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筆者は、ディラック方程式の平面波解を以下の手順に従い、行列を用いて直接に求める方法を開発した。(i)電子の静止系において平面波解を求めた。(ii)静止系に対して、2種類の回転とブースト（並進）を順次施した場合のローレンツ変換の変換行列の一般形を求め、静止系のディラック方程式を変換する一般式を導いた。(iii)これらの手順によって、ディラック作用素とディラックスピノルが、どのように変換されるかを調べた結果、求めたディラックスピノルがディラック方程式の平面波解を直接に与えることを確証した。

Microcavity systems with organic luminescent materials have a hot prospect for room-temperature cavity-polariton devices. The polariton dispersion relation of organic microcavities is significantly different from that of inorganic microcavities due to the strong localization of Frenkel excitons. Also photoexcited particles will undergo a different cooling mechanism until they reach the polariton ground state. In the characterization of efficient polariton condensates, therefore, the polariton cooling dynamics as well as the kinetics of the polariton eigenstate should be measured. Here we present experimental studies on ultrafast dynamics of cavity polaritons in an organic single-crystal microcavity under nonresonant pumping. In time-resolved photoluminescence measurements we observed, for the first time, an ultrafast dynamics of stimulated cooling of the organic cavity polariton. Transient transmission measurement enabled us to investigate the detailed renormalization dynamics of the polariton eigenstate....

We demonstrated direct fabrication of a diffraction grating onto organic oligomer crystals by focused ion beam (FIB) lithography followed by argon/oxygen plasma etching. Surface analysis suggested that FIB irradiation broke the oligomer molecules near the crystal surface to form a carbonized layer resulting in emission quenching. The plasma etching removed the damaged layer near the crystal surface and successfully recovered the emission. This technique was applied to fabricate the diffraction grating onto organic oligomer crystals and provided diffracted peaks in their fluorescence spectra without significant emission quenching.

We improved carrier injection of organic field-effect transistors (OFETs) by utilizing organic/inorganic hybrid electrodes. One of thiophene/phenylene co-oligomers (TPCOs) was employed as a...

Green single crystal-based light-emitting devices based on 1,4-bis(5-phenylthiophen-2-yl)benzene (AC5) crystals have been realized by template stripping method. And double-layered structure with a molybdenum oxide (MoO 3 ) layer is employed as anodic buffer layer to lower the carrier injection barrier. On the basis of this structure, the AC5 crystal-based OLEDs exhibit a significantly lower turn-on voltage of 5 V and higher luminance reaching 28.96 cd/m 2 , which open a realistic route towards their practical application in optoelectronic devices.

We investigated solar cells in which various thiophene/phenylene co-oligomers (TPCOs) were used as the major photoelectric-conversion component and obtained enhanced performance. These simple planar heterojunction solar cells consisted of TPCO as the p-type semiconductor and fullerenes as the n-type semiconductor. The variation in the highest occupied molecular orbital levels of the TPCOs caused systematic modulation of the open-circuit voltage ( V OC ) without incurring a large change in the short-circuit current. We obtained a maximum power-conversion efficiency (PCE) of 3.0%, with V OC reaching 1.0 V. This PCE exceeds that obtained for a device containing zinc phthalocyanine, a conventional p-type photoelectric-conversion material.

So far we discussed propagation of light and its reflection and transmission (or refraction) at an interface of dielectric media. We described characteristics of light from the point of view of an electromagnetic wave. In this chapter, we describe properties of light in relation to quantum mechanics. To this end, we start with Planck’s law of radiation that successfully reproduced experimental results related to a blackbody radiation. Before this law had been established, Rayleigh–Jeans law failed to explain the experimental results at a high frequency region of radiation (the ultraviolet catastrophe). The Planck’s law of radiation led to the discovery of light quanta. Einstein interpreted Planck’s law of radiation on the basis of a model of two-level atoms. This model includes so-called Einstein A and B coefficients that are important in optics applications, especially lasers. We derive these coefficients from a classical point of view based on a dipole oscillation. We also consider a close relationship between electromagnetic waves confined in a cavity and a motion of a harmonic oscillator.

