Intrinsic Semiconductor

PHYSICS SEMICONDUCTOR ELECTRONICS INTRINSIC SEMICONDUCTOR INTRINSIC SEMICONDUCTORS: A semiconductor that is devoid of impurities is referred to as an intrinsic semiconductor. Ideally, an intrinsic semiconductor crystal should exclusively consist of atoms of that specific semiconductor. However, achieving such absolute purity in practice is impractical. Nevertheless, if the impurity content is less than 1 in 108 parts of the semiconductor, it can be considered as intrinsic. To illustrate the properties of intrinsic semiconductors, we will use the examples of silicon and germanium. Both silicon and germanium belong to Group IV of the periodic table and are tetravalent. Their electronic configurations are as follows:  Silicon (Si) with an atomic number of 14: 𝑆𝑖(14) = 1𝑠 2 2𝑠 2 2𝑝6 3𝑠 2 3𝑝2  Germanium (Ge) with an atomic number of 32: 𝐺𝑒 (32) = 1𝑠 2 2𝑠 2 2𝑝6 3𝑠 2 3𝑝6 3𝑑10 4𝑠 2 4𝑝2 These electronic configurations depict the distribution of electrons in the respective energy levels and orbitals of silicon and germanium atoms. Both silicon and germanium exhibit a crystalline structure wherein each atom within the crystal is situated inside a tetrahedron formed by the four closest atoms to it. The figure illustrates one of these tetrahedral units. In this arrangement, each atom shares its four valence electrons with its immediate neighbors on a one-to-one basis, resulting in each atom participating in four covalent bonds. For clarity, a two-dimensional representation of the crystal structure for germanium is presented in the figure, which can also be applied to silicon. This representation focuses specifically on the covalent bonds within the structure. 1 PHYSICS At a temperature of 0 Kelvin (0 K), all the valence electrons are actively engaged in bonding, rendering the crystal a perfect insulator. In this state, there are no free electrons available for conduction. However, as the temperature increases, some valence electrons acquire enough energy to break away from the bonds, allowing them to move randomly within the crystal lattice. When an external electric field is applied, these electrons experience a drift, leading to the conduction of electricity. In essence, the conductivity of the crystal increases with higher temperatures as some electrons gain the energy needed to participate in electrical conduction. When an electron escapes from a band within a semiconductor, it results in the creation of a vacancy in the crystal lattice, known as a "hole." This vacancy, caused by the absence of an electron, essentially represents a positive charge of the same magnitude as that of an electron. As elaborated later, holes play a significant role in the conduction of semiconductors. . When a covalent bond is disrupted, it leads to the generation of an electron-hole pair. At room temperature (300 K), numerous electron-hole pairs are present in the crystal lattice. The process of electron-hole pair generation is elucidated in the figure. 2 PHYSICS Consider a scenario where, due to thermal energy, an electron is liberated from the covalent bond at site A, creating a hole at this location. Subsequently, an electron from the covalent bond of a neighboring atom at site B may jump to the vacant site A, completing the bond at A but leaving a hole at B. In this process, the energy involved is relatively small compared to what is required for the generation of an electron-hole pair. This is because the electron is transitioning from one bond to another, and all electrons in the bonding state have, on average, the same energy. As depicted in the figure, when an electron jumps from C to B, a hole is created at C, and this process continues. Effectively, such a vacancy or hole can be considered mobile. Consequently, in a semiconductor, both electrons and holes function as charge carriers, actively contributing to electric conduction. The intrinsic semiconductor exhibits equality in the number of electrons and holes generated through thermal processes. If ne represents the electron concentration and 𝑛ℎ represents the whole concentration, then 𝑛𝑖 (intrinsic charge carriers' concentration) is given by: 𝑛𝑖 = 𝑛𝑒 = 𝑛ℎ Furthermore, the product of the electron and hole concentrations,𝑛𝑒 𝑛ℎ is equal to the square of the intrinsic charge carriers' concentration (𝑛𝑖 ): 𝑛𝑒 𝑛ℎ = 𝑛𝑖 2 Key points to note about intrinsic semiconductors: 1. A pure semiconductor is termed an intrinsic semiconductor, characterized by thermally generated current carriers. 2. These semiconductors have four electrons in the outermost orbit of the atom, and atoms are interconnected by covalent bonds. 3. Both free electrons and holes function as charge carriers, with 𝑛𝑒 (in the conduction band) equal to 𝑛ℎ (in the valence band). 4. The drift velocity of electrons (ve ) is greater than that of holes (vh ). 3 PHYSICS 5. The Fermi energy level in pure semiconductors lies at the center of the conduction band and valence band. 6. Impurity in a pure semiconductor must be less than 1 in 108 parts of the semiconductor. 7. In an intrinsic semiconductor, 𝑛𝑒 (0) = 𝑛ℎ (0) = 𝑛𝑖 , where ne(0) is the electron density in the conduction band, 𝑛ℎ (0) is the hole density in the valence band, and 𝑛𝑖 is the density of intrinsic carriers. 