Concept of Band Theory in Semiconductors

This article introduces the concept of energy bands in semiconductors. In addition to briefly reviewing the periodic table of elements, it also covers "Extrinsic Semiconductors," donors, and acceptors. Furthermore, it provides a brief introduction to the applications of N-type and P-type semiconductors.

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The band theory of semiconductors forms the basis for understanding their electronic properties. In solid-state physics, bands describe the range of energies that electrons in a solid may occupy. For semiconductors, the focus is primarily on two bands: the valence band and the conduction band.

• The valence band is the highest energy band in a solid that is fully occupied by electrons. These electrons participate in forming chemical bonds and are the main contributors to the material's chemical properties.
• The conduction band is the band above the valence band, and in a pure, undoped semiconductor, this band is typically empty. Electrons in the conduction band are free to move, corresponding to the material's conductivity.
• The bandgap is the energy difference between the bottom of the conduction band and the top of the valence band. This difference determines the minimum energy required for an electron to jump from the valence band to the conduction band, thereby influencing the material's electrical and optical properties. The size of the bandgap is an important characteristic of semiconductor materials (such as silicon, germanium, gallium arsenide, etc.).
• Materials with small bandgaps (< 1 eV) are easily excited at room temperature, allowing electrons to transition to the conduction band, resulting in higher conductivity.
• Materials with large bandgaps (> 2 eV) require higher energy to excite electrons, making them almost non-conductive at room temperature. However, they can conduct under high temperature or illumination, or they can be used for specific optoelectronic applications.

The bandgap of semiconductors places them between conductors (with no or very small bandgaps) and insulators (with very large bandgaps). This unique electronic structure makes them crucial in the fields of electronics and optoelectronics. By controlling doping and fabrication processes, semiconductor conductivity can be finely tuned, enabling the production of various electronic and optoelectronic devices. Generally, semiconductors without impurities are called "intrinsic semiconductors," while those with impurities are called "extrinsic semiconductors." Before explaining extrinsic semiconductors, let's review the periodic table.

Periodic Table

(From Wikipedia)

The periodic table is a way of arranging chemical elements based on their atomic number (the number of protons in the atomic nucleus) and their chemical properties, demonstrating the periodicity of elements. The history of the periodic table can be traced back to the mid-19th century when scientists began attempting to find relationships and patterns among elements.

• John Wolfgang Döbereiner: In 1829, Döbereiner proposed the "law of triads," an early observation of periodicity among elements, indicating that certain elements could be grouped into sets of three, where the elements within each triad exhibited similar properties, and the average atomic mass of the three elements was close to that of one element.
• Dmitri Mendeleev: In 1869, Mendeleev published his periodic table of elements, then known as the "periodic law of elements." He arranged elements based on their atomic mass and predicted the existence and properties of some undiscovered elements, such as germanium (Ge), which was later found to match Mendeleev's predictions.
• Lothar Meyer: Meyer independently discovered the periodicity of elements almost simultaneously with Mendeleev, but his work was published slightly later. While his work was very similar to Mendeleev's, Mendeleev gained more fame due to his predictive abilities.

Over time and with advancements in technology, more elements were discovered and confirmed, leading to multiple updates and revisions of the periodic table. Particularly in the 20th century, with the development of atomic structure theory, scientists began to use atomic number rather than atomic mass as the basis for arranging elements.

• Henry Moseley: In 1913, Moseley confirmed through X-ray spectroscopy experiments that atomic number was a more precise method for distinguishing between different elements. Moseley's work provided a solid scientific foundation for the periodic table, establishing the modern form of organizing elements by atomic number.

The periodic table is a grid that arranges chemical elements according to their atomic number (the number of protons in the atomic nucleus), while also reflecting the electron structure of the elements, i.e., how electrons are distributed around the atom. Here are some basic concepts:

Structure of the periodic table

• Period: The horizontal rows of the periodic table are called "periods," which represent the energy levels or shells occupied by electrons in the atom. As the period number increases, it indicates an increase in the size and number of energy levels of the atom.
• Group: The vertical columns of the periodic table are called "groups" or "families," and elements with the same group number have similar chemical properties. This is because they have the same number of valence electrons, i.e., electrons located in the outermost energy level.

Arrangement of electrons

• Valence electrons: Electrons located in the outermost shell of the atom, which determine the chemical properties of the element and how it bonds with other atoms. The group number of elements roughly corresponds to the number of valence electrons.
• Electron shells: Electrons are distributed around the atomic nucleus in different energy levels, called "shells," according to specific rules. Starting from the core outward, these shells are sequentially named as K, L, M shells, and so on.