A group comprises mathematical elements that satisfy four simple definitions. A bunch of groups exists under these simple definitions. This makes the group theory a discriminating field of mathematics. To get familiar with various concepts of groups, we first show several tangible examples. Group elements can be numbers (both real and complex) and matrices.

In Chap. 5, we considered the propagation of electromagnetic waves in an infinite uniform dielectric medium.

In Chap. 16, we classified the groups into finite groups and infinite groups. We focused mostly on finite groups and their representations in the precedent chapters. Of the infinite groups, continuous groups and their properties have been widely investigated. In this chapter we think of the continuous groups as a collective term that include Lie groups and topological groups. The continuous groups are also viewed as the transformation group. Aside from the strict definition, we will see the continuous group as a natural extension of the rotation group or SO(3) that we studied briefly in Chap. 17. Here we reconstruct SO(3) on the basis of the notion of infinitesimal rotation. We make the most of the exponential functions of matrices the theory of which we have explored in Chap. 15. In this context, we study the basic properties and representations of the special unitary group of SU(2) that has close relevance to SO(3). Thus, we focus our attention on the representation theory of SU(2) and SO(3). The results are associated with the (generalized) angular momenta which we dealt with in Chap. 3. In particular, we show that the spherical surface harmonics constitute the basis functions of the representation space of SO(3).

In a history of quantum mechanics, it was first successfully applied to the motion of an electron in a hydrogen atom along with a harmonic oscillator. Unlike the case of a one-dimensional harmonic oscillator we dealt with in Chap. 2 however, with a hydrogen atom we have to consider three-dimensional motion of an electron. Accordingly, it takes somewhat elaborate calculations to constitute the Hamiltonian. The calculation procedures themselves, however, are worth following to understand underlying basic concepts of the quantum mechanics. At the same time, this chapter is a treasure of special functions. In Chap. 2 we have already encountered one of them, i.e., Hermite polynomials. Here, we will deal with Legendre polynomials associated Legendre polynomials etc.

In this chapter, we deal with the theory of finite-dimensional linear vector spaces. Such vector spaces are spanned by a finite number of linearly independent vectors, namely, basis vectors. In conjunction with developing an abstract concept and theory, we mention a notion of mapping among mathematical elements. A linear transformation of a vector is a special kind of mapping. In particular, we focus on endomorphism within a n-dimensional vector space Vn. Here, the endomorphism is defined as a linear transformation: Vn → Vn. The endomorphism is represented by a (n, n) square matrix. This is most often the case with physical and chemical applications, when we deal with matrix algebra. In this book we focus on this type of transformation.

Theory of analytic functions is one of major fields of modern mathematics. Its application covers broad range of topics of natural science. A complex function f (z), or a function that takes a complex number z as a variable, has various properties that often differ from those of functions that take a real number x as a variable. In particular, the analytic functions hold a paramount position in the complex analysis. In this chapter we explore various features of the analytic functions accordingly. From a practical point of view, the theory of analytic functions is very frequently utilized for the calculation of real definite integrals. For this reason, we describe the related topics together with tangible examples.

In Chap. 12, we dealt with a function of matrices. In this chapter we study several important definitions and characteristics of functions of matrices. If elements of matrices consist of analytic functions of a real variable, such matrices are of particular importance. For instance, differentiation can naturally be defined with the functions of matrices. Of these, exponential functions of matrices are widely used in various fields of mathematical physics. These functions frequently appear in a system of differential equations. In Chap. 10, we showed that SOLDEs with suitable BCs can be solved using Green’s functions. In the present chapter, in parallel, we show a solving method based on resolvent matrices. The exponential functions of matrices have broad applications in group theory that we will study in Part IV. In preparation for it, we study how the collection of matrices forms a linear vector space. In accordance with Chap. 13, we introduce basic notions of inner product and norm to the matrices.

In Sect. 1.2, we showed the Schrodinger equation as a function of space coordinates and time. In subsequent sections, we dealt with the time-independent eigenvalue problems of a harmonic oscillator and a hydrogen-like atoms. This implies that the physical system is isolated from the outside world and that there is no interaction between the outside world and physical system we are considering.

