from z0 = 0 is incident on a planar interface at z = 0 between medium 0 (air) and medium 1 (glass) with angle of incidence θ0 = 0 and amplitude E0bp = 1. The lines in the top and bottom panel show the real and imaginary contributions of the total electric field amplitudes, respectively.

23 0 Graphs of the real and imaginary parts of the total electric field in the range z0 to z1 are presented in figure 3.3 and show that both contributions to the electric field amplitude are continuous at the boundary. Intensities of the incoming, reflected, and transmitted waves, calculated from equations (2.32) and normalized by the intensity of the incident electric field, are presented in figure 3.4. As expected for non-absorbing media, the reflected and transmitted intensity add up to the incoming intensity.

0 Figure 3.4: A wave of wavelength λ = 480 nm traveling from z0 = 0 is incident on a planar interface at z = 0 between medium 0 (air) and medium 1 (glass) with angle of incidence θ0 = 0 and amplitude E0bp = 1. The incident (blue), reflected (red) and transmitted (black) intensities in medium 0 and 1 are shown.

3.3 Multilayer System

To understand multilayer systems, we add an interface I1ba to the two-layer system shown in figure (3.2). There is now a reflected wave, E1r, in medium 1 with electric

24 Figure 3.5: A TE wave is incident on the planar interface between medium 0 and medium 1 with complex amplitude E0bp. The wave is partially reflected in the same medium with amplitude E0bn and partially transmitted into medium 1 with amplitude E1ap. The wave is partially reflected from interface I1ba into medium 1 with amplitude E1bn and partially transmitted into medium 2 with amplitude E2ap.

field amplitude E1an at the interface I0ba. Let the incident, reflected and transmitted electric field amplitudes adjacent to I1ba be E1bp, E1bn and E2ap, respectively, where the subscript ‘2’ is for medium 2. Medium 1 constitutes a layer L1 between the two interfaces; as the wave travels across the layer, the change in electric field is described by

iδ1 E1bp = E1ape , (3.31)

−iδ1 E1bn = E1ane , (3.32)

25 with δ1 = k1z(z1 − z0). This change may be written in matrix form as     E E  1bp  1ap   = M01   , (3.33)     E1bn E1an where M01 is the so-called layer matrix and given by   e−iδ1 0   M01 =   . (3.34)   0 eiδ1

The electric field amplitude E2bp in medium 2 at the interface I1ba is determined by an interface matrix, M1, given by   1 r 1  12 M1 =   . (3.35) t12   r12 1

Combining the equations for the incident electric field amplitudes in (3.23) with equations (3.33)–(3.35), we obtain a relation between the electric field amplitudes at the first and last interface     E E  0bp  2ap   = M0 ∗ M01 ∗ M1   . (3.36)     E0bn 0

Equation (3.36) represents the transfer matrix method for a single finite layer between semi-infinite media [6].

The transfer matrix method is particularly useful when we consider systems of m layers. Consider a stratified structure that consists of a stack of 1,2,..,j,..,m − 1 parallel layers sandwiched between an ambient layer (0) and a substrate layer (m) as shown in figure (3.6). The interfaces normal to the z axis are labeled by Ijba for

26 Figure 3.6: Layer system consisting of ambient (0) and substrate (m) semi-infinite layers and m − 1 finite layers. Layer j has thickness dj, refractive index nj, and forward and backward traveling waves with amplitudes Ejbp and Ejbn, respectively, at the interface Ij+1. Ejap and Ejan are the corresponding amplitudes at the interface Ij.

th j ∈ 1, ..., m − 1. Let the complex refractive index of the j layer be nj and it’s thickness dj. n0 and nm represent the refractive indices of ambient and substrate media, respectively. We specify the angle of incidence θ0 and obtain the angles θj in the layers by repeated application of Snell’s law given by equation (3.8). The reflection and the transmission coefficients between layers j and j + 1 are calculated as in equations (3.13) and (3.14) and denoted by rj(j+1) and tj(j+1). The interface matrices, Mj, and the layer matrices Mj(j+1), are calculated with the aid of equations

(3.34) and (3.35), extended to general indices j and j + 1. The total electric field inside the jth layer consists of two plane waves: a forward traveling wave denoted by the subscript ‘p’ and a backward traveling wave denoted by ‘n’.

