Review on Charge Carrier Transport in Inorganic and Organic Semiconductors

Abstract

Inorganic semiconductors like silicon and germanium are the foundation of modern electronic devices. However, they have certain limitations, such as high production costs, limited flexibility, and heavy weight. Additionally, the depletion of natural resources required for inorganic semiconductor production raises concerns about sustainability. Therefore, the exploration and development of organic semiconductors offer a promising solution to overcome these challenges and pave the way for a new era of electronics. New applications for electronic and optoelectronic devices have been made possible by the recent emergence of organic semiconductors. Numerous innovative results on the performance of charge transport have been discovered with the growth of organic electronics. These discoveries have opened up new possibilities for the development of organic electronic devices, such as organic solar cells, organic light-emitting diodes, and organic field-effect transistors. The use of organic materials in these devices has the potential to revolutionise the electronics industry by providing low-cost, flexible, and lightweight alternatives to traditional inorganic materials. The understanding of charge carrier transport in organic semiconductors is crucial for the development of efficient organic electronic devices. This review offers a thorough overview of the charge carrier transport phenomenon in semiconductors with a focus on the underlying physical mechanisms and how it affects device performance. Additionally, the processes of carrier generation and recombination are given special attention. Furthermore, this review provides valuable insights into the fundamental principles that govern the behaviour of charge carriers in these materials, which can inform the design and optimisation of future devices.

1. Introduction

Semiconductor materials play a vital role in entire electronics and the computer industry. They are the building blocks of any electronic devices. The understanding of semiconductors begins in the early 19th century with many scientists undertaking several experiments relating to the electrical properties of materials. Properties such as light sensitivity, rectification, and negative temperature coefficient of resistance were observed in certain materials. Michael Faraday in 1833 reported that the application of heat to the silver sulphide material decreases its resistance. However, metallic substances like copper showed a contradictory behaviour upon heating [1]. A. E. Becquerel in 1839 observed the photovoltaic effect with a generation of voltage between the liquid electrolyte and a solid due to light incidence [2]. Willoughby Smith in 1873 reported the decrease in the resistance of selenium resistors due to light incidence [3]. In 1874, Arthur Schuster observed rectification properties of copper oxide layers on certain wires, and Karl Ferdinand Braun found the rectification and conduction property in metallic sulphides [4,5,6]. The photovoltaic effect in materials like selenium was observed in 1876 by Adams and Day [7]. A theory of solid state physics was necessary to provide a comprehensive explanation of these phenomena which evolved in the first half of the 20th century. Edwin Hall in 1878 explained the Hall effect, where the moving charge carriers experience a force with the application of a magnetic field [8]. In 1897, J.J Thomson discovered the electron, which incited theories relating to the conduction of electrons in solids, and Karl Baedeker observed the Hall effect with a change in sign for positive charge carriers in certain materials like copper iodide [9]. In 1914, a classification of materials—metals, variable conductors, and insulators—was given by Johan Koenigsberger [9]. In 1928, a theory on the motion of electrons through the atomic lattice was published by Ferdinand Bloch [8]. B. Gudden in 1930 showed that a minor concentration of impurities was responsible for conductivity in the case of semiconductors [9]. The concept of band theory and energy band gap was developed in 1931 by Alan Herries Wilson [8]. Walter H. Schottky and Neville Francis Mott developed models of metal semiconductor junction and potential barrier. Boris Davydov in 1938 developed rectifier theory using copper oxide by understanding the effect of the pn junction with the significance of minority charge carriers and the surface states [10]. John Bardeen explained that the behaviour of semiconductors changes with the addition of a small amount of impurities [11]. Before the development of semiconductor theory, which could guide the construction of more reliable and efficient devices, empirical knowledge was used to design semiconductor devices. Alexander Graham Bell in 1880 adopted the light sensitive property of selenium to transmit sound waves over a light beam. Charles Fritts in 1883, with the help of a metal plate coated with a thin gold layer and selenium, constructed a low efficiency solar cell, which in 1930 became prominent in photographic light meters [7]. H.J. Round in 1907 observed the effects of the light emitting diode (LED) in silicon carbide crystals which, on application of electric current, emitted light [12]. A similar light emission was observed by Oleg Losev in 1922 with no practicability of the effect at that time. In 1920, power rectifiers were developed using selenium and copper oxide, which became prominent as vacuum tube rectifiers [11]. Communication devices and infrared detection incited research into materials like lead selenide and lead sulphide. Such devices were used for detecting aircraft and ships, and for voice communications and infrared rangefinders. Julius Edgard Lilenfeld in 1926 patented a device identical to the present field effect transistor. However, the device was not feasible. A solid state amplifier was developed by R. Hilsch and R. W. Pohl in 1938 using a design that replicates the control grid of the vacuum tube [11]. In 1938, William Shockley and A. Holden at Bell Labs began analysing solid state amplifiers. In 1941, Russell Ohl observed the first silicon pn junction. During the war in France, Herbert Mataré observed amplification in contact points of germanium material and, after the war, his group started the ‘Transistron’ amplifier, following the announcement of ‘Transistor’ from Bell Labs. Moreover, the Bell lab scientists John Bardeen, Walter Houser Brattain, and William Shockley won the 1956 Nobel Prize in Physics for their invention of the transistor, a small semiconductor device.

After the invention of transistors in the mid of 20th century, inorganic semiconductors such as silicon (Si) and germanium (Ge) began to take the role as a prominent material in the field of electronics from the formerly presiding metals. During the same period, the vacuum tube-based electronics were replaced by solid state devices, which popularised the application of semiconductor microelectronics by the end of the 20th century. With the development and understanding of organic semiconductors, the world is experiencing a new revolution in electronics at the beginning of the 21st century. Tremendous progress in the field of organic semiconductors has been made by the prospect of new applications such as flexible displays and light sources, large area, plastic solar cells, and low cost integrated circuits from these materials. It can be rigidly stated that organic semiconductors are ancient. Photoconductivity in anthracene crystals, which are a prototype organic semiconductor, was studied for the first time in the early 20th century. Several researchers conducted an extensive investigation of molecular crystals following the discovery of the electroluminescence in the 1960s. As per these investigations, it was possible to understand the fundamental processes involved in charge carrier transport and optical excitation, for further information see [13,14,15]. However, various limitations made it almost impossible to practically exploit these early devices despite of the fact that principal demonstration of organic electroluminescent diodes were successful. For instance, the stability, including the current and light output could not be achieved as desired. The primary issue was the thickness of the crystal, which resulted in high operating voltages including inefficiencies in the crystal growth scaling and in providing stable, efficient injection contacts to them. The second prominent class of organic semiconductors was effectively introduced as a consequence of the synthesis and doping of conjugated polymers in a controlled way since the 1970s, that fetched the Noble Prize in Chemistry in 2000 [16], and the winners of the Noble Prize were Heeger, McDiarmid, and Shirakawa. These polymers with organic semiconductors have opened possibilities to apply the organic materials in electrophotography as photoreceptors [17] and conductive coatings [18]. In the 1980s, the use of conjugated polymers and oligomers [19,20,21] to fabricate thin-film transistors, as well as the introduction of photovoltaic cells including an organic hetero-junction from p and n conducting materials [22], marked the emergence of undoped organic semiconductors. Nonetheless, the demonstration of the highly efficient electroluminescent diodes of conjugated polymers [23] and of vacuum evaporated molecular films [24] have significantly highlighted the undoped organic semiconductors. In the last 15 years, research, including academic and industrial research, has contributed to the substantial development of organic light emitting devices (OLEDs), making them the first commercial products, including OLED displays. Organic semiconductors are likely to be applied in logic circuits with organic photovoltaic cells (OPVCs) or organic field effect transistors (OFETs) in the near future.