On the basis of studies of group theory, now in this last chapter we apply the knowledge to the molecular orbital (MO) calculations (or quantum chemical calculations).

Maxwell’s equations consist of four first-order partial differential equations. First we deal with basic properties of Maxwell’s equations. Next we show how equations of electromagnetic wave motion are derived from Maxwell’s equations along vector analysis. It is important to realize that the generation of the electromagnetic wave is a direct consequence of the interplay between the electric field and magnetic field that both change with time. We deal with behaviors of electromagnetic waves in dielectric media where no true charge exists. At a first glance, this restriction seems to narrow a range of application of principles of electromagnetism. In practice, however, such a situation is universalistic; topics cover a wide range of electromagnetic phenomena, e.g., light propagation in dielectrics including water, glass, polymers. Polarized properties characterize the electromagnetic waves. These include linear, circular, and elliptic polarizations. The characteristics are important both from a fundamental aspect and from the point of view of optical applications.

Quantum mechanics is an indispensable research tool of modern natural science that covers cosmology, atomic physics, molecular science, materials science, and so forth. The basic concept underlying quantum mechanics rests upon Schrodinger equation. The Schrodinger equation is described as a second-order linear differential equation (SOLDE). The equation is analytically solved accordingly. Alternatively, equations of the quantum mechanics are often described in terms of operators and matrices, and physical quantities are represented by those operators and matrices. Normally, they are non-commutative. In particular, the quantum-mechanical formalism requires the canonical commutation relation between position and momentum operators. One of the great characteristics of the quantum mechanics is that physical quantities must be Hermitian. This aspect is deeply related to the requirement that these quantities should be described by real numbers. We deal with the Hermiticity from both an analytical point of view (or coordinate representation) relevant to the differential equations and an algebraic viewpoint (or matrix representation) associated with the operators and matrices. Including these topics, we briefly survey the origin of Schrodinger equation and consider its implications. To get acquainted with the quantum-mechanical formalism, we deal with simple examples of the Schrodinger equation.

In Chaps. 1– 3 we have focused on solving eigenvalue equations with respect to a particle confined within a one-dimensional potential well or a harmonic oscillator along with an electron of a hydrogen-like atom. In each example we obtained exact analytical solutions with the quantum-mechanical states and corresponding eigenvalues (energy, angular momentum, etc.). In most cases of quantum-mechanical problems, however, we are not able to get such analytical solutions or accurately determine the corresponding eigenvalues. Under these circumstances, we need appropriate approximation methods of those problems. Among those methods, the perturbation method and variational method are widely used.

Representation theory is an important pillar of the group theory. As we shall see soon, a word “representation” and its definition sound a bit daunting.

The correlation between molecular orientation and optoelectrical properties is most critical to the future design of molecular materials. We made highly-anisotropic microcrystalline array structures with an organic semiconductor, a methoxy-substituted thiophene/phenylene co-oligomer (TPCO), by depositing it on friction-transferred poly(tetrafluoroethylene) (PTFE) layers fabricated on substrates with several heat treatments. Polarising microscope observation, polarised emission and absorption spectra measurements indicated that the TPCO molecules aligned along the drawing direction of PTFE. Using these films, we fabricated two types of field-effect transistors (FETs) and compared them with those using non-heated TPCO films which provide aligned pleats structures. Ones had the channel length direction parallel to the drawing direction of PTFE and the others had the channel length direction perpendicular to that drawing direction. As for the microcrystalline array films, the mobility ratio of the former FET to that of the latter device was about 27 in the saturation region, while the emission polarisation ratio was 4.5. The heat treatment promoted the crystal growth to enhance the mobility while retaining the high anisotropy. The results demonstrate that the heat treatments of the TPCO films on the friction-transferred layers were useful for controlling crystallinity and orientation of the molecules.