8. The fraction of electrons from the valence band present in the conduction band is given by 𝐸𝑔 𝑓 ∝ 𝑒 − 𝑘𝑇 , where 𝐸𝑔 is the Fermi energy, k is Boltzmann's constant, and T is the absolute temperature. 9. Due to the low number of charge carriers at room temperature, intrinsic semiconductors exhibit low conductivity and limited practical applications. 10. The number of electrons reaching from the valence band to the conduction band, n, is given by 3 𝐸𝑔 𝑛 = 𝐴𝑇 2 𝑒 −2𝑘𝑇 , where A is a positive constant. 11. The net charge of a pure semiconductor is zero. (a) Electrical Conductivity of Intrinsic Semiconductor: At room temperature, a semiconductor consists of electrons in the conduction band and holes in the valence band. When subjected to an external electric field, electrons move in the opposite direction of the field, and holes move in the direction of the field. This motion contributes to the total current, which is the sum of the electron and hole currents. The total current (i) is expressed as the sum of the electron current (ie ) and the whole current(ih ): i = i𝑒 + iℎ This implies that the electrical conductivity of an intrinsic semiconductor is influenced by both the movement of electrons and holes in response to an applied electric field. The behavior of electrons and holes contributes to the overall conductivity of the semiconductor, forming an essential aspect of its electrical properties. Hole Current (𝐢𝐡 ): The movement of electrons within the covalent bond sites in the valence band results in the motion of holes in the opposite direction. When electrons jump from one position to another within the valence band, the whole moves in the opposite direction to the electron jump. It is essential to note that the original electron that was set free is not directly involved in the motion of the hole. 4 PHYSICS The concept of whole motion is introduced as a convenient way to describe the actual movement of the bonded electrons within the valence band. The motion of holes serves as a useful abstraction to understand the behavior of electrons in the valence band of a semiconductor. Electron Current (𝐢𝒆 ): Conduction electrons, which are free electrons, move independently under the influence of an electric field, contributing to the electron current. It is important to highlight that, in addition to the generation of conduction electrons and holes, there is a simultaneous process of recombination. Recombination involves the electrons combining with the holes, and at equilibrium, the rate of generation is balanced by the rate of recombination of charge carriers. Recombination typically occurs when an electron collides with a hole within the semiconductor material. This dynamic process of generation and recombination is crucial for maintaining equilibrium in the semiconductor, ensuring that the number of charge carriers remains constant over time. Let's consider a semiconductor block with a length 𝜆, a cross-sectional area A, and electron concentration 𝑛𝑒 and whole concentration 𝑛ℎ . When an electric field 𝐸 is applied across the ends of the semiconductor, the electric field is given by: E= V Under this electric field, electrons and holes drift in opposite directions, resulting in currents ie and ih , respectively, in the direction of the electric field. The total current flowing through the semiconductor is the sum of these individual currents: i = ie + ih If 𝑣𝑒 is the drift velocity of electrons in the conduction band and 𝑣ℎ is the drift velocity of holes in the valence band, then: ie = ne eAve ih = nh eAvh 5 PHYSICS Here, 𝑒 represents the magnitude of the electron charge. Therefore, the total current (𝑖) can be expressed as: 𝑖 = 𝑒𝐴(𝑛𝑒 𝑣𝑒 + 𝑛ℎ 𝑣ℎ ) This equation illustrates the combined effect of electron and hole drift velocities in response to the applied electric field across the semiconductor. Consider the resistance R of the semiconductor block and ρ as the resistivity of the material, then 𝑅𝐴 𝜌= 𝜆 (equation iii). Dividing equation (i) by equation (iii) yields: 𝑉 =𝑅 𝑖 𝑖 Substituting the value of 𝐴 from equation (ii) into this equation, we obtain: 𝑅= 𝑒 𝜎 Now, let's introduce the mobility 𝜇, defined as the drift velocity per unit field and expressed in meter 2 /(volt − second). The mobilities of electrons and holes are given by: 𝜇𝑒 = 𝑣𝑒 𝐸 And 𝜇ℎ = 𝑣ℎ 𝐸 Introducing 𝜇𝑒 and 𝜇ℎ into equation (iv), we get: 𝜎 = 𝑒(𝑛𝑒𝜇𝑒 + 𝑛ℎ 𝜇ℎ ) The electrical conductivity 𝜎 is the reciprocal of resistivity ρ. Therefore, the electrical conductivity of the semiconductor is given by: 𝜎 = 𝑒(𝑛𝑒𝜇𝑒 + 𝑛ℎ 𝜇ℎ ) Simplifying further, we find: 𝜎 = 𝑒𝑛𝑖 (𝜇𝑒 + 𝜇ℎ ) Where 𝑛𝑖 is the intrinsic carrier concentration, and in intrinsic semiconductors, 𝑛𝑒 = 𝑛ℎ = 𝑛𝑖 . This expression reveals that the electrical conductivity of a semiconductor relies on the concentrations (number densities) of electrons and holes, as well as their respective mobilities. Notably, the electron mobility is greater than the hole mobility. As the temperature increases, both electron (𝑛𝑒 ) and hole (𝑛ℎ ) concentrations rise due to the breakage of more covalent bonds. 6 PHYSICS Although the mobilities (μ𝑒 ) and (𝜇ℎ ) slightly decrease with an increase in temperature, this decrease is counteracted by the substantial increase in 𝑛𝑒 and 𝑛ℎ . Consequently, the conductivity of a semiconductor experiences an increase, or conversely, the resistivity decreases with a rise in temperature. 7