Phthalocyanines are widely used in modern technologies, including inkjet printing, electrophotography, and toners [25]. Additionally, phthalocyanines have been utilised in the field of organic electronics, where they serve as efficient materials for OLEDs and organic thin film transistors (OTFTs) [26,27]. Moreover, their excellent thermal stability and chemical resistance make them suitable for applications in coatings and pigments used in various industries.

Photocatalysis advances are driven by materials science and nanotechnology. These fields have allowed for the development of new photocatalytic materials, such as metal oxides and semiconductors, which have shown promising results in the conversion of solar energy into chemical energy [28,29]. Furthermore, the integration of nanoscale structures and surface modifications has enhanced the efficiency and selectivity of photocatalytic reactions, opening up new possibilities for applications in environmental remediation and solar fuel production.

Thermoelectric materials are another important application of semiconductor materials. These materials generate electricity through temperature differences across the material, making them ideal for waste heat recovery. By capturing and converting this wasted energy into usable electricity, thermoelectric materials have the potential to greatly improve energy efficiency in various industries, such as automotive, manufacturing, and power generation [30]. Additionally, these materials can also be used in portable devices and wearable technology to power small electronic components without the need for traditional batteries.

Thermoelectric materials have the advantage of being scalable and flexible, making them suitable for a wide range of applications. Some of the key challenges in developing thermoelectric materials include improving their efficiency and durability. Another challenge is finding ways to reduce the cost of production for thermoelectric materials, as this would make them more commercially viable for widespread use in various industries. One approach to improving efficiency is through the optimisation of material properties such as electrical conductivity and thermal conductivity. Additionally, advancements in organic semiconductor technology have shown promise for enhancing the performance of thermoelectric devices by manipulating the materials at the atomic level. Recent progress in the development of flexible thermoelectric materials has involved the use of inorganic materials, organic polymers, and hybrid organic-inorganic composites. More details can be found here [31,32,33,34]. These advancements have opened up new possibilities for the design and manufacturing of flexible thermoelectric devices, allowing for their integration into wearable electronics and other flexible applications. Furthermore, the use of organic polymers and hybrid composites offers the potential for improved flexibility, durability, and scalability in thermoelectric technology.

The performance of all electronic devices, including diodes [35,36], bipolar junction transistors [37,38], field effect transistors (FETs) [39,40,41], metal oxide semiconductor field effect transistors (MOSFETs) [42], even optoelectronic devices such as light emitting diodes (LEDs) [43,44,45], lasers [46,47,48], solar cells [49,50,51] and photodetectors [52,53,54] depend upon the charge transport within the device material. The charge carriers in semiconductors are electrons and holes, which are available in the conduction band and valence band respectively. The mechanism by which these charge carriers conduct electricity through the device is known as charge transport. In-depth analysis and better optimisation of charge transport in semiconductors is significant in improving the device performance. The charge transport measurements in device materials is achievable through device modelling. However, charge transport modelling in organic and inorganic semiconductors follow a different approach. In organic semiconductors, charge transport is typically described using a hopping mechanism, where charge carriers move between localised states. This is due to the presence of disorder and structural imperfections in organic materials. On the other hand, inorganic semiconductors often exhibit more ordered crystal structures, allowing for band-like transport where charge carriers move freely through the conduction and valence bands. Understanding and accurately modelling these different charge transport mechanisms is crucial for designing efficient semiconductor devices. Many physicists, chemists, engineers, and material scientists have performed careful experiments, persistent trial and error approach for device fabrication and material synthesis [55,56]. Simulations are not a replacement for experiments although they have an extremely vital role in research to design more meaningful experiments and to test the device operation as demonstrated in the case of inorganic semiconductor devices [57]. Modelling is a powerful tool for researchers in better understanding the processes within materials to enhance device performance. The fundamental properties of semiconductors and key processes involved within the device material, including the choice of materials, are all essential for device modelling and hence to improve device performance.

This review aims to provide a background of device materials such as inorganic and organic semiconductors in Section 2. Section 3 provides an explanation on the generation and recombination of charge carriers in semiconductors. Section 4 describes the charge transport mechanism in semiconductors. Section 5 explains techniques used to investigate charge transport in organic semiconductors with a focus on the Hall effect method, and a summary with future prospects is given in Section 6.

2. Classification of Inorganic and Organic Semiconductors

2.1. Classification of Inorganic Semiconductors

On the basis of resistance, electronic materials generally comprise three major types: conductors, semiconductors, and insulators [58]. Resistance depends upon the dimensions of the sample, while resistivity is independent of sample dimensions. Resistance (R) depends on the applied voltage (V) and, current (I) through Ohm’s law given by V = IR and resistivity is given by $\rho =\frac{RA}{l}$. Where ‘A’ is the area of cross section and ‘l’ is the length of the sample. Conductors or metals have low resistivity while semiconductors have resistivities about 5 to 6 orders of magnitude greater compared to metals. Insulators, on the other hand, have very high resistivity. Depending on the level of impurity and crystalline state, the semiconductors normally have a range of values for resistivity. The range is different for different materials and is related to the electronic properties of the material. To know this range, we need to understand the evolution of the band gap or energy gap within materials. Metals are good conductors of electricity since no band gap exists between filled and empty states. Filled states refer to the valence band and empty states refer to the conduction band. In the case of metals, these bands overlap with each other and, free electrons in the metal conducts electricity with the application of an external electric field. In the case of semiconductors and insulators, a band gap or energy gap exists between the valence band and conduction band (see Figure 1). The difference between these two materials is that the band gap for semiconductors is less than 3 eV while for insulators band gap is greater than 3 eV [58,59,60,61,62]. Some standard values of band gap for semiconductors and insulators are listed below in

Table 1. Standard band gap values for semiconductors and insulators at room temperature. Semiconductors have band gap values less than 3 eV and insulators have band gap values greater than 3 eV. ZnO is normally considered a broader band gap semiconductor rather than an insulator.

Material	Band Gap (eV)
Si	1.11
Ge	0.67
GaAs	1.43
CdS	2.42
ZnO	3.37
${\mathrm{Al}}_2{\mathrm{O}}_3$	7.0
${\mathrm{SiO}}_2$	9.0

.