27 The amplitudes in media 0 and m (ambient and substrate) are related by a product of interface and layer matrices,     E E  0ap  map   = M01 ∗ M12 ∗ M0 ∗ M1...... M(m−1)m ∗ Mm   . (3.37)     E0an 0

The fields in intermediate layers may be obtained by propagation from the ambient or the substrate,     E E  0ap  jap   = M01 ∗ M0 ∗ ...... ∗ M(j−1)j ∗ Mj   , (3.38)     E0an Ejan     E E  jap  map   = Mj(j+1) ∗ Mj+1 ∗ .... ∗ M(m − 1)m ∗ Mm   . (3.39)     Ejan 0

Equations (3.38) – (3.37) suggest how we can find the total electric field as function of position in any given layer.

We have implemented the matrix method for systems of up to six layers in

Matlab scripts that allow us to program efficient calculations of the electric field amplitudes and create good graphical representations of the results.

As an example for a multilayer system, consider the three layer system, air

(40 nm)-glass (160 nm)-air (50 nm) with refractive index 1.0 for air and 1.5 for glass.

A wave of wavelength 480 nm is incident at angle θ0 = 0 from the ambient side with amplitude E0bp = 1. Since the thickness of the glass layer is exactly half the wavelength in glass, λ = 320 nm, the path difference between light reflected at the two interfaces and the phase change of π on reflection at the air-glass interface are expected to lead to destructive interference of the reflected light in air [13, 14, 6]. 28 In Figure 3.7 we show the absolute square value of the total electric field for this air/glass/air system. In figure 3.8 we present the real and imaginary contributions to the electric field of the backward traveling wave (reflected light) in all layers. As expected, the reflected electric field is zero outside the glass layer and the incoming and transmitted fields have the same intensity.

Figure 3.7: The absolute square value |E|2 of the electric field inside an air-glass-air system for normal incidence and light of wavelength λ = 480 nm.

In figure 3.9 we present results for the absolute square of electric field |E|2 for non-normal incidence, where the reflected electric field in air is not zero. For the

0 angle θ0 = π/3 = 60 , figure 3.9 shows that the transmitted intensity is smaller than before and that the superposition of incident and reflected light leads to a position- dependent intensity in the ambient air.

To illustrate how the electric field distribution depends on the angle of in-

29 cidence, we present results for the absolute square electric field |E|2 for a range of angles of incidence 00 − 800 and fixed wavelength λ = 480 nm as a contour plot and a colormap in figure 3.10. The figure shows that the overall intensity inside the glass layer decreases with increasing angle of incidence and that the intensity distribution becomes more asymmetrical as θ0 increases.

30 Figure 3.8: Reflected amplitudes E0r, E01r and E2r in the air (40nm)-glass (160nm)- air (50nm) system for a wavelength of λ = 480 nm and normal incidence. The top and bottom panel show the real and imaginary contributions, respectively.

31 Figure 3.9: The absolute square value |E|2 of the electric field inside the air-glass-air system for θ0 = π/3 and light of wavelength λ = 480 nm.

32 Figure 3.10: The absolute square value |E|2 of the electric field as a function of position z for a range of angles of incidence θ0 for the air-glass-air system. (a) The contour lines tagged by values represent lines of constant |E|2; the vertical lines show the interfaces. (b) In the colormap of |E|2, the shades represent values indicated in the colorbar on the right.