Classification of elements

• Main group elements: Including elements from Group IA to VIIIA, the valence electrons of these elements mainly fill in the s and p orbitals.
• Transition metals: Located in the middle portion of the periodic table, the valence electrons of these elements fill in the d orbitals.
• Noble gases: Located in Group VIIIA (or Group 18), the outer electron shell of these elements is completely filled, exhibiting very low chemical reactivity.

Through the periodic table, we can understand many important characteristics of elements, such as how many bonds they can form, the chemical properties they may possess, and how they react with other elements. The periodic table is not only a fundamental tool in chemistry but also a window to understanding the material world.

The chemical stability of an element is related to whether its valence electron shell satisfies a stable electron configuration, such as the electron configuration of noble gases. Generally, when an element's outermost electron shell is completely filled, the element tends to be more stable. This is because a fully filled outer electron shell provides the lowest energy state, leading to low reactivity of the element in chemical reactions.

Stable Electron Configurations:

• Noble Gas Configuration: Noble gases like helium, neon, argon, etc., have completely filled outer electron shells, making them highly stable. These elements are nearly unreactive under normal conditions due to their valence electron shells achieving the most stable configuration (e.g., helium has 2 valence electrons, while other noble gases have 8 valence electrons).
• Octet Rule: Many elements tend to acquire, lose, or share electrons to achieve the closest noble gas electron configuration, a principle known as the "octet rule." Having 8 valence electrons generally results in a lower energy and higher stability state.

Valence Electrons and Stability:

• Less than 8 Valence Electrons: When the number of valence electrons in an element is less than 8, they typically undergo chemical reactions to achieve a stable configuration satisfying the octet rule, either by donating, accepting, or sharing electrons.
• Elements with Octet Rule Satisfaction: These elements are generally chemically stable and less prone to participating in chemical reactions.

Therefore, the ability of an element to achieve a stable electron configuration (often close to or equal to the octet configuration) determines its stability. The stability of elements is more closely related to how they achieve this stable state through chemical reactions.

Electron Shells

The electron shells, also known as energy levels, are a series of layers within the electron cloud surrounding the atomic nucleus, where electrons are distributed according to specific rules and energy levels. These shells are categorized based on their distance from the nucleus as K, L, M, N, and so on, with the K shell closest to the nucleus, followed by the L shell, and so forth. Each shell can accommodate a certain number of electrons, determined by the principles of quantum mechanics. Understanding these shells and the distribution of electrons is crucial for comprehending the chemical properties of elements and how they react with other elements.

K Shell

• The innermost electron shell, closest to the atomic nucleus.
• Can accommodate a maximum of 2 electrons.
• Since it is the innermost of all electron shells, electrons in the K shell have the lowest energy state.

L Shell

• The second electron shell.
• Can accommodate a maximum of 8 electrons.
• Electrons in the L shell have higher energy compared to those in the K shell, as they are farther away from the nucleus.

M Shell

• The third electron shell.
• In the absence of electrons in the d orbital, theoretically, it can accommodate a maximum of 18 electrons.
• Electrons in the M shell have higher energy than those in the L shell as they are located further outward.

Electron Accommodation Rules

The distribution of electrons in these shells follows specific rules, the most famous of which are the Pauli Exclusion Principle and the Hund's Rule. The Pauli Exclusion Principle states that no two electrons in an atom can have identical sets of four quantum numbers, while Hund's Rule guides how electrons are distributed among different orbitals within the same energy level.

General Formula for Electron Accommodation

The maximum number of electrons a shell can accommodate can be calculated using the formula 2(2n)^2, where n is the shell's number (K=1, L=2, M=3, and so on). Therefore:

• K Shell (n=1): Maximum of 2 electrons.
• L Shell (n=2): Maximum of 8 electrons.
• M Shell (n=3): Maximum of 18 electrons (considering electrons in the d orbital).

Orbitals

In quantum mechanics, electrons in atoms are believed to exist not in fixed paths around the atomic nucleus but rather in probability clouds known as "orbitals." These orbitals describe the probability of finding an electron at a specific point around the atom at a certain energy level. Orbitals are classified into several types based on their shape and energy, primarily including s, p, d (and higher-level orbitals like f, g, etc., although they are less common in chemistry). 