There are several ways of classifying semiconductors and one of the most basic classifications is given by 1. Elemental semiconductors and 2. Compound semiconductors [64]. Silicon (Si) and germanium (Ge) are the most common elemental semiconductors that belong to the group IVA of the periodic table (see

Table 2. Group IV elements with their commonly used dopants are shown in this portion of the periodic table.

Group IIB	Group IIIA	Group IVA	Group VA	Group VIA
	B	C	N	O
	Boron	Carbon	Nitrogen	Oxygen
	Al	Si	P	S
	Aluminium	Silicon	Phosphorus	Sulfur
Zn	Ga	Ge	As	Se
Zinc	Gallium	Germanium	Arsenic	Selenium
Cd	In	Sn	Sb	Te
Cadmium	Indium	Tin	Antimony	Tellurium
Hg	Tl	Pb	Bi	Po
Mercury	Thallium	Lead	Bismuth	Polonium

). Carbon (C) is an insulator with band gap $E_g$ = 5.5 eV whereas, tin (Sn) and lead (Pb) are metals that also belong to the group IVA. Combining the elements of groups IIIA and VA, IIB and VIA we obtain compound semiconductors [65,66,67,68]. Some examples of III-V compound semiconductors are gallium arsenide (GaAs), gallium phosphide (GaP), gallium nitride (GaN), and indium antimonide (InSb). Examples of II-VI compound semiconductors include zinc oxide (ZnO), zinc sulphide (ZnS), cadmium sulphide (CdS), cadmium selenide (CdSe), and cadmium telluride (CdTe).

Depending upon the band structure, another way of categorising semiconductors is 1. Direct band gap semiconductors and 2. Indirect band gap semiconductors [69,70]. In semiconductors, the valence band is full and, the conduction band is empty. Excitation of electrons from the valence band to conduction band takes place either by thermal excitation or by the optical absorption. Electrons upon their return to the valence band release energy either as heat or as photons. For direct band gap semiconductors electrons release energy in the form of photons (electromagnetic radiation) whereas, for indirect band gap semiconductors electrons release energy in the form of heat (phonons). Optoelectronic devices such as lasers and LEDs are the examples of direct band gap semiconductors. Band structure distinguishes direct band gap semiconductors from indirect band gap semiconductors. E vs. $\overrightarrow{k}$ plot helps to understand the electronic band structure in these semiconductors as shown in Figure 2, where E represents energy and $\overrightarrow{k}$ represents the wave vector of an electron.

Both Si and Ge are indirect band gap semiconductors because the maximum of the valence band denoted by ${\Gamma }_{{25}^{\prime }}$ and the minimum of the conduction band denoted by $X_1$ are not on top of each other for Si and a similar structure is for Ge, where ${\Gamma }_8^{+}$ represents the maximum of the valence band and $L_6^{+}$ represents a minimum of the conduction band as shown in Figure 2. GaAs is a direct band gap semiconductor, as the maximum of the valence band and the minimum of the conduction band lie on top of each other.

Semiconductors are also classified into two classes—1. Intrinsic semiconductors and 2. Extrinsic semiconductors [72,73]. Pure semiconductors without any defects are known as intrinsic semiconductors. Si and Ge are examples of this type of semiconductors. Addition of impurities or dopants to intrinsic semiconductors to modify charge carrier concentration and therefore increase the conductivity is known as doping and the materials formed are known as extrinsic semiconductors. Dopants can be of two types: n-type dopants and p-type dopants [74]. The addition of pentavalent impurities such as phosphorus (P), antimony (Sb) and arsenic (As) to the Ge or Si crystal is known as n-type doping. The addition of trivalent impurities such as boron (B), aluminum (Al) and gallium (Ga) to Ge or Si crystal is known as p-type doping. Pentavalent impurity is referred to as donor impurity as they donate extra electrons to the Ge or Si crystal while trivalent impurity is referred to as acceptor impurity as they accept electrons from the Ge or Si crystal [58]. The addition of both types of impurities to the same material is possible and is called compensation doping. This type of doping is necessary when forming the junctions in case of electronic devices.

2.2. Classification of Organic Semiconductors

Organic semiconductors are the materials which comprise the hydrogen and carbon atoms including some heteroatoms like oxygen, sulphur, and nitrogen. These materials exhibit properties similar to those of semiconductor material. For instance, the emission and absorption of light lies within the visible spectrum, and a measure of conductivity is adequate to operate standard semiconductor devices such as FETs, solar cells, and LEDs. Given that the organic semiconductors exhibit semiconducting properties, it has been found that the ‘semiconducting’ behaviour between organic and inorganic materials differ significantly. Since various traditional inorganic semiconductors like Ge, Si, and GaAs have low band gaps of 0.67 eV, 1.1 eV, and 1.4 eV, respectively, it is possible to create free charges by thermal excitation at room temperature moving from the valence band to the conduction band with charge carrier concentration given by $N=N_{eff}{e}^{-E_g/2kT}$, where $N_{eff}$ represents the effective density states of valence or conduction band and $E_g$ represents the band gap. Usually, intrinsic conductivities lie in ${10}^{-8}\mathrm{to}{10}^{-2}{\Omega }^{-1}{\mathrm{cm}}^{-1}$ range. Besides this, the value of dielectric constant is very large ${ϵ}_r=11$ due to which the Coulomb force among electrons and holes is insignificant as a result of dielectric screening and, free electrons and holes are created due to the absorption of light at room temperature. On the other hand, organic semiconductors manifest extrinsic conductivity which occurs because of the injection of charges at the electrodes, by doping intentionally or unintentionally and, by separation of photogenerated electron-hole pairs that are bound through the Coulomb attraction. This is a consequence of the following two features: firstly, the emission and absorption lie between 2–3 eV, which is about 600–400 nm; it prevents the creation of substantial charge carrier concentration through thermal excitation, at room temperature. Secondly, considering the low value of the dielectric constant of about ${ϵ}_r=3.5$ suggests that Coulomb interactions are substantial such that any electron-hole pair generated through thermal or optical excitation possess coulomb energy ranging from 0.5 eV to 1.0 eV. Based upon these briefly described dissimilarities between organic and inorganic semiconductors, it is necessary to deeply analyse and understand the electronic structure of semiconductors such that their photophysical properties can be learned, which would further assist in designing and improving the performance of semiconductor devices.

Organic semiconductors are classified into three categories such as 1. Amorphous molecular films [75,76], 2. Molecular crystals [77,78], and 3. Polymer films [79,80].

1. Amorphous molecular films: by using evaporation or spin-coating techniques, the organic molecules are accumulated as an amorphous film. In device applications like LEDs, thin amorphous molecular films are conventionally used, while in xerography the molecularly doped polymer (MDP) films are effectively used. The molecules commonly employed here have similar chemical units normally phenyl rings, with five- or six-membered rings including some heteroatoms like oxygen, nitrogen, or sulphur within it, these are referred to as ‘heteroaromatic rings’. These rings can occasionally have a few carbon atoms by either single or double bonds when present at the centre of the molecule or with a chain of single bonds when producing a side chain.