33 CHAPTER IV

ORGANIC SOLAR CELL MODEL AND CHARACTERISTICS

4.1 Organic Solar Cell Model

In this work we model a bilayer device investigated by Pettersson et al. [3] and shown in figure 4.1. An organic solar cell with a conjugated polymer, poly(3-(4’-

(1”,4”,7”-trioxaoctyl)phenyl)thiophene) (PEOPT ), as the donor material and C60 as the acceptor material has a transparent electrode made of indium tin oxide (ITO) at the top and a metallic electrode made of aluminum (Al) at the bottom. A poly(3,4- ethylenedioxythiophne)-poly(styrenesulfonate) (PEDOT)-(PSS) layer is inserted be- tween ITO and PEOPT. The chemical composition of PEOPT and PEDOT is shown in figure 4.2.

Light is incident on the organic solar cell through the transparent electrode

ITO. Because of the small reflection coefficient associated with ITO, most of the light is transmitted through the material into the next layers. The interaction of light with

PEOPT or C60 creates excitons, which diffuse through the active layers and, upon reaching the PEOPT/C60 interface, separate into free charges, electrons and holes. A built in electric field due to the difference in work functions of the electrodes, Al and

ITO, causes the holes to diffuse through PEOPT and the electrons to diffuse through

34 Figure 4.1: The arrangement of layers in the solar cell. At the top is the ITO electrode, in the middle are the PEOPT and C60 active layers, and at the bottom is the aluminum electrode. A PEDOT layer is placed between ITO and PEOPT; light is incident from the ITO side.

C60 until being collected by ITO and Al, respectively, giving rise to a photocurrent.

A PEDOT layer is placed between ITO and PEOPT to prevent electrons traveling towards ITO [3]. Since excitons are short lived particles they need to reach the

PEOPT/C60 interface before they decay. Therefore, the thickness of the active layer needs to be chosen considering the exciton diffusion length.

A limiting factor for the performance of organic solar cells is the band gap of the active materials, PEOPT and C60 [2, 16]. An exciton is generated only if PEOPT is excited by photons of energy higher than the band gap of PEOPT and similarly for

C60. The available wavelengths are set by the solar spectrum; therefore, appropriate materials must be chosen to increase the probability of exciton generation. In figure

4.3 we show the absorption spectrum for PEOPT and C60. A comparison with the solar spectrum in figure 1.2 shows that very little intensity available in the range of wavelengths where excitons may be created in C60.

35 Figure 4.2: Chemical composition of the active material poly(3-(4’-(1”,4”,7”- trioxaoctyl)phenyl)thiophene) (PEOPT), of poly(3,4-ethylenedioxythiophne)- poly(styrenesulfonate) (PEDOT), and of the anion (PSS) [3]

4.2 Modeling Parameters

The organic solar cell in our model is characterized by complex refractive indices for each material and the thickness of the layers. The real and imaginary parts of the refractive indices of C60 and PEOPT are presented in figure 4.4. The imaginary part ni of the refractive index determines the absorption coefficient χ defined in equation

(2.35) and shown in figure 4.3. For ITO, PEDOT, and Al, the real and imaginary parts of the complex refractive indices as a function of wavelength are shown in figure

4.5.

In this work, we consider two devices that differ in the thickness of the C60 layer, 35 nm for geometry (a) and 80 nm for geometry (b). The dimensions of the

36 Figure 4.3: The absorption coefficient χ as function of wavelength λ for PEOPT (solid line) and C60 (dashed line) calculated from index of refraction data of Ref. [3].

cells are shown in Table 4.1. We investigate the effect of the angle of incidence for a

fixed wavelength and the effect of the wavelength of the light.

It should be noted that we treat the layers as optically flat even though, in reality, the layer thickness is not quite uniform over the whole sample and there may be some roughness. One of the effects of uneven layer thickness is the angle of incidence of the light not being constant for the whole device. This is not expected to change the field distribution significantly since, as our results in Chapter 5 show, even large changes in the angle of incidence lead to minor changes in the distribution.

