A revision of the semiconductor theory from history to applications

Abstract

Semiconductors play a crucial role in modern technology across various fields. The term “semiconductor” was introduced in the XVIII century, marking the beginning of a journey filled with discoveries and technological advancements. This article offers a comprehensive review of the historical landmarks in semiconductor development and explores the associated phenomena concerning different types of photodetectors. It also examines the key performance metrics of commonly used semiconductor materials, considering the structural variations. Additionally, the article highlights various applications of semiconductors, illustrating their significance in everyday life. By doing so, it aims to engage new readers while providing a foundational understanding for those interested in delving into this field.

1 Introduction

Semiconductors are an essential part of everyday technology, however, there were a few milestones along the way, until they reached this worldwide impact. In order to highlight them, their history is presented in the following sections, starting from the first half of the XVIII century, with the discovery of the phenomena, until today. Further sections delve into the minute of the aforementioned semiconductors, related phenomena, types, models and applications.

1.1 XVIII and XIX centuries

In 1731, Stephen Gray wrote a letter to Cromwell Mortimer, the secretary of the Royal Society, containing a description of several experiments concerning electricity. He discovered that some materials were able to transfer electrical charges. Thus, the phenomenon of electric conductivity was discovered. In 1742, Desagulliers in a “Dissertation sur I’electricite des corps” established the differences between electric and non-electric bodies/entities, leading to the introduction of the term “electric conductor”. According to G. Busch, the term “semiconducting” was introduced by Alessandro Volta in 1782, who constructed the electric battery for the first time, the famous pile Volta. He discovered that, by touching a charge electrometer with different materials, according to the material, its contact caused a discharge of the electrometer at different speeds. Metals do it instantaneously, semiconductors slowly and insulators not at all. Later, Humphry Davy found that “the conducting power of metallic bodies varied with temperature, and was lower, in some inverse ratio, as the temperature was higher”. However, in 1833, Michael Faraday concluded that silver sulphide’s resistivity ($\hbox {Ag}_2$S), which he called sulphurette of silver, decreases with temperature. This phenomenon was different from the dependence observed in metals. He verified that at room temperature its resistivity was rather high, but at $175^{\circ }$ the resistivity decreased abruptly to nearly ’metallic’ magnitude [1,2,3,4].

In the first half of the XIX century, in 1839, Alexandre-Edmond Becquerel verified the existance of an electric current when two plates of platinum or gold immersed in an acid, neutral, or alkaline solution are exposed to solar radiation [1,2,3, 5]. In solids, the photoconductivity was discovered, in 1873, by Willoughby Smith. He observed an abrupt decrease in resistivity for selenium (Se) sample irradiation by light [1,2,3, 5]. Until then, the photovoltaic effect had not yet been verified, although it led to one of the major strands of this area: the photovoltaic cell. Three years later, William Grylls Adams and Richard Evans Day observed the photovoltaic effect in a solid material: selenium (Se) [1, 3, 5]. In 1874, Karl Ferdinand Braun figured out the rectifier effect in contacts between some oxides and metals, leading to the discovery of the first semiconductor rectifier, and later the simplest electronic device: the diode. Arthur Schuster observed rectification in a circuit made of copper wires, but only when the circuit was not used for some time. Thus, a new semiconductor was found (copper oxide) [3]. In the later years of the 70th decade of the XIX century, an important phenomenon to study semiconductors’ properties was discovered: the Hall effect by Edwin Herbert Hall. This phenomenon demonstrates that, in solids, charge carriers are deflected when a magnetic field is applied [1, 4]. An in detail explanation of this phenomenon is conducted further, in Sect. 2.8. In 1883, Charles Fritts developed the first working photovoltaic solar cell, with an efficiency lower than 1% [1, 3]. Four years later, Hertz’s experiments showed the existence of radio waves, through a coherer, which proved to be difficult to use and not very sensitive. It was through these experiments that the novel rectifiers started to be used [3].

In the later years of the referred century, J. J. Thomson was the first to suggest that one of the fundamental units of the atom was more than 1000 times smaller than an atom, hinting at a subatomic particle, now known as the electron. Hence, several scientists proposed theories of electron-based conduction in metals [1, 2, 4].

1.2 XX century

At the beginning of the XX century, the first patent for a semiconductor rectifier of galena (PbS) was registered by Sir J.C. Bose. In 1904, Sir John Ambrose Fleming, based on the Edison effect, developed a diode, which is a two-electrode vacuum tube rectifier [3, 6]. Increasing the radio’s use and the development of semiconductor rectifiers, G. Pickard tested many materials. The aim was discovering the most effective material to develop a detector of radio waves. In 1906, G. Pickard used fused silicon that he obtained from the Westinghouse Electric Company and concluded that this material was the most stable of the rectifiers tested by him. And so, the emergence of silicon and germanium semiconductors led to a huge leap in the development of technologies [1, 3]. Around this time, selenium (Se) was shown conclusively as a semiconductor. Karl Baedeker observed that copper iodide (CuI) had positive charge carriers, translating into a Hall effect with the reverse sign to that in metals. In 1907, Lee de Forest took out a patent for a triode, which he called Audion. In this device, the additional electrode (placed between the anode and cathode) enabled the apparatus to function as a power amplifier for analogue signals. Later, in 1914, Johann Koenigsberger, divided materials, according to their conductivity into three groups: metals, insulators and semiconductors [1, 4].

In 1917, in an article related to radiation’s quantum theory, Albert Einstein introduced the foundations that, years later, rose to the maser and, later, the laser. For the first time in quantum theory, Einstein introduced the probability of three processes related to the interaction between matter and radiation, like Einstein’s coefficients for spontaneous absorption and stimulated emission [7].

In the 20th and 30th, the development of barrier layer rectifiers and photovoltaic cells increased the interest in this field. Felix Blochh formulated the principles concerning electrons within lattices, while Bernhard Gudden highlighted that the characteristics observed in semiconductors stemmed solely from impurities. Rudolf Peierls introduced the notion of bandgap, which was subsequently applied to solids by Brillouin. Alan Wilson further expanded upon this band theory [1]. In 1928, Rudolph W. Landenburg confirmed the existence of the stimulated emission [7]. It was about this time that Julius Lilienfeld obtained patents for devices which are known nowadays as MESFET and MOSFET [1]. In 1931, Werner Heisenberg developed the concept of hole, and seven years later, Walter Schottky and Neville F. Mott, recipients of the 1977 Nobel Prize in Physics [8], pioneered advancements in understanding the potential barrier and properties of metal–semiconductor junctions. Schottky later refined his model to account for the influence of space charge [1, 3, 4]. In 1938, Valentin A. Fabrikant predicted that emission for electromagnetic waves’ amplification [7], and Boris Davydov presented a theory of a copper-oxide rectifier. He identified the p-n junction’s effect on the oxide, excess carriers, and recombination, as well as the surface states [1]. In the same year, Walter Schottky described the physics of rectifying junctions [9]. Four years later, Hans Bethe developed the theory of thermionic emission, which led to the Nobel Prize, in 1967 [1, 10]. It was around this time that Russell Shoemaker Ohl, during his work on the detection of radio waves, accidentally discovered, through his technological experiments, the impurities that created the p-n junction, leading to four patents on silicon detectors and p-n junction [1, 2, 11].

In the middle of the century, Alfred Kastler, who won the Nobel Prize for Physics in 1966 [12], proposed the method of optical pumping, which was very important for obtaining the maser and laser effects [7]. In 1945, William Shockley introduced the notion of a semiconductor amplifier functioning via the field-effect principle [1]. Thus, when a transverse electric field is applied, the conductance of a semiconductor layer would change, which has not been verified experimentally. John Bardeen thought that this was due to surface states screening the bulk of the material from the field and, two years later, John Bardeen published it [1]. In the same year, he and Walter Brattain constructed the first transistor, which is a germanium (Ge) point-contact transistor. William Shockley and his team continued with this research and, a year later, it was announced the first mechanically solid junction-type transistor [1, 3, 6], which led to the Nobel Prize for Physics, in 1956 [3, 13]. A. I. Gubanov pioneered the theoretical analysis of current–voltage patterns in both isotypic and anisotypic heterojunctions. Yet, significant theoretical strides during this nascent phase of heterostructure exploration were also made by H. Kroemer. He conceptualized quasi-electric and quasi-magnetic fields within seamless heterojunctions and speculated about their potential for significantly enhanced injection efficiencies compared to homojunctions [14].

In 1952, the first grown junction transistors were manufactured and were reported an alloyed junction transistor with indium (I) and silicon (Si), which had a simpler production, having a base width of 10 $\mu$m. This allowed the device to operate up to a few MHz [1]. In 1953, the first microwave amplifier was built by Charles H. Townes. This device worked with similar principles to the laser’s but with microwave, known as Microwave Amplification by Stimulated Emission of Radiation. However, it did not work in continuous mode, a problem that was solved and led to a Nobel Prize in Physics in 1964 [7, 15,16,17,18,19]. In the following year, the first diffused germanium (Ge) transistor, with a base width of 1 $\mu$m and a cut-off frequency of 500 MHz, was produced as well as the first commercially available silicon device by Gordon Teal. The first diffused silicon (Si) transistor appeared in 1955 [1]. In 1955, Tokyo Tsushin Kogyo (Tokyo Telecommunications Engineering Corporation), which is known nowadays as Sony, created the first Japanese transistor TR-55 radio. The Japanese team, while trying to improve the performance of a transistor manufacturing process, found the quantum mechanical effect, which is known as the tunnelling effect [6]. Then, Leo Esaki, as part of a project, constructed a structure with two different semiconductor layers and a thin dielectric layer and discovered the tunnelling effect occurred within this structure. Esaki diode was the first device developed where this quantum effect was applied, being the first made of germanium (Ge) and the second made of silicon (Si) [1, 6]. These experiments led them to the Nobel Prize in Physics, in 1973 [1, 20]. In 1958, the laser’s theoretical birth was reported by Charles Townes and Arthur Schawlow, known as Optical Maser [21], and, in the following year, the term laser was proposed for Light Amplification by Stimulated Emission of Radiation [7, 18, 19, 22]. Yet, the advent of the semiconductor laser in 1962, particularly the GaAs semiconductor laser diode hinged on a p-n junction, sparked a transformative wave in laser technology. This marked a pivotal stride in optoelectronics, showcasing the feasibility of crafting compact, cost-effective lasers, akin to electronic devices, thus laying the groundwork for mass production with promising prospects [22]. And, since then, it has been a powerful source for communication systems. Jack Kilby showcased the inaugural integrated circuit, consolidating multiple components onto a single silicon substrate and linking them through wire bonding. However, this represented a disadvantage, with the formation of interconnects using the deposition of aluminium (Al) on a layer of silicon dioxide ($\hbox {SiO}_2$) covering the semiconductor material [1].

In May 1960, the first optical laser was created. It used a synthetic ruby and obtained a red laser beam, with 694 nm of wavelength [7, 18, 19, 22], also, in the same year, a transistor with an epitaxial layer added was reported, and Jean Hoerni proposed the planar transistor, where the oxide that served as a mask was not removed and acted as a passivating layer [1]. In 1963, Zhores Alferov, who later won the Nobel Prize in Physics in 2000, pioneered the development of semiconductor lasers, creating a heterostructure that emitted laser light independently of Herbert Kroemer’s work during the same period. These semiconductor lasers have since played a crucial role in signal transmission through optical fibers and the storage as well as retrieval of data [14, 23].

During the 1960 s, optical fibers were available to use in devices that needed a short fiber length. However, they presented high-loss values. In 1966, with the intent of reducing this loss, it was suggested that if the impurities of the silica glass (part of its constituency) were removed, then the optical fibers might be the best choice for optical communications [24]. The idea of using glass fibers was revolutionary, but, to use them in optical communications, their losses had to be much lower, which was achieved in the 1970 s. At that time, it was reported that fiber losses were reduced to values lower than 10dB/km, in the wavelength region near $\sim$800/900 nm [24,25,26], where the best optical sources developed emit. In this wavelength region, silicon proved to be the optimal detector material. In the last year of the decade, the optical fiber’s losses were reduced to 0.2 dB/km, in the infrared region, near 1.55 $\mu$m, which is close to the fundamental limit set by Rayleigh scattering’s phenomenon. Until then, several changes were implemented to achieve lower loss values, like replacing titanium with germanium as a dopant inside the fibre’s silica core [24, 25].

In 1971, Masatoshi Shima at Busicom and Federico Faggin at Intel developed a universal logic processor design called Intel 4004. It was the world’s first single-chip microprocessor and six years later, Steve Jobs released the world’s first personal computer Apple II [6].

At the beginning of the 1980 s, Fujio Masuoka, employed by Toshiba at the time, invented the flash memory. This groundbreaking invention led to the development of a rewritable semiconductor memory device that maintains data even when power is shut off, making it non-volatile. Concurrently, advancements in semiconductor technology enabled Nintendo to introduce a video game console equipped with an 8-bit CPU. Subsequently, the company launched the Super Famicom (Family Computer), a 16-bit game console, further showcasing the progression in gaming technology [6]. The concept of photonics and optoelectronic devices emerged in the late 1980s. Consequently, there is a strong push towards developing photonic devices that operate at 1.55 $\mu$m, allowing direct connection to external servers without the need for wavelength conversion. So far, various types of photonic devices, such as light emitters, optical modulators, and photodetectors, have been successfully developed within this wavelength range [27]

1.3 Important historical milestones

Over the years, several important historical milestones occurred in the semiconductor field. Firstly, the theory was introduced, theoretical discoveries which, years later, led to experimental proof. Devices using semiconductors were initially constructed using empirical knowledge before semiconductor theory provided a guide to building more capable and reliable devices. As in any field of science, the first discoveries and proofs are the basis for what is to come. The semiconductors’ history rewards 1782, when the term “semiconductor” was introduced. From there, several effects and devices were discovered and developed. And they both came hand in hand, as advancements in theory led to new and improved devices that in turn propelled the boundaries of the applications and the theory further as well. To point out the main important historical events, a timeline is depicted, which is presented in Fig. 1.

Fig. 1

Timeline of the most important historical milestones throughout the last centuries

As can be seen, it was throughout the last century that the most extraordinary discoveries in the history of semiconductors took place. It was during this century that technology appeared and took its first steps. Indeed, we’re constantly evolving, but since the beginning of the XXI century and until then, evolution has occurred mainly in terms of applications, efficiencies, and optimisation.

The XXI century has seen the integration of electronic devices into people’s daily lives, with greater or lesser capacity, more or less speed, and greater or lesser definition. Numerous digital consumer goods rely on semiconductors to enhance their performance and lifespan, spanning from smartphones, digital cameras, and televisions to washing machines, dryers, ovens, refrigerators, and even LEDs. In the first decade of the century, research in quantum bits, algorithms, and optics made progress. It was here that the era of nanotechnology was ushered in. In 2007, the announcement of the iPhone by Steve Jobs changed completely the landscape of the global mobile phone industry. Three years later, it was announced Apple’s fourth innovation after Macintosh, iPod, and iPhone: iPad. Then, wearable devices started to appear, starting with the Google Glass and later the Apple Watch. Wireless communications like deep learning and artificial intelligence began to develop, which allowed the development of devices such as drones, and autonomous vehicles [6]. Semiconductors play a vital role in both economic competitiveness and national security. According to the GSM Association, it is projected that by 2025, 5 G networks will encompass approximately one-third of the global population, with 5 G connections surpassing 1 billion by the close of 2022 and 2 billion by 2025. This trend indicates a significant portion, over a fifth, of mobile connections. Consequently, chipmakers anticipate a lucrative new market in the forthcoming years [28].

Besides that, it is due to the semiconductors’ evolution that the fight against climate change can be realised, as they play a crucial role in the field of renewable energy. Forbes states unequivocally that “semiconductors, particularly power semiconductors, will play a central role in the transition of our electrical grid from a fossil fuel-dependent economy to one driven by sustainable, renewable energy sources.” [29].

2 Semiconductors theory

Semiconductors are materials that present unique electronic properties. There are two types of semiconductors: elemental and compound semiconductors. The first one can be found in group IV of the periodic table, such as silicon (Si) and germanium (Ge), while the second one is formed from combinations of materials from groups III and V (III-V semiconductors) or II and VI (II-VI semiconductors) of the periodic table. Gallium arsenide (GaAs), indium gallium arsenide (InGaAs) and indium gallium arsenide phosphide (InGaAsP) are examples of compound semiconductors.

Semiconductors have many optoelectronic properties, which makes them, beyond their unique position in the electronics industry, key materials for many important optoelectronic devices, such as light-emitting diodes, semiconductor lasers, and photodetectors [30].

2.1 Material structure

Solid semiconductors can appear as amorphous, polycrystalline or single crystal materials. An amorphous material has no organization, meaning that the atoms are sited in random positions, and it is characterised by atomic or molecular structures that are relatively complex. [31, 32]. Polycrystalline materials are solids that consist of many small crystals, with a size ranging from nanometres to millimetres, called micro-crystals, which are separated by their boundaries and normally have random crystallographic orientations. The atoms are packed loosely in the region of micro-crystals boundaries, making them mechanically and chemically unstable. Thus, cracks and corrosion occur more frequently at grain boundaries. [32]. Single crystal materials exhibit a systematic three-dimensional configuration of atoms, ions, or molecules throughout their entirety.