We introduced a crystal of an oligomer material for p-type emission layer into an organic light-emitting diode (OLED), and compared it with the device composed of thin films. To fabricate the devices, we laminated the oligomer crystal or deposited the oligomer thin film on a substrate covered with patterned indium tin oxide electrodes. We completed the devices by depositing an n-type material, an electron injection layer, and an Al electrode. For the current-voltage characteristics, both of the crystal and thin-film devices indicated rectification properties. At a certain current density, estimated applied electric fields for the crystal device was almost a quarter of that for the thin-film device. Both devices showed the current-injected emission. Comparison of the current-injected emission spectra to photoluminescence spectra revealed that the oligomer layers dominated emissions from the devices. The rough estimation of the input power for the current-injected emissions showed the crystal device required the lower input power than the thin-film device. These results suggest an advantage of the crystal device for current injection and light emission.

We have formed one-dimensional (1D) diffraction gratings on a photoresist polymer (SU-8) by interference exposure using a Lloyd mirror. A single crystal of thiophene/phenylene co-oligomers (TPCOs) grown in a vapor phase was placed on the 1D diffraction grating. Excitation of the single crystal by an ultraviolet light of a mercury lamp produced spectrally-narrowed fluorescence. On the basis of the grating periods and emission peak locations, we evaluated effective refractive indices and diffraction orders. The relevant information enables us to design and fabricate the 1D diffraction gratings that cause the spectrally-narrowed emission lines to superimpose inherently occurring high-intensity fluorescence peaks.

This paper reviews optically and electrically excited emissions from organic semiconducting oligomers in the form of their single crystals. Of the semiconducting oligomers, the review focuses on thiophene/phenylene co-oligomers (TPCOs). The topics cover crystal growth and laser oscillation along with related spectrally narrowed emissions of TPCO crystals. Aside from the strong excitation with a laser beam, weak excitation using a mercury lamp produces optical fringes superimposed on broadband emission spectra. The laser oscillation spectra accompanied by longitudinal multimode and optical fringes observed from the weak excitation have the same origin. This enables us to determine optical constants (i.e. refractive indices) and their dispersion of the crystals and provides sufficient information for the construction of optoelectronic devices based on the organic crystals. As a typical example, the review outlines the improved device constitution and performance as well as device operation methods with light-emitting field-effect transistors (LEFETs), because the device configuration is suited for cutting-edge devices including lasers. Finally the review presents device performance of LEFETs having a diffraction grating. These devices are suited for exploring the possibility of constructing a next-generation current-injected laser device. © 2016 The Authors. Polymer International published by John Wiley & Sons Ltd on behalf of Society of Chemical Industry.

We fabricated organic light-emitting transistors (OLETs) characterized by an Ag layer deposited on a one-dimensional (1D) or two-dimensional (2D) diffraction grating that acts as a combined gate insulator with SiO2. The Ag layer was entirely covered with an organic crystal. Upon photoexcitation that crystal showed narrow linewidth emissions (NLEs) parallel to the substrate plane. The narrowed lines were either redshifted or blueshifted with rotation of the crystal around a normal to its surface with respect to the grating wave vector. Strong emissions (∼104–106 cd m−2) accompanied by current-injected NLEs were observed from the 1D and 2D grating OLETs.

We evaluated emission behaviors of crystallized films of 1,4-bis(5-phenylthiophene-2-yl)benzene (AC5) in detail which was a representative thiophene/phenylene co-oligomer. The crystallized AC5 films were prepared by vapor deposition onto a substrate and thermal treatment. The AC5 films consisted of a crystalline domain with the size of several tens of micrometers. We used femtosecond laser pulses for the excitation of the AC5 films. As a result, the femtosecond laser pulses did not induce re-absorption above excitation energy densities of their laser threshold. The obtained gain value for AC5 crystallized film was large, over 150 cm-1. Furthermore, the emission cross section of the crystallized AC5 film was nearly 10-16 cm2.