Roughness of the interfaces, on the other hand, may lead to incoherent of light, which reduces the intensity variation expected from interference effects. Results from the literature suggest that incoherent scattering does not have a dominant effect

37 Figure 4.4: Complex refractive indices of PEOPT and C60 as a function of wavelength λ. The nr (dashed lines) and ni (solid lines) represent the real and imaginary part of the refractive index n, respectively. [3]

on the electric field distribution [3, 7].

38 Table 4.1: Thickness of layers in the solar cell for two different geometries.

Layer Names Geometry (a) Geometry (b)

ITO 120 nm 120 nm

PEDOT 110 nm 40 nm

PEOPT 40 nm 110 nm

C60 35 nm 80 nm

Al 95 nm 50 nm

39 Figure 4.5: Complex refractive indices of ITO, PEDOT and Al as a function of wave- length. [1] The top and bottom panel show the real and imaginary parts, respectively.

40 CHAPTER V

RESULTS

In this work, we investigate the propagation of light with TE polarization in solar cells of two geometries that differ in the thickness of the C60 layer adjacent to the metal electrode, 35 nm for geometry (a) and 80 nm for geometry (b). Figure 5.1 shows the absolute square value of the electric field |E|2 normalized by the absolute square

2 value of the incident electric field from the ambient side (air) |E0| , as a function of position inside the solar cell for light of wavelength λ = 460 nm incident at normal incidence from the ITO side. The results shown in figure 5.1 are in agreement with the results presented by Petterson et al. [3], considering that the system investigated in Ref. [3] has a 1 mm thick glass layer covering ITO, which we do not consider in this thesis.

2 2 The distribution of |E| / |E0| inside the solar cell is not uniform. Since the aluminum electrode on the right allows very little light to enter, the electric field is very small at the C60/Al interface. Interference effects and refractive index differences between the materials lead to the spatial variation of |E|2 which shows local maxima

2 2 and minima. The value of |E| / |E0| near the PEOPT/C60 interface is of particular interest because excitons generated near the interface have a high probability to reach the interface and dissociate into charge carriers in their lifetime. Figure 5.1 shows

41 Figure 5.1: Distribution of the absolute square value of the electric field |E|2 normal- 2 ized by the absolute square value of the incident electric field |E0| as a function of position inside the solar cell for geometry (a) and geometry (b). The wavelength is λ = 460 nm and the angle of incidence is θ0 = 0 (normal incidence). The layers are, from left to right, ITO, PEDOT, PEOPT, C60, and Al.

42 2 2 that the value of |E| / |E0| at the interface is much higher for geometry (a) than for geometry (b). Therefore, we may conclude that it is important to consider the effect of geometry when studying the electric field distribution inside a solar cell.

To investigate the effect of wavelength on the electric field distribution we

2 2 calculate |E| / |E0| for the range of wavelengths 300 nm - 700 nm. In the colormap

2 2 in figure 5.2, the shades of color represent the value of |E| / |E0| as indicated by the colorbar on the side. Graph (a) and (b) show that the distribution pattern

2 2 of |E| / |E0| becomes wider and shifts to the left as the wavelength increases. The number of interface maxima increases with decreasing wavelength from two at 700 nm to four at 300 nm, consistent with the observation that diffraction patterns generally vary as 1/λ.

To show the wavelength dependence in more detail, we present in figure 5.3

2 2 results for |E| / |E0| for three different wavelengths, λ = 340 nm, 460 nm, and 540 nm, selected from the data shown in figure 5.2 for geometry (a) and (b). Graph

2 2 (a) shows that the value of |E| / |E0| at the active interface is highest for 540 nm,

2 2 similar for 460 nm, and lowest for 340 nm. In graph (b), the value of |E| / |E0| is highest for 540 nm and lowest for 460 nm, but even the highest value in (b) barely exceeds the lowest value in (a).

2 2 To investigate the effect of the angle of incidence we calculate |E| / |E0| for a range of angles, 00 − 800, for fixed wavelength 460 nm for geometry (a) and (b).