The semiconductor properties are strongly dependent on the spatial arrangement of their atoms. Elemental semiconductors have an electronic configuration with 4 valence electrons, which means that to obtain a stable electronic configuration, 4 other electrons have to be added. These electrons create covalent bonds with neighbouring atoms, and their spatial arrangement is equivalent to having one of them in the centre of the tetrahedron and the other atoms added to its vertices [9, 33]. There are some basic crystalline structures, some represented in Fig. 2, in which the simple cubic has 1 atom/cell, the body centred cubic has 2 atoms/cell, and the face centred cubic and the simple hexagonal has 4 atoms/cell and 3/atoms/cell, respectively [34, 35].

Fig. 2

Basic crystalline structures

Compound semiconductors can be binary, if they are composed of 2 different elements, ternary and quaternary, if they are composed of 3 and 4 different elements, respectively. Ternary semiconductors are composed of two different binary semiconductors, which have one element in common. The other two elements are from the same group of the periodic table (e.g. AB and CB are binary semiconductors, and the obtained ternary’s is $\hbox {A}_x\hbox {C}_{1-x}\hbox {B}$, being $0 \le x \le 1$, being x the concentration of A in relation to C). Following the same line of thought, a quaternary semiconductor is written as $\hbox {A}_x\hbox {B}_{1-x}\hbox {C}_y\hbox {D}_{1-y}$, where $0 \le x,y \le 1$ and the binary semiconductors AC, AD, BC and BD [36, 37].

There are four types of crystalline lattice: Diamond, Zincoblenda, Wurtzite and Rock salt, which are represented in Fig. 3.

Fig. 3

Crystalline Lattice

A diamond lattice is a cube characterised by the value of its edge, known as the lattice constant, represented in Fig. 3 as a. It is the periodic repetition of this cube in the three directions of space that gives rise to the crystal. Note that this lattice corresponds to two body centred cubic (Fig. 2b), in which one of them has moved along an interior diagonal to the other. The Zincblende lattice is the most common in compound semiconductors. Its structure is identical to the diamond’s, however, according to the type of compound semiconductor, the atoms inside the cube are different from the others or are replaced by another from the same group, which is the case of the binary semiconductors and ternary semiconductors, respectively. In both crystalline lattice, the atoms are joined together to form a tetrahedron The Wurtzite lattice can be considered as two compact hexagonal networks displaced relative to each other along the height, each one with its type of atoms. This lattice presents two lattice constants: a and c. Lastly, the rock salt is a simple cubic structure (Fig. 2a), in which the two types of atoms are alternately equispaced. This is also two face centred cubic (Fig. 2c), each one with its type of atoms [33, 34, 36].

Table 1 presents the crystalline lattice and constants for several semiconductors [30, 36]. Some of them present more than one crystalline lattice due to its fabrication conditions.

Two crystals with the same lattice structure and the same lattice constant are known as latticed matched.

Table 1 Material structure of several elemental or compound semiconductors

2.2 Compounds semiconductors’ properties

Compound semiconductors’ properties, namely ternary and quaternary semiconductors, vary according to the binary semiconductors’. The lattice constant, as well as the electrical permittivity, present a linear dependence on the concentration, in opposition with energy bandgap, charge carriers mobility and thermal conductivity, which vary non-linearly with the concentration.

For ternary and quaternary semiconductors, the lattice constant varies according to the values of x and y. Considering $a_{AB}$ and $a_{CB}$ as the lattice constant of the binary semiconductor AB and CB, respectively, the lattice constant of the ternary semiconductor by given by Eq. 1. For quaternary semiconductors, the lattice constant is described by equation 2, where $a_{AD}$, $a_{CB}$ and $a_{CD}$ are the lattice constants of the binary semiconductor [36, 37]. Figure 4 presents the lattice constant as a function of the percentage of x and y for several ternary and quaternary semiconductors.

$$\begin{aligned} a_{A_xC_{1-x}B}= & {} a_{AB}x + a_{CB}(1-x) \end{aligned}$$

(1)

$$\begin{aligned} a_{A_xC_{1-x}B_yD_{1-y}}= & {} a_{AB}xy + a_{AD}(1-y) \nonumber \\{} & {} +a_{CB}(1-x)y+a_{CD}(1-x)(1-y) \end{aligned}$$

(2)

Fig. 4

Lattice constant a as a function of the percentage of x and y

The energy bandgap of a compound semiconductor depends on the band type of each binary semiconductor. Considering $E_{G1}$ and $E_{G2}$ as the bandgap energy of the two binary semiconductors, the energy bandgap of the compound semiconductor is given by Eq. 3, being $E_{G2}$ defined by Eq. 4.

$$\begin{aligned} E_G(x)= & {} E_{G1}(1-x) + E_{G2}x + c_2(x^2-x) \end{aligned}$$

(3)

$$\begin{aligned} E_{G2}= & {} E_{G1} + c_1 + c_2 \end{aligned}$$

(4)

If both binary semiconductors are direct bandgap, then the bandgap energy could be obtained using Eq. 3. Otherwise, the bandgap energy function is a branch function. The value that defines the separation of both branches is the value from which the transition from direct to indirect band binary occurs [30, 36].

The bandgap energy of a semiconductor can be correlated with the wavelength and translates the minimum energy of the absorbed photons. This relationship is based on the definition of the photon energy and considering $E_G$ as its value, Eq. 5 translates the relation between the energy bandgap and the wavelength $\lambda$, being h the Planck’s constant and c the light’s speed.

$$\begin{aligned} \lambda = \frac{hc}{E_G} \end{aligned}$$

(5)

Fig. 5 shows the relation between the bandgap energy and the lattice constant for some semiconductors. Both parameters can be related to the wavelength, as can be seen in the background of the figure.

Fig. 5

Lattice constant vs Bandgap Energy for some semiconductors

2.3 Band model

The electron is a quantum particle, having corpuscular and wave characteristics, both linked through Broglie’s relation, defined by Eq. 6, which was proposed in 1924 [9, 34, 36, 38]. In Eq. 6, h corresponds to the Plank’s constant, $\hbar$ is defined by $\frac{h}{2\pi }$, p is the electron linear momentum, k the wave number and $\lambda$ the Broglie’s wavelength.

$$\begin{aligned} p = \frac{h}{\lambda } = \hbar k \end{aligned}$$

(6)

In solids, like semiconductors, the electrons are exposed to interactions over very short distances. Because of that, the quantum mechanics formalism and the Schrödinger equation are used [34]. In a semiconductor, the allowed states of electrons from its building blocks’ atoms create uninterrupted energy bands rather than distinct levels. The optical phenomena linked with these electrons heavily depend on the properties of these energy bands. In semiconductors, the energy bands are separated by an energy bandgap defined as the difference in energy between the minimum of the conduction band, $E_C$, which is empty at T = 0 K, and the maximum of the valence band, $E_V$, which is completely filled at T = 0 K [30, 34, 36].

Regarding the conduction and valence bands, the semiconductors are divided into direct and indirect semiconductors, which diagrams are presented in Fig. 6.

Fig. 6

Energy as a function of momentum’s diagram for both types of semiconductors

In a direct semiconductor, the top of the valence band, i.e. the highest point of the valence band, is aligned with the bottom of the conduction band, i.e. the lowest point of the conduction band. This means that both points have the same momentum. In this case, it is also necessary to ensure energy conservation. In opposition, in an indirect semiconductor, the top of the valence band is not aligned with the bottom of the conduction band, having different momentum values. [9, 25, 30, 33, 36, 39]. So, stimulating the movement of an electron from the valence to the conduction band requires energy supplied by a photon and momentum facilitated by oscillations of the crystal lattice, which are delineated both as waves (vibrations) and as particles (phonons). A phonon is a quantized mode of lattice vibrations and, as is shown in Fig. 6b, it can transfer momentum (from the lattice) to the electron useful to give it the energy to pass to the conduction band [9].

The energy of the electrons in the conduction and valence band (at least in direct semiconductors), whose values are close to $E_C$ and $E_V$, respectively, can be described through Eqs. 7a and 7b, respectively, being $m_n^*$ and $m_v^*$ the effective masses of electrons. The $m_n^*$ value is obtained through the Eq. 8.

$$\begin{aligned} E= & {} E_C + \frac{p^2}{2m_n^*} \end{aligned}$$

(7a)

$$\begin{aligned} E= & {} E_V + \frac{p^2}{2m_v^*} \end{aligned}$$

(7b)

$$\begin{aligned} m_n^* = \frac{1}{\frac{\partial ^2 E}{\partial p^2}} \end{aligned}$$

(8)

The variation in effective mass is due to the concavity of the E(p) curve, i.e. it is related to the “opening” of the parabola that describes the conduction band. However, the concavity of the valence band is negative, as well as the $m_v^*$ value. This is where the concept of holes comes in, considering particles with positive mass, $m_p^* = - m_v^*$. Thus, the electron in the conduction band is a free particle with constant potential energy $E_C$ and, similarly, the hole is a free particle with potential energy $E_V$ and null kinetic energy on top of the valence band [9, 30, 34, 36].

2.4 Density of states

According to Pauli’s exclusion principle, only one electron can occupy a given quantum-mechanical state at a time. In other words, a maximum of two electrons can inhabit the same orbital, each with opposite spins. Thus, the number of electrons in a particular energy band is determined by the number of available states in that band and the probability of occupancy for each state [34].

The electrons’ behaviour in the semiconductor is defined by the Schrödinger equation, which results in the probability of finding a particle at a given instant in a given region of space. The density of states quantifies the number of allowed electron (or hole) states within a given volume at a specific energy level. Its derivation stems from fundamental principles of quantum mechanics. The density of states at an energy E in the conduction band close to $E_C$ and in the valence band close to $E_V$ are given by Eqs. 9a and 9b. Figure 7 shows the density of states function for the conduction, $g_C(E)$, and valence bands, $g_V(E)$. As can be seen, Eq. 9a is applied for values of energy E higher than $E_C$ and Eq. 9b is valid for energy values E lower than $E_V$ [9, 30, 34, 36, 40].

$$\begin{aligned} g_C(E)= & {} 4\pi \left( \frac{2m_n^*}{h^2}\right) ^\frac{3}{2}\sqrt{E-E_C} \end{aligned}$$

(9a)

$$\begin{aligned} g_V(E)= & {} 4\pi \left( \frac{2m_p^*}{h^2}\right) ^\frac{3}{2}\sqrt{E-E_V} \end{aligned}$$

(9b)

Fig. 7

Density of states functions

For a solid material in thermal equilibrium at a temperature T, the probability of any electronic state at an energy E being occupied by an electron is determined by the Fermi-Dirac distribution function, considering Pauli’s exclusion principle and the electrons cannot be distinguished. Mathematically, it is described by Eq. 10, where $E_F$ is the Fermi energy or Fermi level and $k_B$ is the Boltzmann constant [9, 30, 33, 34, 36, 40].

$$\begin{aligned} f(E) = \frac{1}{1+e^{\left( \frac{E-E_F}{k_BT}\right) }} \end{aligned}$$

(10)

2.5 Semiconductors in thermal equilibrium

The performance of a semiconductor device depends on the carrier concentration within the semiconductor material. They are charge carriers and their movement is electrical current. In thermal equilibrium, the semiconductor devices are under an unperturbed state since all entities are stalled, meaning that no external voltage, magnetic field, or illumination, among others are applied.

2.5.1 Equilibrium distribution of carrier concentration

To obtain the concentration of electrons and holes in the conduction and valence bands, are applied the density of states functions, as defined in Eq. 9, and the distribution function (Fermi-Dirac distribution function), outlined in Eq. 10. Thus, the total electrons $n_0$ and holes $p_0$ concentrations in the conduction and valence bands, respectively, are given by Eq. 11. Within this equation, $N_C$ and $N_V$ represent the effective densities of states in the conduction and valence bands, respectively, obtainable via Eq. 12 [9, 30, 33, 34, 36, 40].

$$\begin{aligned} n_0= & {} \int _{E_C}^{E_{top}} n(E) \,dE \nonumber \\= & {} \int _{E_C}^{E_{top}} g_c(E)f(E) \,dE = N_Ce^{\frac{E_F-E_C}{k_BT}} \end{aligned}$$

(11a)

$$\begin{aligned} p_0= & {} \int _{E_{bottom}}^{E_V} p(E) \,dE \nonumber \\= & {} \int _{E_{bottom}}^{E_V} g_V(E)[1-f(E)] \,dE = N_Ve^{\frac{E_V-E_F}{k_BT}} \end{aligned}$$

(11b)

$$\begin{aligned} N_C= & {} 2\left( \frac{2\pi m_n^*k_BT}{h^2}\right) ^\frac{3}{2} \end{aligned}$$

(12a)

$$\begin{aligned} N_C= & {} 2\left( \frac{2\pi m_p^*k_BT}{h^2}\right) ^\frac{3}{2} \end{aligned}$$

(12b)

Fig. 8 illustrates a graphical representation of obtaining the total number of electrons $n_0$ and holes $p_0$ in the conduction and valence bands, respectively.

Fig. 8

Graphical representation of obtaining $n_0$ and $p_0$

A semiconductor presents an electric conductivity with a value ranging from $10^{-8}$ S/m to $10^{6}$ S/m, which is a temperature-dependent parameter. The higher the temperature, the higher the electric conductivity, rising exponentially. At 0 K, the semiconductor behaves as an insulator, meaning that the electric current does not flow freely. Besides that, its magnitude depends strongly on the impurities’ concentration. Thus, the semiconductors can be intrinsic semiconductors or extrinsic semiconductors. Also, they are considered non-degenerated when the doping concentration is much lower than the density of the semiconductor’s base atoms, meaning that energy levels are well defined and separated. Each one will be explored in the next sections.

2.5.1.1 Intrinsic semiconductors

An intrinsic semiconductor refers to a semiconductor without significant doping. As previously noted, at T = 0 K, the valence band is entirely occupied, contrasting with the empty conduction band. When an electromagnetic field is applied, there is no current, since all the states of the valence band are filled. This situation is illustrated in Fig. 9a. At higher temperatures, the bonds start to break due to the absorption of thermal energy, leading to the creation of free-charge carriers. Some valence electrons have sufficient energy to move to the conduction band, leaving, in the valence band, the same number of incomplete bonds. Thus, breaking a covalent bond leads to an electron–hole pair formation. This is shown in Fig. 9b. When an electromagnetic field is applied, there is an electrical current, since there are free states and the electrons can change their state of movement.

Fig. 9

Band model for an intrinsic semiconductor

In thermal equilibrium, the hole density $p_0$ is equal to the electron’s $n_0$ in the conduction band. Equation 13 translates the situation of an intrinsic semiconductor at thermal equilibrium, being $n_i$ the intrinsic density, whose value varies with temperature and from semiconductor to semiconductor [9, 30, 34, 36, 41].

$$\begin{aligned} n_0 = p_0 = n_i \end{aligned}$$

(13)

At thermal equilibrium conditions, Eq. 14 is verified.

$$\begin{aligned} n_0p_0 = n_i^2 \end{aligned}$$

(14)

By multiplying Eqs. 11a and 11b it is demonstrated that the intrinsic density $n_i$ is independent of the Fermi level’s position, as can be verified in Eq. 15, which means that this can be applied for both types of semiconductors [9, 30, 36].

$$\begin{aligned} n_i = \sqrt{N_CN_V}e^\frac{-E_G}{2k_BT} \end{aligned}$$

(15)

Finally, the Fermi level’s position, in intrinsic semiconductors, can be found through Eq. 16. Figure 10 shows the band diagram and the Fermi energy level which, according to Eq. 16, is close to the midgap [9, 33, 34].

$$\begin{aligned} E_{Fi} = \frac{E_C+E_V}{2}+\frac{k_BT}{2}\ln {\left( \frac{N_V}{N_C}\right) } \end{aligned}$$

(16)

According to Eq. 12, the effective densities of both the conduction and valence bands change with temperature. Thus, the intrinsic density $n_i$ is also a temperature dependence variable (Eq. 17).

$$\begin{aligned} n_i \propto T^\frac{3}{2}e^\frac{-E_G}{k_BT} \end{aligned}$$

(17)

Fig. 10

Fermi level’s position in intrinsic semiconductors

2.5.1.2 Extrinsic semiconductors

An extrinsic semiconductor is a semiconductor doped by a specific impurity, which can deeply modify its electrical properties. Doping concentrations can be of two types: (i) donor atoms, which donate electrons. Donor’s concentration is denoted as $N_D$; (ii) acceptor atoms, which accept electrons. Acceptor’s concentration is denoted as $N_A$. In thermal equilibrium, the semiconductor local charge density $\rho$ is null, meaning that the semiconductor is locally electrically neutral. So Eq. 18 is verified.

$$\begin{aligned} \rho = 0 \leftrightarrow p_0 + N_D^+ - n_0 - N_A^- = 0 \end{aligned}$$

(18)

At room temperature, the dopant atoms are ionized, so Eq. 18 is rewritten as Eq. 19, which is known as electrical neutrality Eq. [9, 30, 34, 36, 40].

$$\begin{aligned} p_0 + N_D - n_0 - N_A = 0 \end{aligned}$$

(19)

If $N_D > N_A$, the semiconductor is an n-type semiconductor having $n_0 > p_0$ and, if $N_D>> n_i$, then $n_0 \approx N_D$. Otherwise, if $N_A > N_D$, the semiconductor is a p-type semiconductor having $p_0 > n_0$ and if $N_A>> n_i$, then $p_0 \approx N_A$. In the case of $N_D = N_A$, Eq. 13 is valid, and the semiconductor behaves as an intrinsic semiconductor.