An organic laser device is constructed as distributed feedback (DFB) type using an anisotropic single crystal of an organic semiconductor oligomer BP3T. The DFB structure is formed on top of the crystal via an interference exposure technique of a positive type photoresist. In the DFB structure, both the phase matching condition and waveguide condition must be fulfilled. Under these conditions, the effective refractive indices can be precisely predicted for a BP3T crystal, and they vary depending upon the crystal thickness. As a result, the device produces a single-mode laser oscillation at an intended wavelength (typically ∼569 nm). The present study enables us to design a down-to-earth laser device based on an anisotropic organic semiconductor crystal toward practical use.An organic laser device is constructed as distributed feedback (DFB) type using an anisotropic single crystal of an organic semiconductor oligomer BP3T. The DFB structure is formed on top of the crystal via an interference exposure technique of a positive type photoresist. In the DFB structure, both the phase matching condition and waveguide condition must be fulfilled. Under these conditions, the effective refractive indices can be precisely predicted for a BP3T crystal, and they vary depending upon the crystal thickness. As a result, the device produces a single-mode laser oscillation at an intended wavelength (typically ∼569 nm). The present study enables us to design a down-to-earth laser device based on an anisotropic organic semiconductor crystal toward practical use.

Organic single-crystalline semiconductors with long-range periodic order have attracted much attention for potential applications in electronic and optoelectronic devices due to their high carrier mobility, highly thermal stability, and low impurity content. Molecular doping has been proposed as a valuable strategy for improving the performance of organic semiconductors and semiconductor-based devices. However, a fundamental understanding of the inherent doping mechanism is still a key challenge impeding its practical application. In this study, solid evidence for the "perfect" substitutional doping mechanism of the stacking mode between the guest and host molecules in organic single-crystalline semiconductors using polarized photoluminescence spectrum measurements and first-principles calculations is provided. The molecular host-guest doping is further exploited for efficient color-tunable and even white organic single-crystal-based light-emitting devices by controlling the doping concentration. The clarification of the molecular doping mechanism in organic single-crystalline semiconductor host-guest system paves the way for their practical application in high-performance electronic and optoelectronic devices.

Doping effects in thiophene/phenylene co-oligomers (TPCOs) have been studied by means of extreme-UV (EUV) excited photoelectron spectroscopy (EUPS) and electro-luminescence experiments. Doped poly-crystalline films were fabricated by p-type doping with MoO3 to inherent p-type TPCO, BP1T (2,5-bis(4-biphenylyl)thiophene), and by n-type doping with Cs2CO3 to n-type AC5-CF3 (1,4-bis{5-[4-(trifluoromethyl)phenyl]thiophen-2-yl}benzene), respectively. Doping concentrations were 2% in both cases. The work function of BP1T was shifted from 4.0 to 4.3 eV and that of AC5-CF3 was shifted from 4.8 to 3.9 eV. The energy shifts are reasonable directions for p- and n-type doping. We also confirmed higher current injection with the doping films.

Hermitian operators and unitary operators are quite often encountered in mathematical physics and, in particular, quantum physics.

In this chapter, we deal with various properties and characteristics of differential equations, especially first-order linear differential equations (FOLDEs) and second-order linear differential equations (SOLDEs). These differential equations are characterized by differential operators and boundary conditions (BCs). Of these, differential operators appearing in SOLDEs are particularly important. Under appropriate conditions, the said operators can be converted to Hermitian operators. The SOLDEs associated to classical orthogonal polynomials play a central role in many fields of mathematical physics including quantum mechanics and electromagnetism. We study the general principle of SOLDEs in relation to several specific SOLDEs we have studied in Part I and examine general features of an eigenvalue problem and an initial value problem (IVP). In this context, Green’s functions provide a powerful tool for solving SOLDEs. For a practical purpose, we deal with actual construction of Green’s functions. In Sect. 6.9, we dealt with steady-state characteristics of electromagnetic waves in dielectrics in terms of propagation, reflection, and transmission. When we consider transient characteristics of electromagnetic and optical phenomena, we often need to deal with SOLDEs having constant coefficients. This is well known in connection with a motion of a damped harmonic oscillator. In the latter part of this chapter, we treat the initial value problem of a SOLDE of this type.