The existence and structure of these orbitals are crucial for understanding the chemical behavior of elements. They determine how elements form chemical bonds, their electron configurations, and their chemical and physical properties. For example, most of an element's chemical properties can be explained by the configuration of its valence electrons (those in the outermost orbital), which are primarily located in s and p orbitals.

s Orbital

• Shape: Spherical.
• Number of s orbitals per energy level: Each principal energy level (n) has 1 s orbital.
• Electron capacity: Each s orbital can accommodate a maximum of 2 electrons.
• s orbitals are the lowest-energy orbitals on each principal energy level.

p Orbital

• Shape: Dumbbell or figure-eight.
• Number of p orbitals per energy level: Starting from n=2, each principal energy level has 3 mutually perpendicular p orbitals (px, py, pz).
• Electron capacity: Each p orbital can accommodate a maximum of 2 electrons, so a set of p orbitals (three in total) can accommodate a maximum of 6 electrons.
• p orbitals have higher energy compared to s orbitals on the same principal energy level.

d Orbital

• Shape: More complex, including cloverleaf shapes and others.
• Number of d orbitals per energy level: Starting from n=3, each principal energy level has 5 d orbitals.
• Electron capacity: Each d orbital can accommodate a maximum of 2 electrons, so a set of d orbitals (five in total) can accommodate a maximum of 10 electrons.
• d orbitals have higher energy compared to both s and p orbitals on the same principal energy level.

Donors

The addition of phosphorus (P), arsenic (As), and antimony (Sb) to silicon (Si) semiconductors increases the conductivity by introducing additional electrons without generating electron holes in the valence band. This is primarily due to their electronic and band structure relationships rather than just atomic arrangement. These elements are called donors because they can donate extra electrons to the semiconductor. Here are some key points to explain this phenomenon:

• Atomic Structure: Phosphorus, arsenic, and antimony belong to Group V of the periodic table, meaning they have 5 electrons in their outermost valence electron shell. In contrast, silicon is a Group IV element with 4 valence electrons. When these Group V elements are doped into silicon, the extra electron only requires a small amount of energy to be released from the atom, becoming a free electron, thereby increasing the material's conductivity.
• Band Structure: In solid-state physics, the energy states of materials are divided into bands, where the valence band is the highest-energy band containing valence electrons, and the conduction band is a higher-energy band where electrons can move freely. When Group V elements are doped into silicon, the extra electron requires very little energy, making it easily excited into the conduction band, becoming a free electron that enhances conductivity. The energy levels of these donor impurities lie between the valence and conduction bands but closer to the conduction band.
• Hole Generation: Holes are positively charged carriers formed due to electron deficiencies primarily in the valence band. When electrons are excited from the valence band to the conduction band, they leave behind a hole in the valence band. For Group V dopants, since they provide additional electrons rather than removing electrons from the valence band, they do not directly generate holes in the valence band. In contrast, Group III elements (such as boron, B) when doped into silicon, lack an electron, creating holes in the valence band, resulting in behavior similar to P-type semiconductors.

In general, the reason why phosphorus, arsenic, and antimony increase the conductivity of electrons without generating holes in the valence band is because their electron structure and band structure allow them to donate additional electrons to the conduction band, thereby enhancing the N-type conductivity of the semiconductor. But can nitrogen (N), bismuth (Bi), and moscovium (Mc) also serve as donors, being members of Group V?

• Nitrogen (N): Nitrogen is not typically considered a donor in the traditional sense because its behavior in the crystal differs significantly from elements like phosphorus, arsenic, or antimony, despite having the same number of valence electrons. Nitrogen is more commonly found in specific material manufacturing processes, such as gallium nitride (GaN), where it does not directly contribute additional free electrons but forms different types of semiconductor materials.
• Bismuth (Bi): While bismuth can theoretically act as a donor, its large atomic radius and unique electron structure may limit its application as a donor in conventional silicon-based semiconductors. However, in certain specialized applications such as high-speed or high-frequency electronic devices, exploring the possibility of using bismuth as a donor may be warranted.
• Moscovium (Mc): Moscovium is a synthetic radioactive element with an extremely short half-life, making it impractical for consideration as a donor in practical semiconductor processes. Its chemical properties are difficult to study in detail due to its instability and rarity, rendering its application in practical semiconductor technology unfeasible.

In summary, while nitrogen, bismuth, and moscovium belong to Group V of the periodic table, their practicality as donors differs significantly from elements like phosphorus, arsenic, and antimony due to their unique physical and chemical properties. Particularly for moscovium, its radioactive and highly unstable nature makes it unsuitable for use as a donor in semiconductor materials.

Acceptor

Acceptors in semiconductor physics refer to impurity atoms capable of accepting free electrons, thus creating holes in the valence band. These holes can act as positively charged carriers, thereby increasing the p-type conductivity of the semiconductor. Acceptor atoms typically have one fewer valence electron than the host semiconductor material. In IV-group semiconductors such as silicon (Si) or germanium (Ge), Group III elements like boron (B), gallium (Ga), and indium (In) are common acceptor materials.