The concentration of electrons and holes in both the conduction and valence bands can be obtained through the intrinsic density $n_i$. Mathematically, Eq. 20 allows obtaining both the concentration of electrons and holes [9, 34].

$$\begin{aligned} n_0= & {} n_ie^\frac{E_F-E_{Fi}}{k_BT} \end{aligned}$$

(20a)

$$\begin{aligned} p_0= & {} n_ie^\frac{E_{Fi}-E_F}{k_BT} \end{aligned}$$

(20b)

N-type semiconductors

Let’s consider a semiconductor to which impurities of donor type were added. Figure 11 shows the band model for an n-type semiconductor with the increase of temperature.

Fig. 11

Band model for an n-type semiconductor

At T = 0 K, the electrons have their valence bonds and the donor electron, which is not involved in covalent bonds has a higher energy compared to the ones that are linked. The energy required to move the donor electron to the conduction band is considerably lower than the energy required to move one electron involved in the covalent bonding. The value of $E_C - E_D$ is a very small value and so a small temperature variation leads to an electron transition to the conduction band, ionising it. Since the energy required to move an electron from the valence band to the conduction band is substantial, the conduction band has a surplus of electrons compared to the deficiency of holes in the valence band, as illustrated in Fig. 11b [9, 34, 36, 40].

Given that the energy needed to transition an electron from the valence band to the conduction band is substantial, the conduction band harbours a surplus of electrons compared to the deficiency of holes in the valence band, as depicted in Fig. 11b.

For an n-type semiconductor, assuming that $N_A$ is null, and taking into account that $N_D \approx n_0$, the holes’ concentration can be obtained through Eq. 21. It is expected that the electrons’ concentration is higher than the holes’, which is verified.

$$\begin{aligned} p_0 = \frac{n_i^2}{n_0} \approx \frac{n_i^2}{N_D}<< n_0 \end{aligned}$$

(21)

If it is known the intrinsic density and the donor’s density at a given temperature T, the $n_0$ and $p_0$ values could be obtained through both Eqs. 14 and 19.

Fig. 12

Fermi level’s position in an n-type semiconductor

The Fermi level’s position in an n-type semiconductor is close to the conduction band, as illustrated in Fig. 12. The higher the electrons’ concentration, the closer the Fermi energy will be to the conduction band. The Fermi level’s position can be calculated through Eq. 22 [9, 30, 33, 34, 36].

$$\begin{aligned} E_C - E_F = k_BT\ln {\left( \frac{N_C}{N_D}\right) } \end{aligned}$$

(22)

As already mentioned, the carriers’ density varies with the temperature. Figure 13 shows the carriers’ density as a temperature function for an n-type semiconductor. There are three operating zones, numbered 1 to 3 in the figure. In the x-axis, it is plotted the inverse of the temperature, meaning that the temperature increases from right to left in Fig. 13. Thus, the analysis of each zone is done 3 to 1, to follow the temperature increase.

3:

Ionisation zone At low temperatures, the increase in the temperature leads to an increase in the carriers’ density, in this case, holes. In this zone, the slope m of the graphic follows the behaviour of Eq. 23.

$$\begin{aligned} m = -\frac{E_C-E_D}{2k_B} \end{aligned}$$

(23)

There is only enough latent energy in the material to push a few donor carriers into the conduction band since $E_C-E_D<< E_G$ [36, 41].

2:

Saturation zone As the temperature increases, the density of the electrons is approximately constant, but the density of holes continues to increase with the increase of the temperature due to the increase of the intrinsic density. In this zone, all the donor carriers are ionised. [36, 41].

1:

Intrinsic zone At a certain temperature value, $T_i$, known as intrinsic temperature, the intrinsic density is higher than the dopant’s density and the electrons and holes densities are similar. In this zone, the slope m of the graphic follows the behaviour of Eq. 24 [36, 41].

$$\begin{aligned} m = -\frac{E_G}{2k_B} \end{aligned}$$

(24)

Fig. 13

Carriers’ density as a temperature function for an n-type semiconductor

P-type semiconductors

Let’s consider a semiconductor to which impurities of acceptor type were added. Figure 14 shows the band model for a p-type semiconductor with the increase of temperature.

Fig. 14

Band model for a p-type semiconductor

At T = 0 K, there is a covalent bonding incomplete and its energy is higher than $E_V$. A small temperature variation is enough to move the acceptor electron to the conduction band, ionising it. Thus, in the valence band, there are many more holes than electrons in the conduction band, which is illustrated in Fig. 14b [9, 33, 34, 36, 40].

For a p-type semiconductor, assuming that $N_D$ is null, and considering that $N_A \approx p_0$, the holes’ concentration can be obtained through Eq. 25. In this case, it is expected that the electrons’ concentration is lower than the holes’, which is verified.

$$\begin{aligned} n_0 = \frac{n_i^2}{p_0} \approx \frac{n_i^2}{N_A}<< p_0 \end{aligned}$$

(25)

As in the case of the n-type semiconductor, if it is known the intrinsic density and the donor’s density at a given temperature T, the $n_0$ and $p_0$ values could be obtained through both Eqs. 14 and 19.

Fig. 15

Fermi level's position in a p-type semiconductor

The Fermi level’s position in a p-type semiconductor is close to the valence band, as illustrated in Fig. 15, and can be calculated through Eq. 26 [9, 30, 34, 36].

$$\begin{aligned} E_F - E_V = k_BT\ln {\left( \frac{N_V}{N_A}\right) } \end{aligned}$$

(26)

Fig. 16 shows the carriers' density as a temperature function for a p-type semiconductor. As for the n-type semiconductors, there are three operating zones, numbered 1 to 3 in the figure. The analysis of each of the zones for p-type semiconductors is identical to the n-type semiconductors’.

Fig. 16

Carriers’ density as a temperature function for a p-type semiconductor

For both extrinsic semiconductors, to obtain the intrinsic temperature value, Eq. 27 has to be applied, knowing that $n_i(T_i) = N_D$, for n-type semiconductor and $n_i(T_i) = N_A$ for p-type semiconductor.

$$\begin{aligned} \frac{n_i(T_i)}{n_i(T)} = \left( \frac{T_i}{T}\right) ^\frac{3}{2}\frac{e^\frac{-E_G}{2k_BT_i}}{e^\frac{-E_G}{2k_BT}} \end{aligned}$$

(27)

2.6 Semiconductors in non-equilibrium

When a semiconductor is illuminated, additional electron–hole pairs are generated in the material by the absorption of photons, leading to the increase of holes’ concentration $p > p_0$ in the valence band as well as electrons’ concentration $n > n_0$ in the conduction band. The photogenerated carriers interact with the semiconductor lattice and the extra energy that the electron–hole pairs receive from the photons with energies larger than the semiconductor’s bandgap $E_G$ is released into the lattice in other forms, such as heat or vibrations [9].

2.6.1 Excess carrier generation and recombination

When additional charge carriers are created, the concentration of electrons and holes in the conduction and valence bands, respectively, are given by Eq. 28, being $n_0$ and $p_0$ the thermal equilibrium concentrations, and $\Delta n$ and $\Delta p$ the excess carriers concentrations [9, 34, 36].

$$\begin{aligned} n= & {} n_0 + \Delta n \end{aligned}$$

(28a)

$$\begin{aligned} p= & {} p_0 + \Delta p \end{aligned}$$

(28b)

To obtain the electron and hole concentrations, in non-equilibrium states, two Fermi distributions are used to describe the state occupation in the conduction band and in the valence band with electron. The electrons n and holes p concentrations are given by Eq. 29, being $N_C$ and $N_V$ obtained through Eq. 12, and $E_{Fc}$ and $E_{Fv}$ are known as quasi-Fermi energy for electrons and quasi-Fermi energy for holes, respectively [9, 30, 34].

$$\begin{aligned} n= & {} N_Ce^\frac{E_{Fc}-E_C}{k_BT} \end{aligned}$$

(29a)

$$\begin{aligned} p= & {} N_Ve^\frac{E_V-E_{Fv}}{k_BT} \end{aligned}$$

(29b)

Multiplying Eqs. 29a and 29b is possible to obtain the result presented in Eq. 30, which shows that, in non-equilibrium condition, $np = n_i^2$, presented in Eq. 14, it’s no longer true [9, 30, 34].

$$\begin{aligned} np = N_CN_Ve^{\frac{E_V-E_C}{k_BT}}e^\frac{E_{Fc}-E_{Fv}}{k_BT} = n_i^2e^\frac{E_{Fc}-E_{Fv}}{k_BT} \end{aligned}$$

(30)

When the light turns off, the excess electrons will recombine with holes until the equilibrium state is reached again. Depending on the properties of the semiconductor, different types of recombination will occur. This topic is discussed further in section 2.9.

2.7 Transport properties

In a semiconductor, the electrical current is generated due to the transport of charge by charge carriers. The two basic transport mechanisms are drift and diffusion. In a semiconductor, the total density current J takes into account both transport properties for both charge carriers. Then, the total density current is given by Eq. 31 [9, 30, 33, 34, 36]. Both transport properties are explained in the following sections.

$$\begin{aligned}&J = J_{drift} + J_{diff} = (J_{n_{drift}} + J_{p_{drift}}) + (J_{n_{diff}} + J_{p_{diff}}) \nonumber \\&\quad \leftrightarrow J = q(n\mu _n+p\mu _p)E + q(D_n\nabla n - D_p \nabla p) \end{aligned}$$

(31)

2.7.1 Drift

Drift is charged particle motion in response to an electric field $\vec {E}$. When is applied an electrical field, there is an electrical force $\vec {F}$, which is negative for electrons, i.e. with an opposite direction of the electrical field, and positive for holes, i.e. with the electrical field’s direction. The charged particles are involved in collisions with the thermally vibrating lattice atoms and ionized impurity atoms. The resulting motion of electrons and holes can be described by average drift velocities $v_{dn}$ and $v_{dp}$ for electrons and holes, respectively, which are given by Eq. 32, being $\mu _n$ and $\mu _p$ the electrons and holes mobilities, respectively [9, 30, 34, 36].

$$\begin{aligned} \langle \vec {v}_{dn} \rangle= & {} -\mu _n \vec {E} \end{aligned}$$

(32a)

$$\begin{aligned} \langle \vec {v}_{dp} \rangle= & {} \mu _p \vec {E} \end{aligned}$$

(32b)

The higher the temperature, the lower the mobilities, since when temperature increases, the shock frequency with the lattice increases too. However, when a semiconductor has a higher dopant density, in some cases, the higher the temperature, the higher the mobility [9, 36].

The existence of average drift velocities with the electrical field direction results in an electrical current, flowing charged particles in that direction. The drift current density is given by Eq. 33, where $\rho$ is the charge density. Thus, for both charge carriers, the drift current density is given by Eq. 34.

$$\begin{aligned} \vec {J}_{drift} = \rho \langle \vec {v}_d \rangle \end{aligned}$$

(33)

$$\begin{aligned} \vec {J}_{n_{drift}}= & {} qn\mu _n\vec {E} \end{aligned}$$

(34a)

$$\begin{aligned} \vec {J}_{p_{drift}}= & {} qp\mu _p\vec {E} \end{aligned}$$

(34b)

In a semiconductor, the total drift current density corresponds to the sum of Eqs. 34a and 34b. According to this result and knowing Ohm’s law, conductivity, $\sigma$, can be obtained according to Eq. 35 [9, 30, 34, 36].

$$\begin{aligned} \vec {J}_{drift} = \sigma \vec {E} \leftrightarrow \sigma = q(n\mu _n+p\mu _p) \end{aligned}$$

(35)

2.7.2 Diffusion

Diffusion is the phenomenon where particles naturally disperse from areas with a high concentration to those with a lower concentration, driven by random thermal movement. The movement of the particles is random and takes place in the three directions of space [9, 34, 36]. Both charge carriers behave according to the Maxwell-Boltzmann statistic and, because of that, in non-degenerated semiconductors, both particles are considered as a thin gas of charged particles. The driving force of diffusion is a gradient in the particle concentration, and the resulting currents are proportional to it. Then, the current densities for electrons and holes are described by Eq. 36, being $D_n$ and $D_p$ the electrons and holes proportional diffusion coefficients, respectively.

$$\begin{aligned} J_{n_{diff}}= & {} qD_n\nabla n \end{aligned}$$

(36a)

$$\begin{aligned} J_{p_{diff}}= & {} -qD_p\nabla p \end{aligned}$$

(36b)

The diffusion coefficients and mobilities of both charge carriers are linked through Einstein relationship, given by Eq. 37 [9, 30, 34,35,36, 41], where $u_T$ is the thermal voltage.

$$\begin{aligned} \frac{D_n}{\mu _n} = \frac{D_p}{\mu _p} = \frac{k_BT}{q} = u_T \end{aligned}$$

(37)

In a semiconductor, the total diffusion current density corresponds to the sum of Eqs. 36a and 36a.

2.8 Hall effect

The Hall effect is a consequence of both electric and magnetic fields, which together are exerted on moving charges. Its configuration is represented in Fig. 17. It is used to measure the majority carrier.

Fig. 17

Hall effect configuration

An applied magnetic field $\vec {B}$ in the z direction translates into a force, known as Lorentz’s force, given by Eq. 38a and 38b, for electrons and holes, respectively.

$$\begin{aligned} \vec {F}_B= & {} -q[\langle \vec {v}_n \rangle \vec {B}] \end{aligned}$$

(38a)

$$\begin{aligned} \vec {F}_B= & {} q[\langle \vec {v}_p \rangle \vec {B}] \end{aligned}$$

(38b)

In steady-state, the magnetic field force, $\vec {F}_B$ will be exactly balanced by the induced electric field force, $\vec {F}_{\mathcal {E}}$. Thus, the condition 39 is fulfilled, being $\vec {F}_{\mathcal {E}}$ given by Eq. 40.

$$\begin{aligned}{} & {} \vec {F}_{\mathcal {E}} + \vec {F}_B = 0 \end{aligned}$$

(39)

$$\begin{aligned}{} & {} \vec {F}_{\mathcal {E}} = -q\vec {{\mathcal {E}}} = -q {\mathcal {E}} \vec {u}_y \end{aligned}$$

(40)

Finally, the induced electric field in the y direction is called the Hall effect, described by Eq. 41, which causes an electric voltage, known as Hall voltage, defined in Eq. 42.

$$\begin{aligned} {\mathcal {E}}_H= & {} Bv_x \end{aligned}$$

(41)

$$\begin{aligned} V_H= & {} {\mathcal {E}}_HW = Bv_xW \end{aligned}$$

(42)

In a semiconductor of p-type, where the majority carriers are holes, the Hall voltage $V_H$ is positive. This situation is illustrated in Fig. 17 though the charge . In opposition, within an n-type semiconductor where electrons dominate as majority carriers, the Hall voltage exhibits a negative polarity, being illustrated through the charge in the same figure. Thus, through the Hall voltage, the semiconductor type can be identified [33, 34, 36].

Knowing that the current I is given by Eq. 43 and combining Eqs. 43 and 41, the Hall effect ${\mathcal {E}}$ can be obtained through Eq. 44, as well as the Hall voltage [33, 34, 36].

$$\begin{aligned} I= & {} -qnv_xWd \end{aligned}$$

(43a)

$$\begin{aligned} I= & {} qpv_xWd \end{aligned}$$

(43b)

$$\begin{aligned} {\mathcal {E}}= & {} \left( \frac{-1}{qn}\right) \frac{IB}{Wd} \leftrightarrow V_H = \left( \frac{-1}{qn}\right) \frac{IB}{d} \end{aligned}$$

(44a)

$$\begin{aligned} {\mathcal {E}}= & {} \left( \frac{1}{qp}\right) \frac{IB}{Wd} \leftrightarrow V_H = \left( \frac{1}{qp}\right) \frac{IB}{d} \end{aligned}$$

(44b)

2.9 Generation and recombination process

Incident photons on the surface of a semiconductor can undergo one of three outcomes: reflection from the top surface, absorption within the material, or transmission through the material. The emission and absorption processes can be explained through the interaction between the electrons in the materials and the photons. The absorption process depends on the photon’s energy, which is given by Eq. 45, where f is the optical frequency.

$$\begin{aligned} E = hf = \frac{hc}{\lambda } \end{aligned}$$

(45)

Therefore, three cases are possible. If $E > E_G$, i.e. the incident photon presents energy greater than the energy band gap of the material, the photons are strongly absorbed. As the energy of the photon increases, so does the potential for a greater number of electrons to engage with it, leading to absorption of the photon. If $E < E_G$, i.e. the incident photon presents an energy lower than the energy band gap of the material, the photons interact weakly with the material, passing through it. In this case, the material is known as transparent to the radiation. In this case, electron–hole pairs aren’t generated because the photon lacks adequate energy to elevate an electron from the valence band to the conduction band. Otherwise, if $E = E_G$, the photons have enough energy to create an electron–hole pair, being absorbed efficiently [9, 25, 30, 33, 36, 39]. Consequently, there will be a long wavelength threshold, which is also known as cut-off wavelength, beyond which no photoconductive response is obtained, and this is given by Eq. 46, being h the Planck’s constant and c the velocity of light [9, 25, 30, 39].

$$\begin{aligned} \lambda _0 = \frac{hc}{E_G} \Rightarrow \lambda _0 \ [nm] = \frac{1.24}{E_G}\ [eV] \end{aligned}$$

(46)

The absorption process in the material depends on its absorption coefficient, $\alpha (\lambda )$, which varies with wavelength. Using Lambert-Beer’s law, it is possible to describe the absorption profile of photons in semiconductor devices, which is given by Eq. 47, where $\Phi _0$ represents the incoming photon flux, while x denotes the depth within the material where the intensity of light is being assessed.

$$\begin{aligned} \Phi (x) = \Phi _0e^{-\alpha (\lambda )x} \end{aligned}$$

(47)

Fig. 18 shows the dependence of the absorption coefficient on the wavelength for several materials, which can be obtained through Eq. 48, where k is the extinction coefficient [9, 25, 27, 30, 36, 39, 42].

$$\begin{aligned} \alpha (\lambda ) = \frac{4\pi k}{\lambda } \end{aligned}$$

(48)

Light is poorly absorbed in a material with a low absorption coefficient, and if the material is thin enough, it will appear transparent to that wavelength. The absorption coefficient determines how far into a material, light can penetrate before it is absorbed. Thus, a new term can be stated: absorption depth, which is the inverse of the absorption coefficient. Shorter wavelength light, which corresponds to greater energy light, presents a greater absorption coefficient and a shorter absorption depth. The opposite is also true. The variation in the absorption depth for higher and lower photons’ energies is shown in Fig. 19.