2. Molecular crystals: a crystal is a combination of basis and a lattice. In general, the way a silicon crystal is obtained using silicon atoms held together by a covalent bond or sodium chloride crystal obtained using Na and Cl atoms joined together by an ionic bond, similarly, naphthalene or anthracene molecules join together through van der Waals interactions to form a crystal. Since the molecular crystals allow greater charge mobilities than non-crystalline organic materials, they are primarily considered for transistor applications. In the case of inorganic crystals such as Si and Ge, atoms form the basis, whereas the molecules form the basis in the molecular crystals. Molecules used for crystal formation are typically large, aromatic and flat like polyacenes specifically anthracene, naphthalene, tetracene, and pentacene including perylene, pyrene, and similar compounds.

3. Polymer films: polymers are formed by the repetition of covalently held molecular chain units. A variety of deposition techniques such as ink-jet deposition, simple spin-coating, or industrial reel-to-reel coating are utilised in solution processing to produce these polymers. Blending is easier with polymers as compared to the molecules since the thermodynamic stability of polymer blends is high with low susceptibility to crystallise. Polymer is derived from two Greek words—‘many’ and ‘part’—which means many repeated single units resulting in a macromolecule. These repeating units are known as monomers. In the organic semiconductor’s terminology, ‘many’ indicates 100 or more repeating units. Semiconducting polymers like polyfluorenes and polyphenylene vinylene (PPV) derivatives have a molecular weight ranging from 50,000–100,000 Da, indicating that these polymers contain about 200–400 repeating units. Any chain with 20 repeating units is known as oligomer (meaning ‘a few’). Molecules with 20–100 repeating units are known as either long oligomer or a short polymer. A few organic semiconductors with their chemical structure are shown in the Figure 3. These include Regioregular poly(3-hexylthiophene-2,5-diyl) (P3HT), tetrathiophene (4T), poly(2,5-bis(3-alkylthiophen-2-yl)thieno[3,2-b]-thiophene) (PBTTT), Poly(4,4-dialkyl-cyclopenta[2,1-b:3,4-$b{}^{\prime }$]dithiophene-alt-2,1,3-benzothiadiazole) (PCPDTBT), Fullerene ($C_{60}$), Pentacene, Naphthalene Diimide–Bithiophene (P(NDI2OD-T2)), [6,6]-phenyl-$C_{61}$-butyric acid methyl ester (PCBM), 6,13-bis(triisopropylsilylethynyl) pentacene (TIPS-pentacene), indacenodithiophene -co-benzothiadiazole (IDT-BT), [1]benzothieno[3,2-b][1]benzothiophene (BTBT), poly(3,4-ethylenedioxythiophene) polystyrene sulfonate (PEDOT:PSS) [81,82,83,84,85].

3. Carrier Generation and Recombination

The charge carrier generation and recombination are essential processes for the operation of many electronic devices. The process through which charge carriers are created is known as carrier generation, while the annihilation of charge carriers is known as carrier recombination. In the case of inorganic semiconductors, the fundamental unit of carrier generation and recombination process is the electron-hole pair. The creation of electron-hole pairs correspond to the transition of electrons from the valence band to the conduction band, and a reverse transition leads to the recombination process. Generation and recombination of carriers is possible in semiconductors, both thermally and optically. According to thermodynamics, materials which are at thermal equilibrium have the balance of generation and recombination rates such that the total charge carrier density will remain constant. Therefore, at equilibrium, the product of electron and hole concentration remains constant $np=n_i^2$ and maintains the generation and recombination rates to occur equally. The recombination rate will be greater than the generation rate if there are excess carriers $np>n_i^2$, bringing the system back to the equilibrium state. Similarly, generation rate will be greater than the recombination rate if there is a deficit of carriers $np<n_i^2$, again bringing the system to the equilibrium state [86]. Carrier generation due to the absorption of light occurs if the energy of the photon is greater than the band gap energy of the semiconductor $E_{ph}>E_g$, such that an electron is excited from valence band to the conduction band with the generation of one electron-hole pair. Even if the energy of the photon is just equal to the band gap energy $E_{ph}=E_g$, photon gets efficiently absorbed and generates one electron-hole pair. However, if the energy of the photon is less than the band gap energy $E_{ph}<E_g$, then photons interact weakly with semiconductor material and pass through it as if they are transparent. Light absorption is an active process in the case of solar cells, photodiodes, and photodetectors [87,88,89]. For instance, we can calculate the generation of electron-hole pair for a solar cell at any location and any wavelength of light or for the standard solar spectrum. Maximum amount of light gets absorbed at the surface of the material, so the generation rate is maximum at the surface. Since light is comprised of many wavelengths, it is necessary to consider different generation rates corresponding to different wavelengths while designing a solar cell. Therefore, the generation rate provides the number of carriers generated at each point within the device with the photon absorption. The light intensity at any point within the device can be calculated using Beer–Lambert’s law [90] given by

$$I=I_0{e}^{-\alpha x},$$

(1)

where $\alpha$ represents the absorption coefficient in cm${}^{-1}$, x represents the distance at which light intensity is measured within the material and $I_0$ is the intensity of light on the top surface of the material. Assuming the absorption of photons (decrease in light intensity) directly creates an electron-hole pair, the differentiation of Equation (1) will provide the carrier generation (G) at any point in the device.

$$G=\alpha N_0{e}^{-\alpha x},$$

(2)

where $N_0$ is the photon flux at the material surface and $\alpha$ is the absorption coefficient. Equation (1) shows the intensity of light decreases exponentially throughout the material and, Equation (2) shows the carrier generation rate is maximum at the surface of the material. Carrier generation in semiconductors is also possible through the application of an external electric field. Such type of generation can be seen in the case of transistors and LEDs.