In Sect. 9.4, we saw that the transformation matrices are altered depending on the basis vectors we choose. Then a following question arises. Can we convert a (transformation) matrix to as simple a form as possible by similarity transformation (s)? In Sect. 9.4, we have also shown that if we have two sets of basis vectors in a linear vector space \( V^{n} \) we can always find a non-singular transformation matrix between the two. In conjunction with the transformation of the basis vectors, the matrix undergoes similarity transformation . It is our task in this chapter to find a simple form or a specific form (i.e., canonical form) of a matrix as a result of the similarity transformation . For this purpose, we should first find eigenvalue(s) and corresponding eigenvector(s) of the matrix. Depending upon the nature of matrices, we get various canonical forms of matrices such as a triangle matrix and a diagonal matrix . Regarding any form of matrices, we can treat these matrices under a unified form called the Jordan canonical form.

Quantum-mechanical treatment of a harmonic oscillator has been a well-studied topic from the beginning of the history of quantum mechanics. This topic is a standard subject in classical mechanics as well. In this chapter, first we briefly survey characteristics of a classical harmonic oscillator.

Thus far we have treated the theory of linear vector spaces. The vector spaces, however, were somewhat “structureless,” and so it will be desirable to introduce a concept of metric or measure into the linear vector spaces. We call a linear vector space where the inner product is defined an inner product space.

We synthesized a novel small-molecule organic semiconductor, which is soluble in organic solvents at room temperature under normal pressure. We demonstrated that its high-quality crystalline films can be directly grown on substrates using various solution techniques such as solution casting, slow evaporation, and edge casting. We applied crystals to FETs with a bottom- or top-contact configuration, revealing that the carrier mobility ranged from ~10−4 to ~10−2 cm2 V−1 s−1.

Entanglement of states is one of the most surprising and counter-intuitive consequences of quantum mechanics, with potent applications in cryptography and computing. In organic materials, one particularly significant manifestation is the spin-entangled triplet-pair state, which mediates the spin-conserving fission of one spin-0 singlet exciton into two spin-1 triplet excitons. Despite long theoretical and experimental exploration, the nature of the triplet-pair state and inter-triplet interactions have proved elusive. Here we use a range of organic semiconductors that undergo singlet exciton fission to reveal the photophysical properties of entangled triplet-pair states. We find that the triplet pair is bound with respect to free triplets with an energy that is largely material independent (∼30 meV). During its lifetime, the component triplets behave cooperatively as a singlet and emit light through a Herzberg–Teller-type mechanism, resulting in vibronically structured photoluminescence. In photovoltaic blends, charge transfer can occur from the bound triplet pairs with >100% photon-to-charge conversion efficiency., Singlet fission in organic semiconductors can generate triplet exciton pairs that are crucial to the charge generation in a photovoltaic process, whilst their nature remains elusive. Here, Yong et al. show that the immediate triplet pair is bound and emissive in a range of acene and heteroacene materials.

We have evaluated the emission behaviors of crystalline thin films of thiophene/phenylene co-oligomers. Crystalline films of 2,5-bis(4-biphenylyl)thiophene (BP1T) were prepared on a substrate by vapor deposition followed by thermal treatment. The BP1T films contained crystalline domains several hundred micrometers in size. We examined the emission behavior of crystalline BP1T thin films by femtosecond laser excitation. Very high gains of over 500 cm −1 were obtained for the crystalline BP1T film. Furthermore, the emission cross section of the crystalline BP1T film was on the order of 10 –16 cm 2 , rivaling that of dilute solutions of organic laser dyes.

We applied thiophene/phenylene co-oligomers (TPCOs) to a major component of organic solar cells. We made sandwich-type devices combining a p-type TPCO with fullerene (C60) or an n-type TPCO. As an option we made a device that combined a mixed layer of p-type and n-type TPCOs with the C60 layer. All devices indicated rectification properties in the dark and generated electric powers under the white light illumination. The device comprising methoxy-substituted biphenyl-capped bithiophene and C60 achieved the best power conversion efficiency of 1.20% with a short-circuit current of 4.28 mA/cm2 and fill factor of 0.46. The results demonstrate that the TPCO materials are useful as the major component of the organic solar cells.