Fig. 18

Absorption coefficients for several materials in two different wavelength ranges. Figure 18b) is the zoom-in on the absorption spectrum in the visible zone. For SF10, Soda Lime Glass and Quartz present a null absorption coefficient

Fig. 19

Absorption depth for higher and lower photons’ energies

Considering that each absorption photon originates an electron–hole pair, the generation rate by the internal photoelectric effect, $G_{ph}$ is described by Eq. 49. This occurs when the semiconductor is strongly illuminated with monochromatic radiation of energy $E > E_G$.

$$\begin{aligned} G_{ph} = -\left( \frac{d\Phi }{dx}\right) = \alpha \Phi _0e^{-\alpha x} \end{aligned}$$

(49)

In thermal equilibrium, the charge carriers’ densities are time-independent, being constants. If the semiconductor is homogeneous, these densities are constants in space. For each material, there is a generation rate, $G_{th}$, defined as the number of electrons that move from the valence to the conduction bands due to temperature, for each volume unit and time unit. Similarly, an inverse process, known as recombination rate, $R_{th}$, occurs. This process is the number of electrons that move from the conduction to the valence bands, for each volume unit and time unit [9, 36]. In thermal equilibrium, the Eq. 50 is fulfilled, where $\beta$ is a proportionality factor.

$$\begin{aligned} G_{th} = R_{th} = \beta n_0p_0 \end{aligned}$$

(50)

According to the semiconductors’ application and the conditions prevailing, one of the recombination mechanisms becomes active and more efficient than other types. For solar cell applications, the recombination rate strongly determines its performance.

2.9.1 Direct recombination

If the semiconductor is strongly illuminated, then the total generation rate, $G_T$, is given by Eq. 51, where n and p are given by Eq. 28, and $R_d$ is the radiative recombination rate [9, 36].

$$\begin{aligned} \left\{ \begin{array}{l} G_{T} = R_{T} = \beta np \\ G_{T} = G_{th} + G_{ph} \end{array}\right. \leftrightarrow G_{ph} = R_T - G_{th} = R_d \end{aligned}$$

(51)

For an n-type and p-type semiconductor under low-level injection, the recombination rate is given by Eqs. 52a and 52b, respectively, where the lifetime of the minor carriers is represented as $\tau _{xd}$, being x the representation of n/p according to the semiconductor type.

$$\begin{aligned} R_d\approx & {} \beta n_0(p-p_o) = \frac{p-p_0}{\tau _{pd}} \end{aligned}$$

(52a)

$$\begin{aligned} R_d\approx & {} \beta p_0(n-n_o) = \frac{n-n_0}{\tau _{nd}} \end{aligned}$$

(52b)

The excess carrier concentration is given by the product of the generation rate $G_{ph}$ and the lifetime $\tau _{xd}$. If carrier generation is not uniform across the semiconductor, the diffusion of excess carriers $D_n/D_p$ occurs until they recombine with the dominant carrier. Subsequently, the minority carrier diffusion lengths $L_n/L_p$ are the average distance a carrier can move from its generation point to recombination, which are mathematically described by Eq. 53 [9].

$$\begin{aligned} L_n= & {} \sqrt{D_n\tau _n} \end{aligned}$$

(53a)

$$\begin{aligned} L_p= & {} \sqrt{D_p\tau _p} \end{aligned}$$

(53b)

With higher minority carrier diffusion lengths, the material presents longer lifetimes.

2.9.2 Shockley-read-hall recombination

The Shockley-Read-Hall (SRH) recombination process was introduced in 1952. In this process, the recombination of charge carriers does not occur directly from bandgap to bandgap. This process takes place through the capture of one carrier at a time by recombination centres, which are created by an impurity atom or lattice defects [9, 30, 43]. Figure 20 illustrates the SRH recombination process, where $E_T$ is an energy level within the bandgap energy known as trap states. At the trap states, the Fermi-Dirac function is given by Eq. 54.

$$\begin{aligned} f(E_T) = \frac{1}{1+e^\frac{E_T-E_F}{k_BT}} \end{aligned}$$

(54)

There are two types of traps: donor-type traps, which are neutral in the presence of an electron and become positively charged in its absence, and acceptor-type traps, which are negatively charged when they contain an electron and neutral when they do not.

Fig. 20

SRH’s energy band diagram

There are four processes associated with the recombination centres: (i) electron capture from the conduction band; (ii) hole capture from the valence band; (iii) electron emission to the conduction band; and (iv) hole emission to the valence band. In (i), the electron is captured by the trap with a rate $R_{cn}$ proportional to the density of electrons in the conduction band and to the density of empty trap states. The rate $R_{cn}$ is expressed by Eq. 55, being $v_{th}$ the thermal velocity, $\sigma _n^{+/-}$ the electron capture cross-section of the traps, where $^{+/-}$ corresponds to the donor/acceptor type traps, respectively, $N_T$ is the total concentration of trapping centres and, finally, n is the electron concentration in the conduction band [9, 30, 34].

$$\begin{aligned} R_{cn} = v_{th}\sigma _n^{+/-}N_T[1-f(E_T)]n \end{aligned}$$

(55)

In (ii), the hole is captured by the trap at a rate $R_{cp}$, being mathematically defined by Eq. 56, where $\sigma _p^{+/-}$ is the hole capture cross-section of the traps and p is the hole concentration in the valence band [9, 30, 34].

$$\begin{aligned} R_{cp} = v_{th}\sigma _p^{+/-}N_Tf(E_T)p \end{aligned}$$

(56)

In (iii), the electron is emitted to the conduction band at a rate $R_{en}$ proportional to the number of filled traps. This is given by Eq. 57. In (iv), the hole is emitted to the valence band at a rate $R_{ep}$, mathematically given by Eq. 58. Consider $E_n$ and $E_p$ the electrons’ and holes’ emission coefficients, respectively [9, 30, 34].

$$\begin{aligned} R_{en}= \,& {} E_nN_Tf(E_T) \end{aligned}$$

(57)

$$\begin{aligned} R_{ep}= \,& {} E_pN_Tf(E_T) \end{aligned}$$

(58)

In thermal equilibrium, $R_{cn} = R_{en}$ and $R_{cp} = R_{ep}$ and taking into consideration Eqs. 29 and 54, the obtained result, presented in Eq. 59, is $E_n$ and $E_p$ expressions [9, 30, 34].

$$\begin{aligned} E_n= \,& {} v_{th}\sigma _nN_Ce^\frac{E_T-E_C}{k_BT} \end{aligned}$$

(59a)

$$\begin{aligned} E_p= \,& {} v_{th}\sigma _pN_Ve^\frac{E_V-E_T}{k_BT} \end{aligned}$$

(59b)

In a non-equilibrium situation, it is assumed that the emission coefficients are approximately equal to the emission coefficients under equilibrium. As recombination involves exactly one electron and one hole, in a stable condition, the departure rate of electrons from the conduction band matches that of holes from the valence band. Consequently, the recombination rate $R_{SRH}$ is determined by Eq. 60 [9, 30, 34].

$$\begin{aligned} R_{SRH} = \frac{dn}{dt} = \frac{dp}{dt} = R_{cn} - R_{en} = R_{cp} - R_{ep} \end{aligned}$$

(60)

Assuming the same capture cross-sections and holes and considering both Eqs. 60 and 15, it is possible to rewrite the recombination rate as the one presented in Eq. 61.

$$\begin{aligned} R_{SRH} = v_{th}\sigma N_T\frac{np-n_i^2}{n+p+2n_i\cosh {\left( \frac{E_T-E_{Fi}}{k_BT}\right) }} \end{aligned}$$

(61)

Thus, for n-type and p-type semiconductors, considering a low injection rate, the recombination rate is described by Eq. 62, where $c_n$ and $c_p$ are the hole and electron capture coefficients, respectively. The recombination rate is inversely proportional to the lifetime of the minority carriers $\tau _p^{SRH}$ and $\tau _n^{SRH}$, respectively, and these are indirectly proportional to the trap density $N_T$. Thus, a good semiconductor device must have a lower trap density [9, 30, 34].

$$\begin{aligned} R_{SRH}= \,& {} v_{th}\sigma N_T\frac{p-p_0}{1+2\frac{n_i}{n_0}\cosh {\left( \frac{E_T-E_{Fi}}{k_BT}\right) }} \nonumber \\= & {} c_pN_T(p-p_0) =\, \frac{p-p_0}{\tau _p^{SRH}} \end{aligned}$$

(62a)

$$\begin{aligned} R_{SRH}=\, & {} v_{th}\sigma N_T\frac{n-n_0}{1+2\frac{n_i}{p_0}\cosh {\left( \frac{E_T-E_{Fi}}{k_BT}\right) }} \nonumber \\= & {} c_nN_T(n-n_0) =\, \frac{n-n_0}{\tau _n^{SRH}} \end{aligned}$$

(62b)

2.9.3 Auger recombination

The Auger recombination process is a three-particle process, as illustrated in Fig. 21. In this process, both momentum and hole’s and electron’s recombination energy are conserved, transferring both quantities to another particle (electron or hole).

Fig. 21

Auger’s energy band diagram

Equation 63 illustrates the Auger recombination rate $R_{Aug}$, which depends on the charge carriers densities. Within Eq. 63, consider the recombination rates for electron/electron/hole, $R_{eeh}$, and for electron/hole/hole, $R_{ehh}$, where $C_n$ and $C_p$ are proportionality constants strongly temperature-dependent [9, 30, 34].

$$\begin{aligned} R_{Aug} = R_{eeh} + R_{ehh} = C_nn^2p + C_pnp^2 \end{aligned}$$

(63)

Auger recombination becomes increasingly significant at high carrier concentrations caused by intense doping or high-level injection under concentrated sunlight. Considering strongly doped n-type and p-type semiconductors under low-level injection, $n\approx N_D$ and $p \approx N_A$, respectively. Thus, for each semiconductor, there is a dominant process (electron/electron/hole for n-type and electron/hole/hole for p-type) and the lifetime for each process is given by Eq. 64 [9].

$$\begin{aligned} \left\{ \begin{array}{l} \tau _{eeh} = \frac{1}{C_nN_D^2} \\ \tau _{ehh} = \frac{1}{C_pN_A^2} \end{array}\right. \end{aligned}$$

(64)

3 Photodetectors

Photodetectors transform an optical signal into a different form, namely electrical signals. To do that, several physical effects can be used, such as the photoconductive, photovoltaic, photoemission and thermal effects. Therefore, several varieties of photodetectors exist, categorized into two main groups: thermal detectors and photon detectors. Thermal detectors are based on the photothermal effect, converting optical energy into heat. In contrast, the photon detectors convert a photon into an emitted electron or an electron–hole pair [25, 30, 33]. Between both types of photodetectors, there are some general differences, which are summarised in table 2 [30]. Most thermal detectors have a slow speed of response because their speed is limited by thermalisation through heat diffusion and heat dissipation when the power of an optical signal varies [30].

Table 2 Differences between thermal and photon detectors

3.1 Photodetectors performance parameters

As presented in table 2, some key parameters can characterise photodetectors, among them are sensitivity, quantum efficiency, responsivity, speed of response, cut-off frequency, spectral response, frequency bandwidth and signal-to-noise ratio. These parameters can be considered figures of merit of a photodetector and are applicable to various other photodetector types.

Radiation intensity can be expressed in two ways: photon flux, $\phi$, and radiant power, $P_s$. Both terms are related to each other through the Eq. 65, where $\lambda$ is the wavelength in vacuum, h is the Planck constant and c corresponds to the light velocity in vacuum. The photon flux is defined by the number of incident photons in each unit of time and the radiant power is the radiant energy of the incident radiation per unit of time [44, 45].

$$\begin{aligned} \phi = \frac{\lambda P_s}{hc} \end{aligned}$$

(65)

3.1.1 Sensitivity

The sensitivity of the device can be expressed in two ways: quantum efficiency and responsivity. Quantum efficiency, $\eta$, is the probability of generating a charge carrier in a photodetector for each photon that is incident on the detector. Mathematically, it is described by the number of photogenerated carrier pairs divided by the number of photons, given by Eq. 66, where $i_{ph}$ is the photocurrent [25, 30, 44].

$$\begin{aligned} \eta = \frac{\frac{i_{ph}}{q}}{\frac{P_s}{h\upsilon }} = \frac{hc}{q\lambda }\frac{i_{ph}}{P_s} \end{aligned}$$

(66)

Responsivity is defined as the ratio of the photodetector output and the radiant power [44], which is described through Eq. 67, being s the photodetector output. The responsivity can be given in A/W, if $s = i_s$, or V/W, if $s = v_s$, depending on the output.

$$\begin{aligned} {\mathcal {R}} = \frac{s}{P_s} \end{aligned}$$

(67)

If the photodetector has no internal gain, then $i_s$ = $i_{ph}$, which allows us to obtain the quantum efficiency as a function of responsivity, given by Eq. 68 [25, 30, 44].

$$\begin{aligned} {\mathcal {R}} = \frac{i_{ph}}{P_s} = \eta \frac{q\lambda }{hc} \end{aligned}$$

(68)

Otherwise, for a photodetector with an internal gain, the signal current is amplified by a gain G and the responsivity is given by Eq. 69, being ${\mathcal {R}}_0$ the intrinsic responsivity of the detector defined by Eq. 68.

$$\begin{aligned} {\mathcal {R}} = \frac{Gi_{ph}}{P_s} = G\eta \frac{q\lambda }{hc} = G {\mathcal {R}}_0 \end{aligned}$$

(69)

When a photodetector is irradiated by monochromatic radiation, the term spectral is added to each term, which means a function of wavelength and not a spectrally integrated quantity. In this situation and when the photodetector output is expressed by photocurrent, Eq. 70 translated the relation between spectral quantum efficiency, $\eta (\lambda )$, and spectral responsivity, ${\mathcal {R}}(\lambda )$, in A/W. In this Eq., q is the module of the electron charge, E is the photon energy, in eV, and $\lambda$ is the wavelength, in nm [30, 44].

$$\begin{aligned} {\mathcal {R}}(\lambda ) = \frac{q\lambda \eta }{hc} = \frac{\eta (\lambda )}{E} \end{aligned}$$

(70)

For non-monochromatic radiation input, Eq. 70 is not applicable, since it’s necessary to understand the spectral makeup of the radiation being inputted. In what concerns the quantum efficiency and responsivity, two different definitions for the input can be considered. On the one hand, the input radiation is defined by the one incident to the detector. On the other hand, the input radiation is defined by the one absorbed in the detector. To distinguish both cases, terms external (sometimes omitted) and internal are added in each term and, sometimes, the two terms associated with the same concept can be related through Eq. 71, where R is the system reflectance, A is the absorptance of the surface layer and T is the transmittance of the surface layer, into the sensitive substrate [44].

$$\begin{aligned} \left\{ \begin{array}{l} {\mathcal {R}}_{ext} = (1-R-A){\mathcal {R}}_{int} = T{\mathcal {R}}_{int}\\ \eta _{ext} = (1-R-A)\eta _{int} = T\eta _{int}\\ \end{array}\right. \end{aligned}$$

(71)

3.1.2 Speed and frequency response

Frequency response is characterised by the frequency dependence of the responsivity ${\mathcal {R}}(f)$ at a given optical wavelength. It can be determined using two methods: using the Fourier transformation of the impulse response, or recording the photodetector’s response at one signal frequency at a time while sweeping through them. The output electrical power spectrum of the detector, ${\mathcal {R}}^2(f)$, defines a 3-dB cut-off frequency or a 3-dB bandwidth for a photodetector as demonstrated in Eq. 72 [30, 36].

$$\begin{aligned} {\mathcal {R}}^2(f_{3\text {dB}}) = \frac{{\mathcal {R}}^2(0)}{2} \end{aligned}$$

(72)

The 3-dB bandwidth of a photodetector depends on several physical factors working together to govern the speed and frequency response of the detector [36].

The speed response of a photodetector is directly related to its frequency response, determining its capability to track rapidly changing optical signals. In the time domain, the speed of a photodetector is the rise time, $t_r$, and the fall time, $t_f$, of its response to an impulse signal or a square-pulse signal. These times represent the duration for the response to ascending from 10% to 90% and descend from 90% to 10% of its maximum amplitude, respectively [30].

3.1.3 Signal-to-noise ratio

An electrical signal in the circuit always has undesirable components, which is known as noise. There are several types of noise and some of them cannot be avoided, which is the case of intrinsic noise sources, like the temperature effect.