Carrier recombination is a reverse process to carrier generation. In the case of recombination, excited charge carriers, such as electrons which are available in the conduction band, recombine with holes in the valence band and liberate energy either as heat or photons (light). An explanation of the recombination mechanism in semiconductors is provided through five different types, and these include 1. Radiative recombination [91,92], 2. Shockley–Read–Hall recombination [93,94], 3. Auger recombination [95], 4. Surface recombination [96], and 5. Langevin recombination [97]. The radiative recombination is also known as band-to-band recombination and is dominant in the direct band gap semiconductors. In band-to-band recombination, electrons from the conduction band directly jump to the valence band and combine with holes emitting photons. This represents spontaneous emission, and emitted photons have energy similar to those initially absorbed. Emission of light from a LED is an example of band-to-band recombination. Since the photon has less momentum, this type of recombination is significant in the direct band gap semiconductors. This process is also known as bimolecular recombination [98] (see Figure 4). The Shockley-Read-Hall recombination (SRH) is named after William Shockley, William Read and Robert Hall [99] who reported this recombination in 1962. SRH recombination is also known as trap-assisted recombination, and it is a two step process. In this type of recombination, the electron in the conduction band moves to an extra energy level in the forbidden gap with the liberation of energy either as a photon or multiple phonons. The electron from this trapped energy level recombines with the hole in the valence band further liberating energy as a photon or phonons, as shown in Figure 4. The extra energy states available in the forbidden region are localised states which are introduced within the band gap through defects in the crystal lattice. These defects can be introduced either unintentionally or intentionally through the addition of dopants. Therefore, such energy states are also known as traps. Since differences in the momentum between the charge carriers are absorbed by these traps, SRH recombination is dominant in silicon and in the indirect band gap semiconductors. However, SRH recombination can be dominant in the direct band gap semiconductors that have low carrier density (low injection level) or in the materials such as perovskites that have very high trap density. The auger recombination process involves three charge carriers. In this type of recombination, an electron in the conduction band recombines with a hole in the valence band, but instead of emitting the energy as a photon or heat, the energy is given to the third charge carrier which is another electron available in the conduction band. This energy pushes the electron very high into the conduction band, and then gradually this electron gives off its energy thermally and relaxes back to the conduction band edge as depicted in Figure 4. Auger recombination is significant in heavily doped materials that have high carrier concentration. In the case of silicon-based solar cells, Auger recombination reduces the lifetime and the efficiency. Therefore, more heavily the doping in the material, the shorter is the Auger recombination lifetime. Surface recombination is a trap-assisted recombination that occurs at the surface of the semiconductor material. Due to sudden discontinuity in a semiconductor crystal, dangling bonds are created. These bonds form traps at or near the surface of the semiconductor assisting, the surface recombination process. Surface recombination is described by surface recombination velocity that depends upon the densities of surface defects [100]. Surface recombination is dominant in solar cells, where the collection and extraction of free charge carriers takes place at the surface. A window layer comprising a thin transparent material with a wider band gap is used to reduce the surface recombination in the case of solar cells. The passivation process is also used to minimise this type of recombination [101]. The Langevin recombination mechanism occurs in organic semiconductors as a consequence of opposite polarity charge carriers having low mobilities interfering with each other.

4. Charge Transport in Semiconductors

Semiconductors are capable of conducting electricity and can act as both conductors and insulators. These properties have made semiconductors valuable in the field of electronics and computers. The conductivity in semiconductors generally depend upon temperature, magnetic field, illumination and small amount of dopants. The sensitivity of conductivity plays a significant role in electronics applications. The conductivity in semiconductors is due to the motion of electrons in the conduction band and holes in the valence band with the application of an external electric field [65]. The charge carriers move in the opposite direction, as hole movement in the valence band is due to the electron movement in the opposite direction. Eventually, electrons recombine with holes and are annihilated. The production of electron-hole pairs, including their recombination, is a dynamic process. This is temperature-dependent for intrinsic semiconductors, such that concentrations must be in the equilibrium at a given temperature. The conductivity within semiconductor depends on two main factors: 1. Electron and hole concentrations denoted by ‘n’ and ‘p’, which depend upon the temperature of the sample; 2. The ability of charge carriers to travel in the lattice without getting scattered. Since electrons and holes undergo multiple scatterings with the atoms of the semiconductor material, they are supposed to drift in the lattice. Electron-electron scatterings are generally ignored if the charge carrier concentrations are very small. Conductivity is related to the mobility ($\mu$), which defines the ability of charge carriers to move within the lattice that is in turn related to the scattering time ($\tau$) and effective mass of the charge carriers $m_e^{\ast }$ or $m_h^{\ast }$ given by

$${\mu }_e=\frac{e{\tau }_e}{m_e^{\ast }}\mathrm{and}{\mu }_h=\frac{e{\tau }_h}{m_h^{\ast }}.$$

(3)

Therefore, the expression for conductivity [103] is written as

$$\sigma =ne{\mu }_e+pe{\mu }_h.$$

(4)

According to Equations (3) and (4), the more time there is between two scattering events, the higher the mobility and the higher the conductivity are. From Equation (4), the higher the charge carrier concentration, the greater the conductivity. The charge carrier concentration within an energy band is related to the occupation probability $f\left(E\right)$ and the density of available states $g\left(E\right)$ given by

$$n=\int g(E)f(E)dE,$$

(5)

where integration is performed over the entire band. f(E) is the Fermi function [103,104] given by

$$\mathrm{f}\left(\mathrm{E}\right)=\frac{1}{1+\exp \left(\frac{\mathrm{E}-{\mathrm{E}}_{\mathrm{F}}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)}\approx \exp (-\frac{\mathrm{E}-{\mathrm{E}}_{\mathrm{F}}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}).$$

(6)

The function is approximated by the Boltzmann function for energies much greater than $k_{B}T$, as given by Equation (6). To find the concentration of electrons (n) in the conduction band, Equation (5) can be written as

$$n={\int }_{E_c}^{E_c+\chi }{g}_{CB}\left(E\right)f\left(E\right)dE.$$

(7)

The actual density of states function of the conduction band depends upon the semiconductor material [104]. However, a three dimensional solid with constant potential can be used as an approximation which is given by

$$g_{CB}\left(E\right)=\frac{8\pi \sqrt{2}}{h^3}{\left(m_e^{\ast }\right)}^{\frac{3}{2}}{(E-E_c)}^{\frac{1}{2}}.$$

(8)

Since most of the electrons in the conduction band are confined to the bottom, while $\chi$ is greater than $k_{B}T$, the limits are changed from $E_c$ to ∞ rather than $E_c+\chi$. Making substitutions in Equation (7) and performing integration, the concentration of electrons in the conduction band is given by

$$n=N_{c}exp[-\frac{(E_c-E_F)}{k_{B}T}]\mathrm{where}{N}_c=2{\left(\frac{2\pi m_e^{\ast }{k}_{B}T}{h^2}\right)}^{\frac{3}{2}}.$$

(9)

$N_c$ represents effective density of states at the conduction band edge. $E_c$ represents bottom of the conduction band and $E_F$ represents Fermi level position. Similarly, the concentration of holes in the valence band is given by

$$p=N_{v}exp[-\frac{(E_F-E_v)}{k_{B}T}]\mathrm{where}{N}_v=2{\left(\frac{2\pi m_h^{\ast }{k}_{B}T}{h^2}\right)}^{\frac{3}{2}}.$$

(10)

$N_c$ represents effective density of states at the valence band edge and $E_v$ represents top of the valence band. Therefore, the concentration of electrons and holes in intrinsic or extrinsic semiconductors is given by Equations (9) and (10). The multiplication of n and p from Equations (9) and (10) can eliminate $E_F$ [104] and is given by

$$np=N_c{N}_{v}exp[-\frac{(E_c-E_v)}{k_{B}T}]=N_c{N}_{v}exp[-\frac{E_g}{k_{B}T}],$$

(11)

where $E_g$ is the band gap of the semiconductor material. In case of intrinsic semiconductors, electron and hole concentrations are equal to $n=p$, as electrons and holes are created in pairs, and according to the law of mass action $np=n_i^2$. This condition is valid for any type of semiconductor at equilibrium and $n_i^2$ is the intrinsic carrier concentration which is a material property that depends on the band gap of the semiconductor. By substituting $n_i^2$ in Equation (11), the intrinsic carrier concentration can be written as

$$n_i=\sqrt{N_c{N}_v}exp[-\frac{E_g}{2k_{B}T}].$$

(12)