2 2 Figures 5.4 and 5.5 show contour plots of constant |E| / |E0| for the whole range of angles and colormaps for a smaller range of angles (00 − 400) for geometries (a)

43 Figure 5.2: Distribution of the normalized absolute square value of the electric field 2 2 |E| / |E0| for a range of wavelengths from λ = 300 nm to λ = 700 nm and angle of incidence θ0 = 0 (normal incidence). In the colormap for geometry (a) and geometry 2 2 (b), different shades correspond to different values of |E| / |E0| as indicated in the colorbar on the side.

44 Figure 5.3: Distribution of the normalized absolute square value of the electric field 2 2 0 |E| / |E0| as a function of position inside the solar cell for angle of incidence θ0 = 0 and the wavelengths λ = 340 nm (solid line), λ = 460 nm (dashed line), λ = 540 nm (dash-dotted line) for geometry (a) and geometry (b).

45 Figure 5.4: Distribution of the normalized absolute square value of the electric field 2 2 |E| / |E0| for geometry (a) and wavelength λ = 460 nm. The tagged lines in the contour plot (1) represent lines of constant values of the normalized electric field 2 2 0 0 |E| / |E0| for a range of angles of incidence from θ0 = 0 to θ0 = 80 . In the colormap 2 2 (2) different shades correspond to different values of |E| / |E0| as indicated in the 0 0 colorbar on the side; the range of angles is θ0 = 0 to θ0 = 40

46 Figure 5.5: Distribution of the normalized absolute square value of the electric field 2 2 |E| / |E0| for geometry (b) and wavelength λ = 460 nm. The tagged lines in the contour plot (1) represent lines of constant values of the normalized electric field 2 2 0 0 |E| / |E0| for a range of angles of incidence from θ0 = 0 to θ0 = 80 . In the colormap 2 2 (2) different shades correspond to different values of |E| / |E0| as indicated in the 0 0 colorbar on the side; the range of angles is θ0 = 0 to θ0 = 40

47 and (b), respectively. The figures show that the pattern of maxima and minima bend slightly to the left with increasing θ0 and that the intensity is largest for the smallest angles of incidence. We should point out that, for a glass-capped solar cell, refraction at the air glass interface restricts the angles of incidence at the glass/ITO interface to about 410.

To understand the combined effect of wavelength and angle of incidence,

2 2 we present results for |E| / |E0| for three different wavelengths and three different angles of incidence in figures 5.6 - 5.8. Figure 5.6 for wavelength λ = 340 nm, shows

2 2 that the value of |E| / |E0| at the active interface decreases with increasing θ0 in

2 2 both geometries. For all angles of incidence, the value of |E| / |E0| at the active interface is higher in geometry (a) than in (b). Figures 5.7 and 5.8 show results for wavelengths λ = 460 nm and λ = 540 nm, respectively. In both figures it can be seen

2 2 that the value of |E| / |E0| at the active interface is highest for normal incidence and is higher for geometry (a) than geometry (b).

To study the absorbance in the active layers we calculate the absorption dis- tribution function Az, defined in equation (2.34), for the PEOPT and C60 layers.

Since the angle of incidence has little effect on the electric field distribution for rele- vant angles, we focus on normal incidence. Az(z)dz describes the amount of energy absorbed in a layer of thickness dz at position z. In figure 5.9, we show values of the scaled absorption function

Az As = 2 . (5.1) c0 |E0| for three wavelengths λ = 340 nm, λ = 460 nm, and λ = 540 nm for geometry (a) 48 Figure 5.6: Distribution of the normalized absolute square value of the electric field 2 2 |E| / |E0| as a function of position inside the solar cell for wavelength λ = 340 nm and three angles of incidence for geometry (a) and geometry (b); the angles are 0 0 0 θ0 = 0 (solid line), θ0 = 30 (dashed line), θ0 = 60 (dash-dotted line).