The signal-to-noise ratio, SNR is defined as the ratio of the mean square of a signal to the mean square of its noise. It can be given by Eq. 73, where s corresponds to a signal. Depending on the photodetection system, the signal can take many forms, like photocurrent or photovoltage as output electrical signals, or photon flux as the input optical signal [30, 36].

$$\begin{aligned} SNR = \frac{\overline{s^2}}{\overline{s_n^2}} \end{aligned}$$

(73)

3.1.4 Noise equivalent power

The noise equivalent power (NEP) is defined as the minimum input power required to obtain a unitary SNR. Mathematically, NEP can be defined by Eq. 74, where $\overline{s^2_n}$ the mean square noise of a signal at an input optical power level for SNR = 1. If the photodetector has an output current, then $\overline{s^2_n} = \overline{i^2_n}$, and if its output corresponds to a voltage signal, then $\overline{s^2_n} = \overline{v^2_n}$. In Eq. 74, ${\mathcal {R}}$ is the responsivity defined in section 3.1.1 [30, 36].

$$\begin{aligned} NEP = \frac{\sqrt{\overline{s^2_n}}}{{\mathcal {R}}} \end{aligned}$$

(74)

The NEP value does not allow comparing photodetectors’ behaviour, since it depends on the exposure area and the bandwidth considered.

3.1.5 Detectivity

The detectivity, D, characterizes the ability of a photodetector to detect a small optical signal, and it is defined as the inverse of NEP value, in W$^{-1}$, which is described in section 3.1.4. There is another figure of merit, called specific detectivity, $D^*$, in Hz$^{1/2}$/W. This applies when the shot noise contributed by the input optical signal is much lower than other sources’ noise. Specific detectivity is described by Eq. 75, where A is the detector’s area and B is the signal bandwidth, in Hz [30, 36].

$$\begin{aligned} D^* = \frac{\sqrt{A B}}{NEP} \end{aligned}$$

(75)

Sensitivity and accuracy are important requisites in what concerns photodetectors applications. However, the circuit is designed to obtain lower values of noise and good stability in the desired passband. Thus, in some cases, it is necessary to implement a circuit called a photodiode amplifier. There are several methods to do that, but the preamplifier must be well-designed.

Figure 22 illustrates some popular photodiode amplifiers based on the use of operational amplifiers to obtain extended performance [46].

Fig. 22

Circuit schemes of popular photodiode amplifiers

3.2 Thermal detectors

Thermal detectors are frequently used in spectral regions where photon sensors face challenges, particularly in long-wavelength infrared light. A thermal detector is characterized by its low speed, moderate sensitivity and low dynamic range, and it is used, for example, as an optical power meter for lasers. Despite their limitations, thermal sensors possess broad spectral sensitivity and maintain nearly constant responsivity across a wide range of wavelengths, contrasting with photon detectors [30].

There are many types of thermal detectors. An example is a thermocouple, which is a temperature-measuring device consisting of two wires of different metals joined at each end. When a temperature gradient occurs between the wire junctions, a thermoelectric voltage emerges due to the Seebeck effect, resulting in small voltage values typically in the low millivolt range. This thermoelectric voltage is roughly proportional to the temperature difference.

3.3 Photon detectors

Photon detectors can be based on the external and internal photoelectric effects. The first one is photoemission devices, like vacuum photodiodes and photomultipliers, and the second one is semiconductor devices, being electron–hole pairs generated through incident photons’ absorption, such as photoconductors and photodiodes [25, 30].

Whereas the photoconductive effect generally is too slow to follow a rapidly changing optical signal, photoemissive devices (photomultiplier) come at a high cost and, like all tubes, have a relatively short lifetime. Clearly, the optimum choice is using the photovoltaic effect in its fastest form, the reverse-biased photodiode [25]. The photodiode offers numerous advantages: it can be mass-produced, boasts extended mean time between failures, occupies minimal space, is durable, sensitive, and can exhibit rapid response times [25].

3.3.1 Vacuum photodiodes

A vacuum photodiode was invented by Sir John Ambrose Fleming in 1904. It is also known as the Fleming valve or the thermionic valve, and the IEEE describes it as “one of the most important developments in the history of electronics” [47]. It consists of a simple device with two electrodes, a photocathode and an anode, enclosed in a vacuum tube. The vacuum maintained in the tube corresponds to the depletion region of the diode. The cathode generates free electrons into the vacuum tube, whereas the anode collects free electrons from the cathode. It can use either a reflection mode photocathode or a transmission mode photocathode. In the first one, the device is opaque and in the second one, the device is semitransparent [39]. A vacuum photodiode can be represented by a circuit like the one illustrated in Fig. 23.

Fig. 23

Basic circuit of a vacuum photodiode

The vacuum photodiode works on the thermionic emission principle. When the cathode absorbs an electron with energy surpassing the bandgap energy of the material, a conduction electron is produced in the semiconductor. This electron diffuses through the material until it reaches the surface, where it may escape into the vacuum. The electrical field between the cathode and anode drives the electrons to the anode, where a signal is produced. Hence, a voltage is applied between the anode and the photocathode of the vacuum photodiode to effectively gather the photoelectron upon irradiation of the photocathode with an optical signal. The obtained current is the result of the level of illumination of the cathode. To eliminate the space charge effect between the photocathode and anode, maintaining a high voltage at the anode is crucial. Thus, efficiency is improved, as well as the electron transit time between the photocathode and anode, which is responsible for increasing the response speed of the device. The high impedance of the physical vacuum that forms its depletion region makes the vacuum photodiode a device with an excellent performance [30, 48].

Concerning the equivalent circuit, the small-signal equivalent circuit is presented in Fig. 24. The photocathode produces a photocurrent in reaction to an optical signal, where the load resistance $R_L$ is essential for converting the photocurrent into an output voltage signal, while the capacitance C represents the equivalent capacitance from the anode to ground [30].

Fig. 24

Small-signal equivalent circuit of a vacuum photodiode

The dark current of a vacuum photodiode arises from the thermionic emission of the photocathode. Typically measured in the realm of femtoamperes at standard room temperature, this current is often negligible, when compared to other sources of noise. The dominant shot-noise sources for a vacuum photodiode are the photocurrent and the background radiation current. Thermal noise, on the other hand, is predominantly influenced by the load resistance, particularly as it tends to be significantly smaller than the internal resistance of the vacuum photodiode. Ignoring the dark current from thermionic emission, the total noise is given by Eq. 76, where $i_s$ is the photocurrent. A vacuum photodiode has no gain, thus $i_s$ = $i_{ph}$ [30, 49].

$$\begin{aligned} \overline{i_n^2} = 2qB(\overline{i_s}+\overline{i_b})+\frac{4k_B}{2R_L} \end{aligned}$$

(76)

The response speed of a vacuum photodiode is determined by:

• (i) the transit time, denoting the duration it takes for a photoelectron to travel from the cathode to the anode;

• (ii) the transit-time spread of the photoelectrons from the photocathode to the anode, which is the spread in the transit time among different photoelectrons. This phenomenon stems mainly from the variance in the initial kinetic energies of the emitted photoelectrons.

• (iii) the RC time constant of its equivalent circuit.

Conditions (i) and (ii) can be reduced if the geometry of the vacuum photodiode is carefully designed and if it is applied a high anode-to-cathode voltage. To ensure optimal performance in high-speed scenarios, selecting a sufficiently low load resistance (usually around 50 $\Omega$) and eliminating any stray capacitances are crucial to prevent the RC time constant from becoming the limiting factor. A fast vacuum photodiode typically presents a speed ranging from 100 ps to 1 ns, with a corresponding 3-dB bandwidth ranging from a few hundred megahertz to about 3 GHz [30, 39].

3.3.2 Photomultipliers

A photomultiplier tube (PMT) is a vacuum photodiode with a built-in high gain, low noise electron multiplier. It is composed of:

• (i) a photocathode for emitting photoelectrons;

• (ii) an electron optics with focus electrodes to accelerate and concentrate the photoelectrons to the first dynode;

• (iii) an electron multiplier with a chain of dynodes for secondary electron emission;

• (iv) an anode to collect the electrons for the output signal.

The electron multiplier mechanism within a PMT comprises a sequence of electrodes known as dynodes. These dynodes are subject to increasingly higher voltages via a voltage-divider setup involving a series of resistors. In instances where the PMT operates in high-current pulse mode, capacitors are placed in parallel to the resistors in the last two or three stages of the divider circuit. This addition ensures consistent voltage levels across the last few dynodes, thereby enabling the PMT to exhibit a broad linear dynamic range [30, 48]. Figure 25 shows the basic circuit of a photomultiplier.

Fig. 25

Basic circuit of a photomultiplier

The electron multiplication is done using a secondary electron emission. A photoelectron emitted from the photocathode is accelerated by the high voltage between the photocathode and the first dynode, reaching energies typically ranging from 100 to 200 eV. When a high-energy electron collides with a dynode, it releases additional secondary electrons. This cascade effect persists through subsequent dynode stages [30, 39, 48].

Fig. 26

Small-signal equivalent circuit of a photomultiplier

The small-signal equivalent circuit is presented in Fig. 26. As in Fig. 24, the capacitance C is the total equivalent capacitance from the anode to the ground [30]. A photomultiplier presents a total current gain G, defined by Eq. 77, which is the ratio between the output signal current at the anode, $i_s$, and the photocurrent at the photocathode, $i_{ph}$. However, this gain can be described through the total electron multiplication gain through the dynode chain, being n the number of the dynodes in the chain and m the electron multiplication factor for each dynode stage [30, 48].

$$\begin{aligned} G = \frac{i_s}{i_{ph}} = m^n \end{aligned}$$

(77)

The dark current in a PMT arises from thermionic emission (from both the photocathode and the dynodes), leakage current, field emission, and electron emission due to cosmic rays. The total amplified dark current, denoted as $i_d$, is measured at the anode of the PMT. It is a very small current, ranging from 10 pA and 10 nA (depending on the materials used), although it cannot be ignored because of its high gain [30, 39]. The total current noise is given by Eq. 78, where F is the excess noise factor.

$$\begin{aligned} \overline{i_n^2} = 2qBGF(\overline{i_s}+\overline{i_b}+\overline{i_d})+\frac{4k_B}{2R_L} \end{aligned}$$

(78)

The sensitivity of PMT depends on the photocathode’s properties. The response speed of a PMT is determined by the same parameters that constrain the speed of a vacuum photodiode. However, due to the dynode chain, the electron transit time from the photocathode to the anode in a PMT is considerably longer compared to a vacuum photodiode. Nevertheless, the transit-time spread is much lower than the transit time. In terms of impulse response, the higher the transit time, the higher the delay in the response. The rise time of the response pulse is influenced by both the spread in transit time and the RC time constant. Thus, PMTs are highly rapid detectors with a frequency bandwidth on the order of a few hundred megahertz [30, 39, 48].

3.3.3 Photoconductors

Photoconductors are devices based on the phenomenon of photoconductivity. The photoresponse of these devices arises from the generation of electron–hole pairs through photogeneration. The conductivity $\sigma$ of a photoconductor increases with optical illumination due to the photogeneration of free carriers, and it can be given by Eq. 35 [9, 30, 39]. If there is no optical illumination, the conductivity is known as dark conductivity $\sigma _0$, since the electron and hole concentrations are in equilibrium concentrations ($n_0$ and $p_0$, respectively). When a semiconductor is illuminated with light with sufficient photon energy, additional carriers are generated. The photoconductivity is the additional conductivity contributed by the additional generated carriers, given by Eq. 79, being $\Delta n$ and $\Delta p$ the photogenerated excess electron and hole concentrations, respectively (described by Eq. 28) [9, 30, 36, 39].

$$\begin{aligned} \Delta \sigma = \sigma - \sigma _0 = q(\mu _n\Delta n + \mu _p\Delta p) \end{aligned}$$

(79)

Figure 27 presents the optical transitions for the different types of photoconductors, where $E_C$ and $E_V$ correspond to the conduction and valence bands, respectively, and $E_D$ and $E_A$ are the donor and acceptor levels, respectively. A deeper analysis of the differences between the different types of semiconductors is done in Sect. 2.

Fig. 27

Optical transitions for different types of photoconductors

These devices, since they present a photoresponse as a consequence of the electron–hole pair generation, have a minimum required energy to excite an electron from the valence band to the conduction band, whose value corresponds to the bandgap energy. Thus, the long-wavelength threshold of the material, which is also known as cut-off wavelength, is defined by Eq. 46. The lower the wavelength, the higher the maximum value of the response. This means that for higher wavelength values, and consequently for lower energies, there is no photoconductive response [9, 25, 30, 39].

Photoconductors cover a wide spectrum, ranging from ultraviolet to far-infrared wavelengths. Notably, numerous photoconductors are highly sensitive in the infrared range extending beyond 1.2 $\mu$m, where no photoemissive detectors are available. Both direct-gap and indirect-gap semiconductors can be used for photoconductor.

A sensitive photoconductor must have a low dark conductivity, in order to, under optical illumination, the total photoconductivity varies significantly. For this reason, according to Eq. 79, it is necessary to minimise the thermal equilibrium concentrations ($n_0$ and $p_0$), of free electrons and free holes in a photoconductor. The operation of a photoconductor requires an applied voltage, and its gain depends on the electrical contacts’ properties and other photoconductor parameters. Figure 28a shows the basic circuit of a photoconductor with an applied voltage, and Fig. 28b illustrates the equivalent circuit for a small-signal.

Fig. 28

Photoconductor’s electrical schemes

The shot noise in a photoconductor is related to the generation and recombination carriers process, contributing from the optical signal, background radiation and the dark current. Furthermore, there is a thermal noise from the resistances (represented in the basic electrical circuit). Thus. the total noise of a photoconductor is given by Eq. 80, where $R_{eq}$ is the equivalent resistance at the output. The dark current comes from the dark conductivity of the device, due to the thermal excitation of free carriers [30].

$$\begin{aligned} \overline{i_n^2} = 4qBG(\overline{i_s} + \overline{i_b}+ \overline{i_d}) + \frac{2k_B}{R_{eq}} \end{aligned}$$

(80)

The photoconductive gain, G, can be generally expressed as the quotient between the carrier lifetime, $\tau$, and the relaxation time constant, $\tau _r$, which depends on the photoconductor’s properties, contacts’ and the applied voltage [30].

3.3.4 Junction photodiodes

Junction photodiodes stand as the prevailing choice among photodetectors in the photonics industry. Spanning a broad spectrum from ultraviolet to infrared, they manifest in various iterations, such as semiconductor homojunctions, semiconductor heterojunctions, and metal–semiconductor junctions. Each one is deeply analysed in section 4.

The photoresponse of a photodiode is based on the photogeneration of electron–hole pairs, similar to the photoconductor devices. A junction photodiode with a greater performance is a photodiode with the lowest possible diffusion current. Its external quantum efficiency, $\eta _{ext}$, represents the proportion of all incident photons absorbed within the active region that actively contribute to the photocurrent. Regarding its photocurrent, as already mentioned, there are two contributions: a drift current in the depletion layer and a diffusion current in the diffusion regions, both from photogeneration. The two homogeneous regions located at the extremities of the photodiode serve as blocking contacts, since carriers neither drift nor diffuse through these regions. Thus, having a unitary gain, the external signal current is equal to the photocurrent, which depends on the optical signal’s power, given by Eq. 81. If a bias voltage is applied to the photodiode, the total current of the device is given by Eq. 82, where the diode’s current is derived from the Eq. 98 ($I = JA$, being A the area’s device) [30, 50].

$$\begin{aligned} i_s= & {} i_{ph} = \eta _{ext}\frac{qP_s\lambda }{hc} \end{aligned}$$

(81)

$$\begin{aligned} i= & {} I_0\left[ e^\frac{qV_x}{nk_BT}-1\right] - i_s \end{aligned}$$

(82)

Figure 29 shows the basic circuit for two modes of operating for a p-n junction photodiode. For a negative applied voltage, the junction photodiode operates in a photoconductive mode. Otherwise, the device works in a photovoltaic mode. For both operating modes, the photodiode presents a different response, which is described in table 3, where $R_L$ and $R_i$ are the load and internal resistances, respectively. It is important to note that the circuit, for the photovoltaic mode, does not require a bias voltage [25, 51].

Fig. 29

Basic circuit for both operating modes’ photodiode

Table 3 Operating mode’s photodiode

Figure 30 illustrates the small-signal equivalent circuit of a junction photodiode.

Fig. 30

Small-signal equivalent circuit of a junction photodiode

In the circuit are presented an internal capacitance and resistance, $C_i$ and $R_i$, respectively, which are photodiode features and depend on its size and structure, varying with the voltage across the junction. Thus, the higher the reverse voltage, the higher $R_i$ and the lower $C_i$, since the depletion’s width increases. Regarding the additional parameters outlined in the circuit of Fig. 30, $R_s$ represents the series resistance and takes into consideration both resistance within homogeneous regions and parasitic resistance from contacts. The external parallel capacitance $C_p$ denotes parasitic capacitance arising from contacts and packaging, while the series inductance $L_s$ signifies parasitic inductance from wiring or transmission-line connections. All these parameters can be reduced through design, processing, and packaging optimization, and collectively influence the device’s frequency response [30].

The total noise of a photodiode is the sum of the shot and thermal noises. Since $i_s = i_{ph}$, the total noise is described by Eq. 83, where $R_{eq}$ is the equivalent resistance of Fig. 30 and G is the gain value, which takes the unitary value [25, 30].

$$\begin{aligned} \overline{i_n^2} = 2qBG(\overline{i_s}+\overline{i_b}+\overline{i_d})+\frac{2k_B}{R_{eq}} \end{aligned}$$

(83)

In photoconductive mode, the dark current $i_d$ takes the value of the reverse saturation current $I_0$ and has a small load resistance. In photovoltaic mode, the dark current could be eliminated, and the load resistance is very large. Thus, the noise of the photodiode under photoconductive mode is higher than the photovoltaic’s [30, 51].