Using Equation (4), the conductivity for intrinsic semiconductors can be written as

$${\sigma }_i=n_{i}e({\mu }_e+{\mu }_h).$$

(13)

The effect of temperature on the intrinsic carrier concentration can be explained by using Equation (12). Both the exponential and the pre-exponential terms include temperature. There are two effects of increasing the temperature—1. Effective density of states at the band edges $N_c$ and $N_v$ increases with increasing temperature; 2. The exponential term of Equation (12) decreases with increasing temperature. From Equation (13), the conductivity of the semiconductor increases with the increasing intrinsic carrier concentration and (or) the mobility. The mobility decreases with the temperature while the intrinsic carrier concentration increases with temperature because of the pre-exponential term in Equation (12). However, increasing the temperature is not a feasible option because most electronic devices need to operate at or near the room temperature. Therefore, another method must be adopted to increase and control the conductivity in semiconductors, and this is achieved through doping. The addition of dopants to the intrinsic semiconductors can have two advantages—1. They increase the overall conductivity of the semiconductor by increasing the charge carrier concentration of a specific polarity either electrons or holes; 2. Charge carrier concentration can be stabilised around room temperature through the addition of dopants. Since there are two types of extrinsic semiconductors, n-type and p-type, as mentioned in Section 2.1, the n-type semiconductor has donor atoms while p-type semiconductor has acceptor atoms [58]. If the concentration of donor atoms within the lattice is denoted by $N_d$, then there are $N_d$ extra electrons, which are available for the conduction. These electrons are located close to the conduction band such that the concentration of electrons at room temperature is given by $n=N_d$, where $N_d>>n_i$ and the hole concentration can be written using the law of mass action $np=n_i^2$.

$$p=\frac{n_i^2}{N_d}<<N_d.$$

(14)

The acceptor atoms in p-type semiconductor accept electrons from the valence band so their energy levels are located near to the valence band. If the concentration of acceptor atoms is denoted by $N_a$ then, the concentration of holes at room temperature is given by $p=N_a$, where $N_a>>n_i$ and the electron concentration can be written using the law of mass action $np=n_i^2$.

$$n=\frac{n_i^2}{N_a}<<N_a.$$

(15)

Therefore, the conductivity for extrinsic semiconductors of n-type and p-type can be written using Equation (4), as

$$\begin{array}{c}\hfill \mathrm{n}-\mathrm{type}:\sigma =ne{\mu }_e\approx N_{d}e{\mu }_e\\ \hfill \mathrm{p}-\mathrm{type}:\sigma =pe{\mu }_h\approx N_{a}e{\mu }_h.\end{array}$$

(16)

Semiconductors have a very large number of charge carriers that we need to follow to operate the device. It is not feasible to track such a large number of charge carriers even using powerful computers. In such cases, it is assumed that the charge carriers would follow the drift-diffusion model. So, rather than tracking each charge carrier, a weighted statistical average that is equivalent to a quasi-fermi level is followed. This approach is much similar to the way of measuring ‘temperature’ instead of finding the energy of the individual atom upon heating the material. The charge carriers move freely within the semiconductor lattice in a random direction with a certain velocity determined by the mass of the charge carrier and the temperature. In the absence of an electric field, charge carriers travel in a random direction for a specific distance, known as the scattering length, before colliding with a lattice atom. After the collision, the charge carriers travel in a different random direction. With the application of an external electric field, electrons travel in the direction opposite of the electric field whereas holes travel in the direction of the electric field, as described earlier in this section. The transport of charge carriers in the presence of an external electric field is known as ‘drift transport’. In the presence of an external electric field, each electron experiences a force given by $\overrightarrow{f}\left(t\right)=-e\overrightarrow{E}$, where $\overrightarrow{E}$ is the applied electric field and e represents the electron charge. According to the equation of motion of electrons [105],

$$\frac{d\overrightarrow{p}}{dt}=-\frac{\overrightarrow{p}}{\tau }+\overrightarrow{f}\left(t\right),$$

(17)

where the first term in Equation (17) represents the drag force due to collision and the second term represents the force due to electric field. This can be rewritten as

$$\frac{d\overrightarrow{p}}{dt}=-\frac{\overrightarrow{p}}{\tau }-e\overrightarrow{E}.$$

(18)

After a very long interval of time, the current inside the semiconductor material will be uniform then $\frac{d\overrightarrow{p}}{dt}=0$. Equation (18) can be written as

$$-\frac{\overrightarrow{p}}{\tau }-e\overrightarrow{E}=0.$$

(19)

From Equation (19), the momentum ($\overrightarrow{p}$) is then given by

$$\overrightarrow{p}=-e\overrightarrow{E}\tau .$$

(20)

The velocity of electrons and the momentum are related to each other

$$\overrightarrow{v}=\frac{\overrightarrow{p}}{m}=\frac{-e\overrightarrow{E}\tau }{m^{\ast }},$$

(21)

where m and $m^{\ast }$ are the mass and effective mass of electron, respectively. The conduction current density ($\overrightarrow{J}$) is proportional to the velocity of electrons ($\overrightarrow{v}$) and is given by

$$\overrightarrow{J}=-ne\overrightarrow{v}.$$

(22)

Substituting the value of velocity from Equation (21), we obtain

$$\overrightarrow{J}=-ne\left(\frac{-e\overrightarrow{E}\tau }{m^{\ast }}\right)=\frac{n{e}^2\tau }{m^{\ast }}\overrightarrow{E}=\sigma \overrightarrow{E},$$

(23)

where $\sigma =\frac{n{e}^2\tau }{m^{\ast }}$ represents the conductivity of the semiconductor which is related to the mobility of charge carriers given by Equation (3). Hence, the conductivity becomes $\sigma =ne\mu$. Considering both electron and hole conduction (see Equation (4)), the current density is given by

$$\overrightarrow{J}=e(n{\mu }_e+p{\mu }_h)\overrightarrow{E}.$$

(24)

Besides the drift current, there is also a diffusion current within the semiconductor that occurs as a consequence of the non-uniform concentration of charge carriers in the semiconductor material. The diffusion current can be in the opposite or the same direction of the drift current. The drift–diffusion equation [106] describes drift current and diffusion current together. When considering semiconductor devices, especially with junctions, it is essential to describe the diffusion current because both drift current and diffusion current are present within the depletion region of the device. The one dimension diffusion current densities for both electrons and holes can be written as follows:

$$\begin{array}{c}\hfill J_n=e{D}_n\frac{dn}{dx}\\ \hfill J_p=-e{D}_p\frac{dp}{dx},\end{array}$$

(25)

where n and p are electron and hole concentrations, respectively, while $D_n$ and $D_p$ are diffusion coefficients for electrons and holes given by Einstein relation [107]

$$\begin{array}{c}\hfill D_n={\mu }_n\frac{kT}{e}\\ \hfill D_p={\mu }_p\frac{kT}{e}.\end{array}$$

(26)