49 Figure 5.7: Distribution of the normalized absolute square value of the electric field 2 2 |E| / |E0| as a function of position inside the solar cell for wavelength λ = 460 nm and three angles of incidence for geometry (a) and geometry (b); the angles are 0 0 0 θ0 = 0 (solid line), θ0 = 30 (dashed line), θ0 = 60 (dash-dotted line).

50 Figure 5.8: Distribution of the normalized absolute square value of the electric field 2 2 |E| / |E0| as a function of position inside the solar cell for wavelength λ = 540 nm and three angles of incidence for geometry (a) and geometry (b); the angles are 0 0 0 θ0 = 0 (solid line), θ0 = 30 (dashed line), θ0 = 60 (dash-dotted line).

51 and (b). Figure 5.9 shows that, near the PEOPT/C60 interface for geometry (a), light of wavelength λ = 340 nm is absorbed best by C60 and least by PEOPT, while light of wavelength λ = 460 nm is absorbed best in PEOPT and moderately well in

C60. For geometry (b), all the wavelengths are absorbed poorly in PEOPT but light of the shortest wavelength is absorbed well in C60.

Figure 5.9 suggests that light of short wavelength makes a significant contri- bution to exciton generation in the device. Unfortunately, sun light has very little intensity at short wavelengths. To illustrate this, we reweight our results by the solar irradiance spectrum and calculate

A I(λ) A∗ = s , (5.2) s I(460nm) where I(λ) is the solar irradiance at wavelength λ [10]. In figure 5.10 we present the reweighted absorbance values for the conditions of figure 5.9. A comparison shows that the short wavelength contribution to the absorbed energy is much reduced in both materials. Even for C60, wavelengths near the maximum of the solar spectrum are expected to make the largest contribution to exciton generation.

52 Figure 5.9: Scaled absorption function As defined in equation (5.1) as a function of position inside the PEOPT (230 nm - 270 nm) and C60 layers for geometry (a) and 0 geometry (b); the angle of incidence is θ0 = 0 , and the wavelengths are λ = 340 nm (solid line), λ = 460 nm (dashed line), and λ = 540 nm (dash dotted line).

53 ∗ Figure 5.10: Reweighted absorption function As defined in equation (5.2) as a function of position inside the PEOPT (230 nm - 270 nm) and C60 layers for geometry (a) and 0 geometry (b); the angle of incidence is θ0 = 0 , and the wavelengths are λ = 340 nm (solid line), λ = 460 nm (dashed line), and λ = 540 nm (dash dotted line).

54 CHAPTER VI

SUMMARY AND CONCLUSION

In this work we investigated the absorption of light in thin-film organic solar cells of two planar geometries for the relevant part of the solar spectrum and a range of angles of incidence with computational methods. The devices consist of a transparent ITO electrode, a stopping layer (PEDOT), two active layers, the conjugated polymer ma- terial PEOPT and the fullerene C60, and an aluminum electrode. Since the conversion of incident photons to charge carriers occurs only in the active layers the intensity distribution of light within the device has an important effect on the efficiency of a solar cell. Since interference effects play an important role in thin-films, we employ a transfer matrix method to calculate the complex amplitude of the electric field at the interfaces and propagate the electromagnetic wave within the layers.

Our results for normal incidence are consistent with results reported by Pet- terson et al. [3] and confirm that the thickness of the C60 layer has a large effect on the electric field distribution near the active interface PEOPT/C60.

In contrast, the angle of incidence θ0 has a small effect on the spatial dis- tribution of the electric field in the active layers for relevant angles. As expected, the electric field at the active interface is largest for the smallest angles of incidence.

The electric field distribution in the device depends strongly on the wavelength of the

55 incident light and shows increasing interference effects with decreasing wavelengths.

Three factors determine the absorption of light energy in the active layers: the lo- cal electric field strength, the refractive indices of the materials, and the spectral distribution of the incoming light. Therefore, the geometry of a device needs to be optimized for given materials to convert as much as possible.

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