The speed of a photodiode is determined by the response time of the photocurrent and the time constant of its equivalent circuit, presented in Fig. 30. Due to its substantial RC time constant, a junction photodiode functioning in photovoltaic mode is unsuitable for high-speed applications, with only those operating in photoconductive mode being appropriate. Thus, the response time of the photocurrent for a photodiode operating in a photoconductive mode depends on:

• (i) the drift of the electrons and holes that are photogenerated in the depletion layer;

• (ii) the diffusion of the electrons and holes that are photogenerated in the diffusion regions.

For a high-speed photodiode, the diffusion mechanism has to be reduced or eliminated, reducing the carriers’ photogeneration outside the depletion layer through the design of the device structure [30]. In terms of applications, photodiodes operating in photoconductive mode are ideal for optical communication systems and fast light detection [51]. A Schottky photodiode typically functions in photoconductive mode across most of its applications, and it can achieve very high speed, particularly when an n-type semiconductor is used. By absorbing the optical signal in a thin layer at the interface, only the majority carriers need to traverse the active region. Its spectral response depends on whether an optical signal is absorbed by the semiconductor or by the metal. In the second case, for a photoresponse to occur, the photon must possess adequate energy to excite an electron across the Schottky barrier. For a Schottky photodiode to operate in this mode, the metallic layer has to be thick and absorbing, but the absorption has to take place at the junction interface. Thus, its spectral response range is $E_b< h\mu < E_G$ for the optical signal to penetrate from the semiconductor side without being absorbed by the semiconductor [30].

3.3.5 Avalanche photodiodes

An avalanche photodiode (APD) is a semiconductor photodiode detector with high sensitivity, utilizing the photoelectric effect to transform light energy into electrical signals. In terms of functionality, they are akin to the solid-state version of the PMT. In an APD, the internal gain is due to the avalanche multiplication of charge carriers through impact ionization, which is caused by a sufficiently large bias applied voltage. This factor differs from the junction photodiode [30, 34, 52].

In the impact ionization process, when an electron or hole possesses significant kinetic energy, it can induce the creation of an additional electron–hole pair by transferring this energy through collisions with the lattice, exciting secondary carriers. When subjected to a strong electric field, the newly generated electron and hole can be accelerated to gain sufficient kinetic energies for impact ionization to generate more electron–hole pairs. Thus, a cascade of these events leads to avalanche multiplication of the photogenerated carriers [25, 30, 36, 46, 52, 53].

The number of ionizing collisions per unit length is termed as the ionization coefficient $\alpha _{n.p}$, for electrons and holes, respectively. Both are characteristics of a semiconductor and are strong functions of both electric field strength (increase rapidly with its increase) and temperature (decrease with increasing it) [25, 30, 54]. If their ratio k, known as the ionization ratio, is higher than 1, then holes dominate the impact ionization. Otherwise, electrons dominate it [30].

The small-signal equivalent circuit of an APD is similar to that of a junction photodiode, which is illustrated in Fig. 30, except that the avalanche multiplication gain is included in the signal current $i_s$, in such a way that $i_s$ = G$i_{ph}$. The avalanche multiplication factor of photogenerated carriers, which is a dependent parameter of its thickness, the structure of the avalanche region in it and the reverse voltage applied to it. To enhance avalanche multiplication, two conditions are required. Firstly, the avalanche region has to be relatively thin to support a very high field without local breakdown. Lastly, it is preferable to inject a single type of carrier into the avalanche region rather than generating both electrons and holes throughout the region [30].

Due to the avalanche multiplication process, all APDs generate excess noise, and the total current noise of an APD is given by Eq. 78 [25, 30, 55, 56]. The APD could respond to light waves modulated at microwave frequencies [34], offering a combination of high sensitivity and high quantum efficiency [52]. The higher the thickness of the absorption region, the higher the quantum efficiency. This type of photodiode presents a higher sensitivity compared to p-i-n photodiodes, which is quite important for applications with low illumination levels. Nevertheless, APDs have a higher noise and higher bias voltage [36].

Fig. 31

n-p-n bipolar phototransistor

3.3.6 Phototransistors

A phototransistor is a type of semiconductor component capable of detecting light intensity and modifying the current passing through from the emitter to the collector based on the amount of light it is exposed to. This device presents a functional structure similar to the classic transistor. Figure 31 shows an n-p-n bipolar phototransistor. The incident radiation generates electron–hole pairs in the base-collector region. In the n-p-n phototransistor, electrons drift to the collector, and holes accumulate in the base. The first ones are directly collected, and the holes are confined due to the electric field at the two junctions. As a result of the hole phenomenon, the potential barrier at the base-emitter junction decreases, facilitating the injection of new electrons from the emitter into the base to neutralize it. When they reach the base-collector junction, they are swept out of the space charge region, producing a photocurrent [34, 46, 57].

This component has optical access from the base-collector junction and its area is large. [34, 46] The frequency response of a phototransistor is limited by the base-collector junction capacitance. However, a phototransistor is a lower-noise device than the APD [34]. The phototransistor can have high gain through the transistor action [34, 57] and its efficiency can be maximized if the transition region of the collector junction is increased, which can be achieved by ensuring a low concentration of replacement impurities in the collector region [36].

4 Types of photodiodes

There are many types of photodiodes in the market, all of them working with the same basic principle. Some of them are explained in section 2. The junction photodiodes can take many forms, including homojunctions, and heterojunctions, which can be semiconductor-semiconductor and metal–semiconductor junctions.

4.1 p-n junction photodiode

A p-n junction corresponds to two regions, one doped with acceptor impurity atoms to form the p region, being $p_{p_0} = N_A$, and another doped with donor atoms to form the n region, being $n_{n_0} = N_D$, with the same semiconductor material. For a simpler explanation, consider a step junction, wherein the doping concentration remains uniform within each area, with a sudden shift in doping occurring at the junction.

The boundary between the n and p regions forms a metallurgical junction, characterized initially by a significant density gradient in both electron and hole concentrations. The majority of charge carriers of each region will begin diffusing to the opposite regions. As electrons and holes diffuse from the n and p regions, respectively, positively charged donor atoms and negatively charged acceptor atoms are left behind, respectively. The net positive and negative charges in the n and p regions are known as the space charge region, which is the constant region. This region, although electrically neutral, has a region near the junction with non-zero charge density, which is responsible for the existence of an electric field, ${\mathcal {E}}$, from n to p. The electric field takes the maximum value in the contact between both regions, and it’s null outside the transition region. The presence of the internal electric field means that there is a potential difference with the opposite direction of the electric field [9, 30, 34, 36].

Figure 32 shows the evolution of the mobile charge carriers in the p-n junction under thermal equilibrium.

Fig. 32

Evolution of mobile charge carriers in a p-n junction under equilibrium

As can be seen, the concentration of charge carriers remains constant over the isolated doped semiconductor, inclusive in the quasi-neutral regions. In the space-charge region, these concentrations decrease very fast, being the density of carriers in the transition region much lower than the density of the majority carriers outside the transition region. Thus, the density of carriers in the transition region depends on the concentration of ionized dopants, assuming that the space charge region is depleted of any mobile charge, which can be called the depletion region. Mathematically, it is described by Eq. 84 [9, 30, 34, 36].

$$\begin{aligned} \rho (x) = \left\{ \begin{array}{ll} 0 &{}\qquad x \le -x_{p_0}\\ -qN_A &{}\qquad -x_{p_0} \le x \le 0\\ qN_D &{}\qquad 0 \le x \le x_{n_0}\\ 0 &{}\qquad x \ge x_{n_0}\\ \end{array}\right. \end{aligned}$$

(84)

The electric field, ${\mathcal {E}}(x)$, in one dimension model, is given by Eq. 85 and knowing Eq. 84, the electric field for each interval can be obtained, as well as, the potential difference. The result is presented in Eq. 86 [9, 30, 33, 34, 36].

$$\begin{aligned} \frac{d^2V}{dx^2}= & {} -\frac{\rho }{\epsilon } = -\frac{d{\mathcal {E}}(x)}{dx} \end{aligned}$$

(85)

$$\begin{aligned}{} & {} {\mathcal {E}}(x) = -\frac{dV}{dx} = \left\{ \begin{array}{cc} -\frac{qN_A}{\epsilon }(x_{p_0} + x)&{} -x_{p_0} \le x \le 0\\ \frac{qN_D}{\epsilon }(x - x_{n_0}) &{} 0 \le x \le x_{n_0} \end{array}\right. \nonumber \\{} & {} \quad \leftrightarrow V(x) = \left\{ \begin{array}{cc} \frac{qN_A}{2\epsilon }(x_{p_0} + x)^2 &{} -x_{p_0} \le x \le 0\\ -\frac{qN_D}{2\epsilon }(x - x_{n_0})^2 &{} 0 \le x \le x_{n_0}\\ \end{array}\right. \end{aligned}$$

(86)

Under equilibrium, a potential difference across the space charge region, known as built-in voltage, $V_{bi}$ appears, and it is calculated through the difference between the electric field at the edges of the transition region, i.e. at $-x_{p_0}$ and $x_{n_0}$. At the junction ($x = 0$) the electric field is continuous meaning that $N_Ax_{p_0} = N_Dx_{n_0}$. Thus, the built-in voltage is obtained through Eq. 87. This parameter can be calculated using the energy-band diagram, presented in Fig. 33.

$$\begin{aligned} V_{bi} = \frac{q}{2\epsilon }(N_Dx_{n_0}^2+N_Ax_{p_0}^2) \end{aligned}$$

(87)

Fig. 33

Band diagram of a p-n junction in thermal equilibrium

Knowing that the built-in voltage is the difference between the bandgap energy $E_G$ and the energies between the fermi level and the conduction and valence bands ($E_1$ and $E_2$) and according to Eqs. 15, 22 and 26, it is possible to obtain the built-in voltage using Eq. 88.

$$\begin{aligned} qV_{bi}&= E_G - k_BT\ln {\left( \frac{N_V}{N_A}\right) }-k_BT\ln {\left( \frac{N_C}{N_D}\right) } \nonumber \\&\leftrightarrow V_{bi} = \frac{k_BT}{q}\ln {\left( \frac{N_AN_D}{n_i^2}\right) } \end{aligned}$$

(88)

The width W of the space charge region of the junction can be calculated. Through Fig. 33, it is clear that the width of this region is the sum of the distances of $x_{p_0}$ and $x_{n_0}$ from the reference point of the graph. Starting with the established equality $N_Ax_{p_0} = N_Dx_{n_0}$ and considering Eq. 87, $x_{p_0}$ and $x_{n_0}$ are obtained through Eq. 89 and the width W is, finally, calculated using Eq. 90.

$$\begin{aligned} x_{p_0}= & {} \sqrt{\frac{2\epsilon V_{bi}}{q}\frac{N_D}{N_A}\left( \frac{1}{N_A+N_D}\right) } \end{aligned}$$

(89a)

$$\begin{aligned} x_{n_0}= & {} \sqrt{\frac{2\epsilon V_{bi}}{q}\frac{N_A}{N_D}\left( \frac{1}{N_A+N_D}\right) } \end{aligned}$$

(89b)

$$\begin{aligned} W = x_{p_0} + x_{n_0} = \sqrt{\frac{2\epsilon }{q}V_{bi}\left( \frac{1}{N_A}+\frac{1}{N_D}\right) } \end{aligned}$$

(90)

As already mentioned, the maximum electric field is obtained in the contact between n and p regions and once the values of $x_{p_0}$ and $x_{n_0}$ have been obtained, the maximum electric field value can be determined, using Eqs. 89 and 86. The result, in module, is stated in Eq. 91 [9, 30, 33, 34, 36].

$$\begin{aligned} {\mathcal {E}}_{max} = \sqrt{\frac{2q}{\epsilon }V_{bi}\left( \frac{N_AN_D}{N_A+N_D}\right) } = \frac{2V_{bi}}{W} \end{aligned}$$

(91)

In the depletion region, the generation of an electron–hole pair can occur. When this happens, the electric field sweeps the electron to the n region and the hole to the opposite side. The photoexcitation can also happen in the diffusion region, i.e. in the regions closest to the depletion zone, which is identified in Fig. 33 as $L_n$ and $L_p$. Thus, the minority carrier can reach the transition zone by diffusion and, through the electric field, the minority carrier sweeps to the opposite side. In both zones, a drift current flows in a reverse direction. However, if the generation process occurs in the homogeneous region, i.e. in the regions furthest from the depletion zone, no current is generated, since there is no electric field to separate the charges and the minority carrier generated in these regions cannot diffuse to the depletion region before recombining with a majority carrier [30]. Under an equilibrium situation, the current density is null, since $div \vec {J} = 0$ is imposed. Considering the current density involved for each charge carrier, Eq. 31 is verified and, finally, can be proved that the energy Fermi level is constant over the junction [9, 36]. Considering the built-in voltage (Eq. 88) and assuming a total ionization, $n_{n_0}\approx N_D$ and $p_{p_0}\approx N_A$, being $n_{n_0}$ the thermal-equilibrium concentration of majority carrier electrons in the n region and $p_{p_0}$ the thermal-equilibrium concentration of majority carrier holes in the p region, the thermal-equilibrium concentration of minority carrier charges, $n_{p_0}$ and $p_{n_0}$, can be obtained through Eq. 14. Thus, the relation established in Eq. 92 can be stated.

$$\begin{aligned} \left\{ \begin{array}{l} n_{p_0} = n_{n_0}e^\frac{-qV_{bi}}{k_BT}\\ p_{n_0} = p_{p_0}e^\frac{-qV_{bi}}{k_BT} \end{array}\right. \end{aligned}$$

(92)

When an external voltage is applied to the p-n junction, the situation under analysis is a non-thermal equilibrium, meaning that the potential difference and the electric field across the transition region will change. In this situation, the Fermi energy level is not constant. Figure 34 shows the band diagram of a p-n junction in non-thermal equilibrium, under a reverse bias voltage and a forward bias voltage, both applied in the p region to the n region.

Fig. 34

Band diagram of a p-n junction in non-thermal equilibrium, under an applied voltage

In the reverse bias situation, a positive voltage is applied from the n to p region and the applied voltage will increase the potential difference across the p-n junction. In opposition, a positive voltage is applied from the p region to the n region, and this voltage will decrease the potential difference in the p-n junction. It is known as a forward bias situation [9, 30, 34, 36, 50].

Regarding the width of the space charge carrier, compared to the equilibrium situation, the following condition is verified: $W_2< W < W_1$. Once the electric field varies, for a given impurity doping concentration, the number of charge carriers in the depletion region also varies, but only if its width also varies. Thus, the higher the electric field, the higher the charge carrier concentration in the depletion region, which translates into a higher width of the region. This also applies to the diffusion regions, meaning that $L_{p_2}< L_p < L_{p_1}$ and $L_{n_2}< L_n < L_{n_1}$. For both conditions, considering $V_{total}$ as $V_{bi}-V_x$, where $V_x$ is negative for reverse condition ($-V_r$) or positive for forward condition ($V_f$), the total width of the space charge carrier is given by Eq. 93. The maximum electric field for a non-thermal equilibrium can be obtained using Eq. 94.

$$\begin{aligned} W= & {} \sqrt{\frac{2\epsilon }{q}V_{total}\left( \frac{1}{N_A}+\frac{1}{N_D}\right) } \end{aligned}$$

(93)

$$\begin{aligned} {\mathcal {E}}_{max}= & {} \sqrt{\frac{2q}{\epsilon }V_{total}\left( \frac{N_AN_D}{N_A+N_D}\right) } = \frac{2V_{total}}{W} \end{aligned}$$

(94)

Regarding the equation that relates the minority and majority charge carrier concentration on the p side of the junction to the majority and minority charge carrier concentration on the n side of the junction, Eq. 92 is rewritten as Eq. 95 [34, 36]. Figure 35 shows the evolution of the mobile charge carriers in the p-n junction under both non-thermal equilibrium conditions.

$$\begin{aligned} \left\{ \begin{array}{l} n_p = n_{n_0}e^\frac{-qV_{total}}{k_BT} = n_{n_0}e^\frac{-qV_{bi}}{k_BT}e^\frac{qV_x}{k_BT} = n_{p_0}e^\frac{qV_x}{k_BT} \\ p_n = p_{p_0}e^\frac{-qV_{total}}{k_BT} = p_{p_0}e^\frac{-qV_{bi}}{k_BT}e^\frac{qV_x}{k_BT} = p_{n_0}e^\frac{qV_x}{k_BT} \end{array}\right. \end{aligned}$$

(95)

Fig. 35

Evolution of mobile charge carriers in a p-n junction under non-thermal equilibrium

Under a non-thermal equilibrium, the density current, given by Eq. 31, takes into consideration the individual electron and hole currents that are constant through the depletion region. Once they are continuous functions, the total p-n junction current will be the minority carrier electron diffusion current at $x = -x_{p_0}$ plus the minority carrier hole diffusion current at $x = x_{n_0}$. Starting from the transport equation for excess minority carrier holes and electrons in n and p regions, which are given by Eqs. 96 and 97, respectively, where $\Delta p = p_n(x)-p_{n_0}$ and $\Delta n = n_p(x)-n_{p_0}$, and considering equations of the transport properties described in section 2.7, the total current density, J. is given by Eq. 98, which is known as the ideal current–voltage relationship of a p-n junction. This relationship is plotted in Fig. 36, where n is the ideality factor, that has a value between 1 and 2, and $J_0$ is the reverse saturation current density [9, 30, 34, 36].

$$\begin{aligned}{} & {} D_p\frac{\partial ^2(\Delta p)}{\partial x^2}-\mu _pE\frac{\partial (\Delta p)}{\partial x} + g' - \frac{\Delta p}{\tau _{p_0}} = \frac{\partial (\Delta p)}{\partial t} \end{aligned}$$

(96)

$$\begin{aligned}{} & {} D_n\frac{\partial ^2(\Delta n)}{\partial x^2}-\mu _nE\frac{\partial (\Delta n)}{\partial x} + g' - \frac{\Delta n}{\tau _{n_0}} = \frac{\partial (\Delta n)}{\partial t} \end{aligned}$$

(97)

$$\begin{aligned}{} & {} J = J_p(x_n) + J_p(-x_p) \nonumber \\{} & {} \quad = \left[ \frac{qD_pp_{n_0}}{L_p}+\frac{qD_nn_{p_0}}{L_n}\right] \left[ e^\frac{qV_x}{nk_BT}-1\right] \nonumber \\{} & {} \quad = J_0\left[ e^\frac{qV_x}{nk_BT}-1\right] \end{aligned}$$

(98)

Fig. 36

Ideal I-V characteristic of a p-n junction diode

In the depletion region, occurs a separation of positive and negative charges. Thus, a capacitance is associated with the junction, known as depletion layer capacitance, and it can be defined as the change in electric charge for a change in applied voltage, given mathematically described by Eq. 99, where dQ is given by Eq. 100. Thus, the junction capacitance per unit area, considering the total potential barrier, is given by Eq. 101 [34, 58]. This expression can be obtained using the space charge region $x_n$ or $x_p$ [34].

$$\begin{aligned} C'=\, & {} \frac{dQ}{dV} \end{aligned}$$

(99)

$$\begin{aligned} dQ=\, & {} qN_Ddx_n = qN_Adx_p \end{aligned}$$

(100)

$$\begin{aligned} C'= & {} \sqrt{\frac{q\epsilon }{2V_{total}}\frac{N_AN_D}{N_A+N_D}} = \frac{\epsilon }{W} \end{aligned}$$

(101)

4.2 p-i-n junction photodiode

A p-i-n photodiode is an intrinsic semiconductor region within two highly doped contact layers, one p$^+$ and the other n$^+$. Figure 37 illustrates the p-i-n junction structure. This structure combines good frequency response with simplicity [59].