The net electron current density is obtained by adding drift and diffusion currents,

$$J_n=en{\mu }_e\overrightarrow{E}+e{D}_n\frac{dn}{dx}.$$

(27)

A similar equation can be written for net hole current density

$$J_p=ep{\mu }_h\overrightarrow{E}-e{D}_p\frac{dp}{dx}.$$

(28)

Hence, the total current density is the sum of electron and hole current densities given by

$$J_{Total}=J_n+J_p=en{\mu }_e\overrightarrow{E}+e{D}_n\frac{dn}{dx}+ep{\mu }_h\overrightarrow{E}-e{D}_p\frac{dp}{dx}.$$

(29)

Charge transport in the case of organic semiconductors is described through the hopping mechanism, which is a phonon-supported tunnelling mechanism, where charge carriers hop from site to site. Since the molecules in the organic semiconductors are kept intact by the van der Waals forces, this prevents the charge transport in the valence and conduction bands. The top of the valence band corresponds to the highest occupied molecular orbital (HOMO) and the bottom of the conduction band corresponds to the lowest unoccupied molecular orbital (LUMO). The carrier site energies have Gaussian distribution and are localized [15]. The notion of transport energy [108] is useful for understanding the hopping mechanism in organic semiconductors. All the above equations from (3) to (29) are applicable even for organic semiconductors.

Three different types of transport mechanisms, namely band-like transport, multiple trapping and release theory, and hopping transport, have generally been accepted [109,110,111]. According to the crystallinity or structural disorder of semiconductors, charge transport theories are classified as follows: (1) Single crystalline materials or some conjugated polymers with very low chain torsion [112,113,114] are the majority of the disorder-free organic semiconductors, which are expected to exhibit classic band-like transport. Some small molecules, like tetracyanoquinodimethane (TCNQ) and pentacene, exhibit higher charge carrier mobility than polymer-based semiconductors [115,116]. With a mean free path that is significantly greater than the distance to the nearest neighbour, the carriers for this class of organic molecules travel as highly delocalised plane waves in a wide carrier band. The transport characteristics of charge carriers typically resemble those of metals. The term “band-like transport" refers to this particular transport mechanism. The band-like transport primarily takes place in crystals with delocalised charge carriers; (2) The mobility edge model, which is akin to the multiple trap and release (MTR) model, is applicable for polycrystalline organic semiconductors with a low degree of the structural disorder [117,118]; (3) Charge transport primarily takes place by hopping or tunnelling between localised states in amorphous or highly disordered semiconductors. The most frequently used models are based on percolation theory [117,118,119], Bassler’s Gaussian disorder model [117,120], and variable range-hopping mechanism [121,122]. Some of them are also taken into account for organic semiconductors that is polycrystalline. Moreover, charge carriers are always found on the localised sites for a wide range of polymer-based semiconductors, including polyaniline (PANI), poly(3,4-ethylenedioxythiophene) polystyrene sulfonate (PEDOT:PSS), and poly(2,5-dimethoxyphenylenevinylene) (PMeOPV) [114,117,123] due to spatial and energetic disorder. The localised carriers are always transported in order to jump between localised sites. Transport of this kind is referred to as “hopping transport." When the electronic wave functions of the two sites overlap, tunnelling occurs from one localised site to another. The absorption (or emission) of a phonon compensates for the energy difference whenever a charge carrier hops to a site with a higher (or lower) site energy than the site from which it originated. More details on charge transport mechanisms in semiconductors can be found here [124,125,126,127]. For the modelling of modern semiconductor devices, quantum transport models are widely used, and interested readers are encouraged to refer to the works of Shin, Dong Hoon, et al. (2023) [128] and Ferry, David K. et al. (2022) [129] for a comprehensive understanding of these models. These models consider the quantum mechanical behaviour of electrons, allowing for accurate predictions of device performance at nanoscale dimensions. Furthermore, they provide insights into phenomena such as quantum tunnelling and ballistic transport, which are crucial for the design and optimisation of advanced semiconductor devices.

5. Techniques Used to Investigate Charge Transport in Organic Semiconductors

A variety of techniques were used to investigate charge transport in disordered organic materials. These include electrical transient techniques that are based on charge extraction by linearly increasing the voltage (CELIV) [130] and time-of-flight (TOF) [131], as well as steady-state electrical methods that are based on charge transport in OFETs [132], space charge-limited current (SCLC) [133], and the Hall effect [132].

The CELIV method measures charge carrier mobility in organic solar cells using a triangle voltage to extract intrinsic or photogenerated charge carriers. It does not differentiate electron or hole charge carriers but measures overall device carrier mobility.

The TOF approach determines trap states by measuring hole and electron mobilities individually using fixed biases. In this method, the thickness of the film should be greater than 500 nm, and light illumination is required to ensure that charge carriers can effectively cross the depletion area. This allows for accurate measurement of the mobilities of electrons and holes.

The FET method studies charge transport dynamics in the in-plane direction without a charge-blocking layer. It requires high crystallisation and quality samples to observe the desired field effect. FET measurements can identify n-type, p-type, or ambipolar semiconductors and measure charge mobility. However, FET mobility depends on the interfacial morphology of the sample, grain size, and operating temperature. OFET experiments have higher mobilities than diode structures due to a thin layer at the interface.

The SCLC method measures the mobility of a hole or electron individually in organic semiconductors using blocking electrodes. However, the presence of leakage, intrinsic doping, or charge traps in the materials can complicate the accuracy and implementation of this method. These factors can introduce errors in the measurement of hole and electron mobilities, making it challenging to obtain reliable results. Therefore, it is crucial to carefully consider and address these issues when using the SCLC method for mobility measurements in organic semiconductors.

The Hall effect method evaluates charge carriers and intrinsic defect concentrations in semiconductors, determining n-type or p-type. This technique measures the voltage induced by a magnetic field perpendicular to the current flow. The Hall effect method is widely used in semiconductor characterisation and device fabrication processes. This paper focuses on the charge carrier dynamics in a two-dimensional organic semiconductor diode, which is investigated using the Hall effect method.

When a voltage is applied across a semiconductor device, current flows through it, and electrons drift in the opposite direction. This voltage is called the longitudinal voltage. An additional voltage known as Hall voltage ($V_H$) is produced when the device is subjected to an external magnetic field. This is transverse voltage, which acts orthogonal to the direction of current flow. As a result, the phenomenon of producing Hall voltage with the application of Lorentz force is known as the Hall effect, named after Edwin Hall, who made this discovery in the 1870s [134]. Consequently, charge carriers (electrons) encounter a Lorentz force given by

$${\overrightarrow{F}}_L=-e\overrightarrow{E}-e(\overrightarrow{v}\times \overrightarrow{B}),$$

(30)

where $\overrightarrow{v}$ is the velocity of electrons and $\overrightarrow{B}$ is the magnetic field.