Boron is one of the most commonly used acceptor materials, especially in silicon semiconductor technology. Boron has several important characteristics and roles:

• Increased p-type conductivity: After boron atoms enter the silicon lattice, they create an electron vacancy in the lattice since boron has only 3 valence electrons, one less than silicon. When a boron atom replaces a silicon atom, it creates an electron vacancy, or a hole, in the lattice. This hole can accept electrons from neighboring silicon atoms, seemingly allowing the hole to move. This movement of holes provides an additional conduction channel, increasing the material's p-type conductivity.
• Low-temperature activation: Boron, as an acceptor material, has a relatively low activation energy, meaning that electrons can be excited from the valence band to the conduction band at lower temperatures, forming holes. This allows boron-doped semiconductors to operate over a wider temperature range.
• Flexibility in manufacturing techniques: Boron can be incorporated into semiconductors through various methods, including ion implantation, vapor deposition, and solution growth. This flexibility makes boron a highly practical acceptor material in semiconductor processes across different types and manufacturing stages.

Boron-doped silicon chips find wide application in various semiconductor devices, including diodes, field-effect transistors (FETs), and bipolar junction transistors (BJTs). It is a key material for realizing p-n junctions, p-type metal-oxide-semiconductor (PMOS) transistors, and complementary metal-oxide-semiconductor (CMOS) technology.

Aluminum (Al), gallium (Ga), and indium (In) are common acceptor dopants in semiconductor technology. They effectively generate holes in silicon or other semiconductor materials, thereby forming p-type semiconductors.

Thallium (Tl) and recently discovered nihonium (Nh) theoretically can also serve as acceptor dopants as they belong to Group III elements. However, due to factors such as the chemical properties, toxicity (especially in the case of thallium), and stability of thallium and nihonium, their practical application in semiconductor processes may be limited.

Especially for newly discovered or less common elements like nihonium (Nh), being a synthetically produced superheavy element, its extremely short lifespan makes it unsuitable as a dopant for industrial semiconductor applications. Research on these elements primarily focuses on fundamental science rather than practical semiconductor manufacturing.

In theory, all Group III elements can serve as acceptor dopants, but in practical applications, aluminum (Al), gallium (Ga), and indium (In) are primarily used due to their proven effectiveness and reliability in manufacturing p-type semiconductor materials.

Applications of N-Type and P-Type Semiconductors

N-type and P-type semiconductors are two fundamental types in semiconductor technology, playing crucial roles in many electronic devices and systems. The main difference between these two types of semiconductors lies in their carrier types: P-type semiconductors mainly conduct through holes (positive charge carriers), while N-type semiconductors mainly conduct through electrons (negative charge carriers). These characteristics give them unique applicability and advantages in different application areas.

Applications of N-Type Semiconductors:

• Digital and analog integrated circuits: N-type semiconductors are widely used in the manufacture of various digital and analog integrated circuits, where NMOS transistors are indispensable basic components in modern electronic devices.
• High-speed electronic devices: Due to the higher mobility of electrons in N-type semiconductors compared to the mobility of holes in P-type semiconductors, they are particularly suitable for applications requiring high-speed switching and high-frequency communication.
• LEDs and laser diodes: In certain types of optoelectronic devices, such as blue LEDs and laser diodes, N-type semiconductors are combined with P-type semiconductors to generate effective electron-hole recombination, thereby emitting light.

Applications of P-Type Semiconductors:

• Solar cells: P-type semiconductors serve as commonly used substrate materials in the manufacture of solar cells, forming P-N junctions with N-type semiconductors to efficiently convert light energy into electrical energy.
• Power semiconductor devices: In some power semiconductor devices, such as power diodes and thyristors, P-type semiconductors are used to control the flow of current, enhancing the efficiency and stability of the devices.
• Complementary metal-oxide-semiconductor (CMOS) technology: In CMOS technology, P-type semiconductors are used to fabricate PMOS transistors, combined with NMOS transistors fabricated from N-type semiconductors, to achieve high-performance and low-power electronic devices.

Integrated Applications:

• P-N junctions: The most famous example of combined applications of P-type and N-type semiconductors is the P-N junction, which forms the basis of many semiconductor devices, including diodes, transistors, and photodiodes. P-N junctions, by combining P-type and N-type semiconductors, can control the direction of current flow and perform functions such as rectification, amplification, and conversion of electrical signals.
• P-type and N-type semiconductors, through their unique electrical properties, play crucial roles in modern electronics and optoelectronics, from basic electronic switches to complex computing and communication systems. The selection and application of these two materials are essential parts of electronic engineering and materials science research.

Reference:

• 半導體元件物理與製程

02/27/2024

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