Fig. 37

p-i-n junction structure

The depletion region is defined almost by the intrinsic layer, and the electric field across it is uniform. The depletion layer width W is pretty much fixed by the width of the intrinsic region, $W_i$, so that $W \approx W_i$. The internal capacitance of a p-i-n photodiode can be determined through Eq. 102, which shows that it is an independent parameter of the bias voltage [30].

$$\begin{aligned} C = \frac{\epsilon A}{W} \approx \frac{\epsilon A}{W_i} \end{aligned}$$

(102)

4.3 Semiconductor heterojunctions

A semiconductor heterojunction is when two different semiconductor materials are used to form a junction and these materials must have lattice constants that match with each other, to guarantee good electrical and optical properties in the interfaces [33, 34]. This type of junction offers flexibility in optimising the performance of these devices.

Fig. 38

Types of semiconductor heterojunctions

4.3.1 Semiconductor-semiconductor junctions

When two different semiconductors are joined together, have discontinuities between the valence band maxima or conduction band minima at their interface, which act as barriers to the charge carriers. Thus, according to their band alignment, heterojunctions can be divided into two major groups: type-I and type-II. However, the literature defines a particular case of type-II’s heterojunction, known as a type-III or broken-gap heterojunction. Figure 38 illustrates the previously mentioned heterojunctions’ types [33, 34, 60,61,62].

Type-I:

The two semiconductor materials present different bandgap energy, in such a way that one material has a lower energy of the conduction band, $E_C$, and a higher energy of the valence band, $E_V$, which translates into a smaller band gap energy.

Type-II:

The charge carriers are confined in different spaces since the lower conduction band, $E_C$ and the higher valence band, $E_V$, are displaced. In this type, one of the band offsets is larger than the difference between the two semiconductor bandgaps, but it is smaller than the largest bandgap.

Type-III:

One of the semiconductor material has a lower $E_C$ comparing to the energy of the valence band $E_V$ of the second semiconductor. Thus, the conduction band overlaps the valence band at the interface.

There are two basic types of heterojunction: those in which the dopant type changes at the junction, known as anisotype, and others in which the dopant type on either side of the junction, called isotype. Thus, anisotype heterojunction can be P-n or p-N junctions, where the capital letter indicates the larger bandgap material semiconductor. Figure 39 illustrates the band diagram of the anisotype heterojunctions in thermal equilibrium. Regarding the isotype heterojunction, there are n-N and p-P heterojunctions [30, 34, 36, 62].

Fig. 39

Band diagram of anisotype heterojunction in thermal equilibrium

At the junction, where the semiconductors are joined together, the differences in both bandgaps, $\Delta E_G$, result in a discontinuity of the conduction and valence bands, $\Delta E_C$ and $\Delta E_V$, respectively. Thus, the bandgap discrepancy is given by Eq. 103, where $E_{G_2}$ and $E_{G_1}$ correspond to the bandgap energy of the semiconductor 2 and 1, respectively.

$$\begin{aligned} \Delta E_G = E_{G_2}-E_{G_1} = \Delta E_C + \Delta E_V \end{aligned}$$

(103)

The total built-in potential barrier $V_{bi}$ can be obtained using the energy-band diagram, illustrated in Fig. 39, or using the physical expressions. Thus. for a heterojunction 1-2, the total built-in potential barrier $V_{bi}$ is generically given by Eq. 104, and for each type of heterojunction takes the form of the expressions in table 4 [9, 30, 34, 36, 62, 63]. On the other hand, this parameter can also be obtained by using Gauss’ Law, through the same methodology used for homojunction (section 4.1), considering the depletion approximation.

$$\begin{aligned} qV_{bi} = E_{F_1}-E_{F_2} \end{aligned}$$

(104)

Table 4 Built-in potential barrier for each type of heterojunction

The width W of the depletion region of a heterojunction 1-2 corresponds to the sum of $x_1$ and $x_2$ values. According to the heterojunction’s type, these values vary, as well as the mathematical expressions, which are described in table 5 [9, 34, 62,63,64]. In non-thermal equilibrium, these expressions can be adapted in the same way as described for homojunction (section 4.1), replacing $V_{bi}$ by $V_{total}$.

Table 5 Widths of the depletion region in each zone for each type of heterojunction

As in the case of a homojunction, a change in depletion width with a change in junction voltage yields a junction capacitance. Mathematically, Eq. 99 is applied [34, 65]. Table 6 presents the final expressions for each type of heterojunctions.

Table 6 Junction capacitance per unit for each type of heterojunction

4.3.2 Metal–semiconductor junctions

The metal–semiconductor junction can be rectifier or ohmic type, depending on the barrier’s height, i.e. if $E_{b_{p,n}}$ $<<$ $k_BT$, the metal–semiconductor junction is rectifying. Otherwise, it is ohmic. Its type also depends on the work functions of metal, $q\phi _m$ (required energy to remove an electron from the Fermi level to a position outside the material), and semiconductor, $q\phi _s$, and the type of semiconductor (n or p). The work functions of several metals, in eV, are given in table 7 [66, 67].

Table 7 Work functions of several metals

The rectifier metal–semiconductor junction allows the Schottky diodes’ manufacture, which has a current–voltage characteristic very similar to the p-n junction’s. Nevertheless, its current is mainly determined by the majority carrier, which allows them to be faster devices [9, 36, 62].

Fig. 40

Band diagrams of a junction between a metal and a semiconductor

In the junctions, i.e. in the ohmic contact, exists a contact potential difference and, consequently, a local electric field. If a semiconductor is an n-type, and if $\phi _{s_n} > \phi _m$, then the ohmic contact is without a potential barrier. In the case of a p-type semiconductor, this happens if $\phi _{s_p} < \phi _m$ [9, 30, 34, 36]. This is illustrated in Fig. 40. Otherwise, there is a potential barrier, known as the Schottky barrier, of a height $E_b$ for electrons or holes to flow from the metal to the semiconductor, depending on whether it is an n-type or a p-type semiconductor, which is described in Eq. 105, respectively, where $\chi$ is the electron affinity. The sum of both barrier heights of n-type and p-type semiconductors is equal to its bandgap $E_G$.

$$\begin{aligned} E_{b_n}= & {} q(\phi _m-\chi ) \end{aligned}$$

(105a)

$$\begin{aligned} E_{b_p}= & {} E_G - q(\phi _m-\chi ) \end{aligned}$$

(105b)

On the semiconductor side, there is a built-in potential barrier, $V_{bi}$ which is given by Eq. 106a, for an n-type semiconductor. For a p-type semiconductor, the built-in potential barrier is given by 106b. Both are independent of the metal, and $\phi _{s_n}$ and $\phi _{s_p}$ are the distances between the lower and higher edges of the conduction and valence bands, respectively, and the Fermi level. These values correspond to the potential difference at $x = 0$, taking the values of expression 107.

$$\begin{aligned} V_{bi}= & {} E_{b_n} -\phi _{s_n} \end{aligned}$$

(106a)

$$\begin{aligned} V_{bi}= & {} \phi _{s_p}-E_{b_p} \end{aligned}$$

(106b)

This situation is under thermal equilibrium. Near the contact, on the semiconductor side, there is a depleted region with a positive/negative charge, mainly determined by the positive/negative ionised impurities ($N_D^+/N_A^-$), for the n-type/p-type semiconductor. This contact is, therefore, a rectifying contact, being the electric field from the semiconductor to the metal or from the metal to the semiconductor, depending on whether it is an n-type or p-type semiconductor, respectively. For an n-type semiconductor, reverse polarisation, i.e. applying a reverse bias voltage, $V_r$, means that the electric field near the contact is strengthened, increasing the potential barrier to $q(V_{bi}+V_r)$. However, when a forward bias voltage, $V_f$, is applied to the metal to the semiconductor, the barrier height for an electron that wants to travel from the semiconductor to the metal decreases to $q(V_{bi}-V_f)$. For a p-type semiconductor, it is applied the same logical thinking, only the voltage polarisation changes [9, 30, 34, 36, 62]. Generalising, under non-thermal equilibrium, the total potential barrier, $V_{total}$, corresponds to $V_{bi}-V_x$, where $V_x$ is the applied voltage, being negative for reverse condition ($-V_r$) or positive for forward condition ($V_f$).

The electrostatic properties of the junction are determined in the same way as the p-n junction. The electric field in the space charge region is given by Eq. 85, reaching a peak value at the metal–semiconductor interface. Thus. the potential difference in each metal–semiconductor type junction is given by Eq. 107.

$$\begin{aligned}{} & {} {\mathcal {E}}(x) = -\frac{dV}{dx} = \left\{ \begin{array}{lc} -\frac{qN_A}{\epsilon }(x_{p_0} + x), &{} \text {p-type semiconductor}\\ \frac{qN_D}{\epsilon }(x - x_{n_0}), &{} \text {n-type semiconductor} \end{array}\right. \nonumber \\{} & {} \quad \leftrightarrow V(x) = \left\{ \begin{array}{ll} \frac{qN_A}{2\epsilon }(x_{p_0} + x)^2, \hspace{0.6 cm} \text {p-type semiconductor}\\ -\frac{qN_D}{2\epsilon }(x - x_{n_0})^2, \hspace{0.3 cm} \text {n-type semiconductor}\\ \end{array}\right. \end{aligned}$$

(107)

Thus, the space charge region width W may be calculated through Eq. 108, for n-type and p-type semiconductors respectively. Under non-thermal equilibrium, the space charge region is obtained in the same way, replacing $V_{bi}$ by $V_{total}$ [9, 34].

$$\begin{aligned} W= & {} x_{n_0} = \sqrt{\frac{2\epsilon V_{bi}}{qN_D}} \end{aligned}$$

(108a)

$$\begin{aligned} W= & {} x_{p_0} = \sqrt{\frac{2\epsilon V_{bi}}{qN_A}} \end{aligned}$$

(108b)

The junction capacitance per unit area can be determined using Eq. 99. The final expression for n-type and p-type semiconductors is given by Eq. 109 [34].

$$\begin{aligned} C'= & {} \sqrt{\frac{q\epsilon }{2V_{total}}N_D} \end{aligned}$$

(109a)

$$\begin{aligned} C'= & {} \sqrt{\frac{q\epsilon }{2V_{total}}N_A} \end{aligned}$$

(109b)

This type of photodiodes presents a low efficiency since the metal does not absorb light efficiently [30]. However, compared to the simplest junction photodiode, it has a simpler manufacturing method, does not present high-temperature diffusion processes, and has a high response speed. Additionally, it has a relatively high dark current [52].

4.4 Summary

In this chapter, different types of photodiodes were introduced, as well as their working principle. Table 8 presents a summary of the main parameter expressions for the different types of photodiodes, at thermal equilibrium. Note that indices 1 and 2 refer to the junction zones so that the expressions can be standardised for any case. As an example, the p-n junction is described in the table as a junction 1-2, where p is 1 and n is 2. Generalising for non-thermal equilibrium, $V_{bi}$ is replaced by $V_{total}$.

Table 8 Summary of parameters expressions for photodiodes in thermal equilibrium

5 Semiconductors materials, structures, and performances

Different semiconductor materials differ in their properties and according to it, a specific material could be more efficient in a certain application. Thus, when the device is conceived or when it is chosen, it is important to bear in mind what is its purpose and application.

In general, an ideal photodiode should present high responsivity or sensitivity, high detection speed, large bandwidth, high quantum efficiency, low dark currents (small standby power consumption and low noise level), and low applied voltage-bias requirements [27]. As illustrated in Fig. 18, different materials have different absorption coefficients in different regions of the spectrum. Besides the semiconductor behaviour at a certain wavelength, it is important to take into consideration the electrical properties of the material. For instance, if the main of a project is to achieve a faster response, then it is important to ensure lower time responses, which implies that the semiconductor has higher mobility values. Whereas, if the goal is to obtain higher responses, then it is important to analyse the material’s conductivity (due to the losses) and the mobilities. For a specific application, the choice of a certain photodiode is mainly determined by the following [25]:

• (i) The type of optical signal, which depends on its wavelength;

• (ii) The semiconductor’s properties and its geometry. Both will influence the quantum efficiency and reliability;

• (iii) Economic aspects.

In an ideal world, the last aspect would not be considered. However, the photodiode’s choice is mainly determined by it. Most of the time, the best option is not the real one, but the one that fulfils the minimum requirements.

Figure 41 shows the spectral sensitivity of several semiconductor photodiodes over two different spectral ranges. The performances shown are representative of the best devices in production [46].

Fig. 41

Spectral sensitivity of typical semiconductors photodiodes

As referred, the electrical properties of the semiconductor are important to consider. Table 9 shows some electrical properties of several semiconductors, at room temperature (300 K) [68,69,70,71,72,73,74,75]. In this table, only the most commonly used materials are shown.

Table 9 Semiconductors electrical properties @ 300 K

The photodetectors’ performance depends on the material systems and structures adopted. Because of this, depending on its purpose, different devices are chosen. Table 10 presents a summary of different structures and materials of photodetectors, as well as some performance details.

Table 10 Performance comparison for photodetectors based on various material systems

Silicon (Si) is the semiconductor material with the most advanced technology. It is a material found in abundance on Earth and presents a relatively easy preparation of pure and high-quality single crystals. Due to its abundance, it is also a lower expensive material [5, 25, 45, 89,90,91]. Si has excellent electrical and thermal properties and, fortunately, it is also an excellent material for photodetectors. Si photodiodes offer a high efficiency in the visible and near IR and the IR cut-off depends on the width of the depletion region [46]. The p-i-n Si photodiode is one of the simplest photodetectors. Typically, their time response exceeds 1 ns both for pulse rise and fall time and their maximum quantum efficiencies are around 60–85%. However, Si crystal has a relatively large indirect band gap, as such, Si is not the most suitable material for photodiodes operating at the telecommunications wavelength compared to other materials that have smaller bandgap energies, such as indium gallium arsenide (InGaAs) or germanium (Ge) [27].

Ge presents a bandgap of 0.66 eV and, in principle, it covers the total wavelength range useful for optical communication. As compared to silicon, however, it suffers from several severe drawbacks, which limit its practical application to wavelengths greater than 1 pm [25, 27]. A typical Ge photodiode has a quantum efficiency directly proportional to the wavelength. However, this is no longer true for wavelengths longer than 1.5 pm, where quantum efficiency decays rapidly since the light penetration depth more and more exceeds the depletion layer width. At wavelength values lower than 1.5 pm and if the RC product is not too high, the time response of the photodiodes is short due to the thin depletion layer. For higher wavelength values, the diffusion currents will impair the time response [25, 92].

Gallium Arsenide (GaAs) has a spectral range for wavelengths shorter than 0.9 $\mu$m, which is also covered by Si. Because of that, its interest in optical communication systems is not very pronounced. As already mentioned, Si presents many advantages, in opposition to GaAs. This single-element processing is less advanced and more complicated, it is a more expensive material, as well as less poor crystal quality results. GaAs p-n junctions photodiodes exhibit a relatively low quantum efficiency, but, in what concerns Schottky photodiodes, they can be rather easily produced with good quality [25, 45]. However, in the literature, it is refereed that GaAs photodiodes display a sufficiently high sensitivity to make them an interesting and useful detector. In the spectral region from 450 nm to 900 nm, this semiconductor may offer superior sensitivity as well as compact size [93]. Due to its higher costs, GaAs is used in solar cells [46].