The Hall effect indicates that if the external magnetic field applied to moving charges is in the z-direction and the applied electric field is in the x-direction, then the moving charges experiencing a magnetic force will be in the y-direction. As a result, the equation for magnetic force in the y-direction is given by

$$\begin{array}{c}\hfill {\mathrm{F}}_{\left(\mathrm{magnetic}\mathrm{force}\right)}=\mathrm{e}({\mathrm{v}}_{\mathrm{dx}}\times {\mathrm{B}}_{\mathrm{z}})\\ \hfill {\mathrm{E}}_{\mathrm{y}}=\frac{{\mathrm{F}}_{\left(\mathrm{magnetic}\mathrm{force}\right)}}{\mathrm{e}}={\mathrm{v}}_{\mathrm{dx}}\times {\mathrm{B}}_{\mathrm{z}}\\ \hfill {\mathrm{E}}_{\mathrm{y}}=\frac{{\mathrm{F}}_{\left(\mathrm{magnetic}\mathrm{force}\right)}}{\mathrm{e}}={\mathrm{v}}_{\mathrm{dx}}{\mathrm{B}}_{\mathrm{z}}\sin \theta ,\end{array}$$

(31)

where ‘e’ is the charge of an electron, $v_{\mathrm{dx}}={\mu }_n{\mathrm{E}}_{\mathrm{x}}$ is the drift velocity of electrons in the x direction, $\theta ={90}^0$ and ${\mathrm{B}}_{\mathrm{z}}$ is the magnetic field that is being applied in the the z direction [135]. The magnetic force acting on electrons and holes is given by

$${\mathrm{F}}_{\left(\mathrm{magnetic}\mathrm{force}\right)}=\mathrm{e}{\mu }_n{\mathrm{E}}_{\mathrm{x}}(-{\mathrm{B}}_{\mathrm{z}})$$

(32)

$${\mathrm{F}}_{\left(\mathrm{magnetic}\mathrm{force}\right)}=\mathrm{e}{\mu }_p{\mathrm{E}}_{\mathrm{x}}(-{\mathrm{B}}_{\mathrm{z}}),$$

(33)

where the −ve sign of magnetic field indicates that the magnetic force is in the +ve y direction. The electron conduction current density in y direction is given by

$$\begin{array}{c}\hfill {\mathrm{j}}_{\mathrm{n}}(\mathrm{y},\mathrm{t})=\mathrm{e}{\mu }_n\mathrm{E}(\mathrm{y},\mathrm{t})\mathrm{n}(\mathrm{y},\mathrm{t})+{\mu }_n{\mathrm{k}}_{\mathrm{B}}\mathrm{T}\frac{\partial \mathrm{n}}{\partial \mathrm{y}}\\ \hfill {\mathrm{j}}_{\mathrm{n}}(\mathrm{y},\mathrm{t})={\mu }_n\mathrm{F}(\mathrm{y},\mathrm{t})\mathrm{n}(\mathrm{y},\mathrm{t})+{\mu }_n{\mathrm{k}}_{\mathrm{B}}\mathrm{T}\frac{\partial \mathrm{n}}{\partial \mathrm{y}}.\end{array}$$

(34)

Similarly, the hole conduction current density in y direction is given by

$$\begin{array}{c}\hfill {\mathrm{j}}_{\mathrm{p}}(\mathrm{y},\mathrm{t})=\mathrm{e}{\mu }_p\mathrm{E}(\mathrm{y},\mathrm{t})\mathrm{p}(\mathrm{y},\mathrm{t})-{\mu }_p{\mathrm{k}}_{\mathrm{B}}\mathrm{T}\frac{\partial \mathrm{p}}{\partial \mathrm{y}}\\ \hfill {\mathrm{j}}_{\mathrm{p}}(\mathrm{y},\mathrm{t})={\mu }_p\mathrm{F}(\mathrm{y},\mathrm{t})\mathrm{p}(\mathrm{y},\mathrm{t})-{\mu }_p{\mathrm{k}}_{\mathrm{B}}\mathrm{T}\frac{\partial \mathrm{p}}{\partial \mathrm{y}}.\end{array}$$

(35)

The Hall voltage generated in y direction is

$${\mathrm{V}}_{\mathrm{H}}\left(\mathrm{t}\right)={\int }_0^{\mathrm{d}}\mathrm{E}(\mathrm{y},\mathrm{t})\mathrm{dy}.$$

(36)

A more detailed study on charge transport in two-dimensional organic semiconductors using the Hall effect method can be found here [136,137,138,139]. To better understand the charge transport properties of organic semiconductors and improve the device’s performance, the charge carrier concentration and Hall voltages obtained using the Hall effect method are significant figures of merit.

Hall effect measurements are significant for demonstrating the intrinsic mobility and fundamental charge transport process in organic semiconductors. The Hall effect measurements provide a non-destructive method to determine the carrier concentration, dopant concentration, and doping types in organic semiconductors, which are important parameters for optimising device performance. These measurements can also help to understand the charge transport mechanisms in these materials and guide the development of new materials with improved properties. This technique can also provide insights into the electronic properties of materials and their potential applications in electronic devices.

6. Summary and Prospects

After the invention of transistors about 50 years ago, inorganic semiconductors such as Si and Ge, which nowadays have formed the basis of all modern electronic devices. Silicon microelectronics has enabled new emerging technologies such as digital computers, smartphones, tablets, laptops, smart accessories, internet, and wireless systems. Electronic circuits comprising semiconductors are present in home appliances, cars, automobiles, and machinery. The evolution of organic semiconductors in recent years have opened various applications of electronic and optoelectronic devices in everyday life, for instance, fibre optic communication for transferring data, organic solid state drives (O-SSD), light detection and ranging (LIDAR), visible light communication (LIFI), light navigation and ranging (LINAR), big data servers, cloud computing and fog computing devices, quantum computing, supercomputers, digital signal processing chip and micro processors, micro controllers, routing devices, system on chip (SoC), embedded system design, ultra large scale integration (U-VLSI) design, high speed-low latency internet of things (IoT) devices, digital cameras, solar cells, solar photovoltaic panels and solar impulse aircraft. In summary, this review has discussed the early research and classification of both inorganic and organic semiconductors, carrier generation and recombination processes in semiconductors, charge transport in semiconductors, and various techniques employed to investigate charge transport in organic semiconductors with an emphasis on the Hall effect method. Furthermore, it has been shown that the notion of charge transport and carrier recombination are essential in understanding the device operation.

In the future, it is imperative to explore the simulation and experimental aspects of the Hall effect method for organic semiconductor devices in the presence of dopants, traps, and different types of carrier recombinations. Moreover, further investigation is needed into the impact of temperature and environmental conditions on the performance of organic semiconductor devices. Understanding how these factors affect device operation will help in designing more robust and stable devices for real-world applications. By combining simulation and experimental approaches, researchers can gain a deeper understanding of the Hall effect method’s capabilities and limitations in the context of organic semiconductor devices. This comprehensive exploration will contribute to the continued advancement and refinement of this technique, ultimately leading to more efficient and reliable electronic devices. Additionally, exploring the simulation and experimental aspects will also enable researchers to uncover novel insights and possibilities for utilising organic semiconductors in emerging electronic applications.