Gallium phosphide (GaP) presents a good sensitivity down to UV-C. It is an interesting material for ultraviolet photodiodes of the solar blind type, with negligible sensitivity to the visible. It is a faster material than Si and presents a smaller dark current [46].

Research into compound semiconductors increased when semiconductor materials were needed that were better suited to detection at wavelengths longer than 0.9 $\mu$m. Both binary semiconductors, Gallium Antimonide (GaSb) and Indium Arsenide (InAs), cover this spectral range, but the quantum efficiency of p-n junction photodiodes would be limited since they are both direct materials and their light penetration depth is small [25].

The $\hbox {In}_x\hbox {Ga}_{1-x}$As ternary system is attractive for its wide energy gap, ranging from 0.36 eV for pure InAs to 1.42 eV for pure GaAs (table 9). By simply changing the alloy composition, the photodetection responsivity can be maximized at the desired wavelength [27, 94]. The $\hbox {In}_{0.53}\hbox {Ga}_{0.47}$As compound presents a bandgap energy of 0.73 eV, which gives a complete lattice-match to the InP substrate, and outstanding photodetection performance for wavelengths ranging from 1.0 to 1.7 $\mu \hbox {m}$ [27]. In optical communication applications, $\hbox {In}_x\hbox {Ga}_{1-x}$As photodiodes currently outperform Si and Ge owing to their direct band gap properties and relatively large absorption coefficient. In terms of material properties, Ge is more suitable for photodetection than InGaAs. On the other hand, the lattice mismatch between Ge and Si is 4.2% which is relatively smaller than that of InGaAs/Si. However, this lattice mismatch causes high surface roughness and threading dislocation densities, which degrades the device performance [27].

In the middle IR (MIR) spectral ranges, both photodiodes and photoresistances are made of indium antimonide (InSb) and indium arsenide (InAs). Regarding the far IR (FIR), the best option is the ternary compound II-VI cadmium-mercury telluride ($\hbox {Hg}_x\hbox {Cd}_{1-x}\hbox {Te}$), also called CMT [27, 46, 76, 95]. This compound has adjustable bandgap energy, due to the changing of the alloy composition. Thus, this compound offers flexibility in spectral response over a wide infrared spectrum [27, 46, 53]. The $\hbox {Hg}_x\hbox {Cd}_{1-x}\hbox {Te}$ system is believed to be suitable for this spectrum range due to several reasons pointed out in the literature, such as a direct bandgap with high photon absorption coefficient, which leads to high quantum efficiency and low noise generation [27]. Similar to CMT, lead tin telluride ($\hbox {Pb}_x\hbox {Sn}_{1-x}\hbox {Te}$), also known as LTT, is easier to grow [46].

Lately, new semiconductors have been discovered, which is the case of 2-D SnO. Transistors made with this semiconducting material could lead to computers and smartphones that are more than 100 times faster than regular devices, according to the researchers, and processors will not get as hot as normal computer chips. They also will require much less power to run [96]. $\hbox {Re}_6\hbox {Se}_8\hbox {Cl}_2$ is another novel semiconductor. At first time, it was thought that this new semiconductor was a bad energy conductor. However, the opposite was true. Electrons passed through the material between 100 and 1000 times faster than Si semiconductor [97], which could lead to the development of faster and more efficient electronic devices. However, this new compound is composed of an extremely rare element. Thus, it is quite impossible, at least for now, that this compound semiconductor become a viable commercial competitor to Si semiconductors.

6 Nanostructures

The properties of the materials have changed substantially as the size of the material approaches the nanoscale. The term ’nanomaterial’ is associated with its size, as well as its structure and its structure manipulation at the atomic and molecular levels, which allows the development of structures more applicable to the desired applications [98]. Different materials’ dimensions or shapes substantially impact the nanomaterials’ performance. In several fields, the geometry and the structure of the materials represent the main topic to achieve a better performance [99]. The nanomaterials can be divided into three different groups, according to their size: zero-dimensional (0D), one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) nanostructured materials, being the 2D nanomaterials the exciting classes with promising features [99,100,101]. Among others, nanowires, nanorods, nanotubes, nano-spheres, nano-leave, hierarchical nanostructures, nano-ribbons, nano-shells, nanoplates, nanocages, nanoflowers, nanobuds, nano-chains, nanosheets, nano-rings and nanoparticles [98, 99, 102].

The 2D nanomaterials, due to their large diameter-length ratio, display exceptional properties which are significantly unique compared to their bulky counterpart. Moreover, they present a high charge carrier capability, which is an excellent feature when the application is related to solar devices, and their flexibility is extremely useful in designing flexible electronics and sensors. Finally, these materials present unique optical properties, which include plasmonic effects, light sensitivity, emission, and absorption [99]. These materials are synthesized using two different approaches: top-down and bottom-up methods [99, 101, 102]. The first one is extensively used for synthesizing ultrathin 2D nanomaterials, which starts from the bulk materials, where the layers have been exfoliated in several ways to reach the atomically ultrathin 2D sheets. Other methods to produce the 2D nanomaterials are mechanical milling, in which the bulk of the material is mechanically milled to separate the sheets of the 2D nanomaterials; lithography, including ion and electron beam lithography; laser ablation; and sputtering. Regarding the bottom-up methods, they are usually used to obtain high-quality 2D sheets of materials. This method includes molecular beam epitaxy, chemical vapour deposition, metal-organic vapour deposition, hydrothermal, and solvothermal methods, which are used due to their operational simplicity and ease of control. Another method to produce 2D metal oxide nanostructures is the sol–gel method. Due to its simplicity and cost-effectiveness, it is a practical process that can be handled at low temperatures [99, 101, 102].

The emerging 2D nanomaterials can be classified according to the groups of materials in the periodic table: Group-III, Group-IV, Group-V and Group-VI. Table11 presents a summary of some emerging 2D nanostructures, according to the groups of materials in the periodic table, including their synthesized methods and major applications [98, 99, 101, 102].

Table 11 2D-Nanomaterials’ classification

In the third group, the 2D-monoelemental forms of materials such as boron, known as borophene (2D-B), and gallium, known as gallenene (2D-Ga), have successfully been grown in several experiments. Borophene is known for its metallic characteristics, high anisotropy and excellent flexibility [99, 101]. Gallenene has shown outstanding prospects in applications like 2D metals in sensors, electrical contacts and plasmonics [101]. In the fourth group are presented 2D-monolayer materials known as silicene (2D-Si), germanene (2D-Ge) and graphene. Silicene is a 2D nanomaterial similar to graphene, but it has an exclusive potential for unusual electronic properties. It presents a strong spin-orbit coupling effect. Germanene has a higher mass number than 2D-Si, producing improved spin-orbit coupling. Consequently, it has more topological insulator properties in the ground state than 2D-Si. Thus, it exhibited semiconductor properties, with the methyl termination inducing a direct electronic bandgap, independent of thickness, which can be useful in optoelectronic applications. The 2D carbon layer, called graphene, namely its isolation from graphite shows that the single or few atom-thick layers of materials can be designed to achieve exceptionally high performance of the materials in various fields [99, 101, 102]. Regarding the fifth group, it can be pointed out phosphorene (2D-P), antimonene (2D-Sb) and arsenene (2D-As). The phosphorene is investigated as electrode material to improve the efficiency of energy storage devices due to its large surface area, high theoretical specific heat capacity, and good electrical conductivity. In Group VI, 2D-monolayer selenene (2D-Se) and tellurene (2D-Te) are the main 2D nanostructures. Unlike the referred elements, group VI is characterised by a 3D-block structure containing a 0D-atomic ring or a 1D-atomic chain with 2D-coordination bonds under ambient conditions [99, 101].

Throughout the years, nanoscale technology, starting from the material, faces a huge challenge regarding the development of efficient materials. First, the presence of defects in nanomaterials can affect their performance. The nanomaterials’ synthesis through cost-effective routes is another major challenge. Another challenge is about the particles’ agglomeration [100, 102]. For instance, carbon nanotubes are one of the strongest materials that are known, but the existence of impurities, discontinuous and defects impair the tensile strength of carbon nanotubes. Moreover, regarding the synthesis method, it is a big challenge to achieve chiral selectivity, conductivity, and precisely controlled diameters [102].

7 Semiconductors ageing

Ageing is a major issue for semiconductors that play a crucial part in applications requiring reliability or safety. The semiconductors’ ageing refers to the slow loss of electrical characteristics of the semiconductor device as a result of continuous use or prolonged exposure to various environmental conditions like temperature, irradiation, and electrical stress. An example of this is the gradually reduced responsivity with the increase of laser energy density [103]. Besides that, important parameters of photodetectors are affected by the temperature [104]. Thus, their lifetime is compromised, as their performance deteriorates over the years.

There are several ageing mechanisms, such as bias temperature instability, hot-carrier injection, time-dependent dielectric breakdown, and electromigration [105, 106]. A recent study analyses a high-efficiency ageing test technique that estimates the voltage stress based on the median deterioration of device electrical indicators rather than the substrate current, therefore reducing the test time while increasing its accuracy [105]. The dominant ageing mechanisms are the bias temperature instability and the hot-carrier injection. Both of them translate into lower speed, lower gain, instability and other figures of merit are affected [105, 107]. Several studies show that the degradation of several devices is more than 5% for tests of 10 000 s [105].

As degradation occurs, the reliability of the devices is questioned. It is therefore measured using a curve, called the “bathtub curve”, which is illustrated in Fig. 42.

Fig. 42

Failure rate curve

The failure rate curve is divided into three regions [108]:

1.:

Early failure period: Failure that occurs at an early stage from when the equipment was first used. The early failure rate tends to decrease as time goes by. In the case of semiconductor devices, most early failures are due to defects that form during the production process and defective materials.

2.:

Random failure period: Failure that occur over a long period of time. These types of failures can be due to excessive stress, such as power surges, and software errors.

3.:

Wear-out failure period: Failure that increases as the device’s lifetime approaches to the end. It is influenced by the usage condition

A metric used to measure the useful lifetime of devices is the mean time to failure (MTTF), which is a maintenance metric that measures the average amount of time a non-repairable asset operates before it fails. Usually, semiconductor manufacturers determine a device’s lifetime by an accelerated life test through the Arrhenius Eq. [108].

8 Applications

Semiconductors play a crucial role in several applications across numerous industries due to their unique electrical properties. They are part of several applications in the most diverse fields: healthcare, biotechnology and medicine equipment, defence and aerospace systems, electronics industry, computing and information technology, communication systems, renewable energy, industrial applications, and automotive electronics, among others. They are part of the new era dominated by technology, which brings everyone closer together.

The term ’semiconductor’ is automatically associated with electronics, since it is the basis of modern electronics, which are used in transistors, diodes and integrated circuits fabrication. Throughout the years, the reduction in the size of devices has given rise to microelectronics, which has enabled the development of ever more powerful microchips, present in a wide range of devices that all over the world are used every day. In several fields, electronics are used to incorporate electronic devices like sensors, providing essential data for monitoring, control and automation. A photodetector might act as a sensor since the material properties must change with certain stimuli. The electrical properties presented in table 9 vary according to the conditions to which the photodetector is exposed. Thus, figures of merit vary, which leads to the photodiode’s function being optimised, to monitor certain stimuli, adding more capabilities to optical communication systems [109,110,111,112].

There are many types of sensors: chemical sensors, physical sensors, electric and magnetic field-based sensors, temperature sensors [109,110,111,112,113,114,115,116,117,118]. Temperature sensor constitutes the vast class of commercially available optical sensors. The semiconductors’ properties are extremely influenced by the temperature, in such a way that their bandwidth increase with higher temperatures and decreases with lower temperature values [109, 110]. In biomedical applications, a typically used sensor is the oxygen and carbon dioxide sensor for blood. Measurements of arterial blood gas are frequently performed on critical patients in both the operating rooms and intensive care unit [117]. However, when the topic is semiconductors, a typical sensor that comes to mind is the Hall sensor, whose operating principle is described in the Sect. 2.8. This sensor is commonly used in speed detection, position detection of permanent magnets in brushless electric DC motors [118], and door open-closed, where a magnet is placed on a door frame while the sensor unit is placed on the door. When shut, the door sensor aligns with the magnet and triggers the Hall effect sensor.

Moreover, the concern for the environment and the climate crisis, which is becoming more and more of a reality, has led to renewable energies and the way they are captured becoming more and more efficient. The use of semiconductor devices such as rectifier diodes and power transistors are essential in power electronics systems, used in inverters, converters, electric motors and energy control in electrical networks. Nowadays, due to the climate era we’re going through, it is essential to talk about the light sensor [119], which can be analysed as a photodiode, represented by a current source controlled by the lighting. In photodetectors, only the electrons and holes generated by the photoelectric effect, which reach the contacts without recombining, contribute to the current. A simple process to achieve this is to take advantage of the depletion zone of a p-n junction, which is deeply described in section 4.1. Under illumination, the I(V) relation is described by Eq. 82, where $i_s$ translates the lighting current [5, 36].

The working principle of a solar cell is based on the photoelectric effect. When the depletion zone is under illumination, electron–hole pairs are generated and due to local electrical field forces, holes and electrons go to opposite sides. Thus, a higher potential difference appears on the semiconductor [5]. A solar cell can be electrically described by the equivalent circuit presented in Fig. 43. This device, as many others, uses electrical models to describe its behaviour, and the one represented in the aforementioned figure is the most commonly used. It is known as the 1M5P model, characterised by 5 unknown parameters, including both resistances, which translate the circuit losses. A simpler model does not include both resistances and because of that is known as an ideal model. It is characterised by 3 unknown parameters (1M3P) [5].

Fig. 43

Electrical equivalent model of a solar cell

Throughout the years, other models have been proposed to describe its behaviour. A recent model, known as d1MxP, proved to be an accurate model, fitting the experimental data with extremely low errors. The model is well described and is analysed in the following articles [120,121,122].

In the optical communications field, the appearance of semiconductors and faster electronic devices has led to the emergence of new forms of communication, such as fibers. This was possible due to the parallel development of low-loss fibers, heterojunction lasers and LEDs emitting in the regions of low fiber loss, and sensitive photodetectors [92]. Optical communications begin and end with a transmitter and receiver. In the transmitter, lasers produce light to travel down the fiber encoded with information. In the end, the receiver converts this light to an electronic signal via a photodiode detector. Thin alloy semiconductor films play an important role in both components. Semiconductor LASERs are popular optical communication light sources for high-speed data transmission [36, 123]. There are five basic types of semiconductor lasers used in optical communication: Fabry-Pérot (FP) Lasers, Distributed Feedback (DFB) lasers, which present a multi-GHz modulation speed, as well as the Multiple Quantum Well (MQW) lasers. The External Cavity Diode Lasers (ECDL) have a slow modulation speed (less than GHz) and Vertical Cavity Surface Emitting Lasers (VCSEL) present a few-GHz modulation speed [123]. For optical communication systems at shorter distances (a few km) and a transmission rate ($\le$10 Mbit/s), LEDs can be used, as they are much cheaper than LASERs. The distinguishing features of LEDs designed for optical communication include high modulation rate capability, high radiance, high reliability, and emission wavelengths restricted to the near-infrared spectral regions of low attenuation in fibers [92].

Semiconductors are present in every aspect of modern communications. From wireless devices such as smartphones and Wi-Fi routers to network infrastructure, semiconductors play a central role in data transmission and processing. In data transmission, optical fibers are the main choice for TV and Internet services, as they allow high data transfer rates over long distances and are fundamental for high-speed communication networks. Thus, the use of semiconductor lasers is crucial in the transmission of light signals in optical fibers. Semiconductors are the foundation of all modern electronic devices which use circuitry. These materials were first introduced to computing to solve issues related to vacuum tubes used in analogue computers. The tubes would often leak, and the metals used to transmit electrons within them would frequently burn out, unlike semiconductors which conduct electrons in a completely different way to metals, which means that they avoid combustion.

9 Closing remarks

In this article, the theory of semiconductors is analysed. Firstly, a historical review is presented to get an overview of their growth. Semiconductors are presented in several devices that are used every day, and it is thanks to them that we now have increasingly advanced technologies that allow us to hold the world in our hands.

From optical communications to renewable energies, semiconductors are used in the most diverse fields. In this article, photon detectors are deeply analysed and since the fight against climate change is a constant concern in this era, the devices most commonly used for energy conversion are analysed in detail, as is the case with junction photodiodes. Its working principle as well as its mathematical approach is presented.

In this article, the most commonly used semiconductor materials, as well as their characteristics, advantages, and disadvantages over others are presented. New semiconductors are also presented which have begun to arouse interest in the scientific community and which seem to have promising characteristics for various applications.

Finally, several applications are examined and are divided into three main groups: sensors, solar cells, and optical communications. In what concern the sensors, they are used in the most diverse areas of application, from medicine to mechanics. Semiconductor materials make possible many of today’s technological advances, from handheld electronics to solar cells and even electric vehicles. New semiconductors, namely the ones with wide bandgap, have opened new opportunities in ultra-high power electronics applications for several applications. The research in this field progresses day by day, intending to develop stronger, faster and more efficient materials. Nevertheless, this industry faces the challenge and opportunities of increasing product demand due to the growth of artificial intelligence, the Internet of Things and the technology market. The challenge will be further complicated by the ongoing international trade disputes, which may drive up semiconductor materials’ costs and interfere with global collaboration within the industry.

Thus, this review aims to clarify readers about the most important milestones in semiconductor history as well as its theory, to catch the attention of new readers and provide a basis for those who want to dedicate themselves to this area of interest.