Module 4 (Semiconductors & Dielectric Materials) PDF

Module 4 Semiconductors & dielectric materials Semiconductors Fermi level in an intrinsic semiconductor For a semiconductor, the energy gap is much less than that for insulators. It is the energy difference between the bottom of the conduction band and the top of the valence band. By convention, the...

Module 4 Semiconductors & dielectric materials Semiconductors Fermi level in an intrinsic semiconductor For a semiconductor, the energy gap is much less than that for insulators. It is the energy difference between the bottom of the conduction band and the top of the valence band. By convention, the energy at the top of the valence band is taken as zero for reference. At T = 0(K), all the energy levels in the valence band are filled, and all the energy level in the conduction band are completely empty. But at ordinary temperatures, such as room temperature, some of the electrons at the top of the valence band can jump the energy gap due to thermal excitation and occupy some energy levels at the bottom part of the conduction band. They return soon to the energy levels which they had left vacant in the valence band. This process of excitation and deexcitation continues, and the electrons involved become conduction electrons. Based upon this picture, one can say that conduction electrons are distributed between the energy levels in the bottom part of the conduction band, and the top portion of the valence band. Due to this, the average energy of the conduction electrons will be almost equal to Eg/2. Thus, the Fermi level lies in the mid-part of the forbidden gap for an intrinsic semiconductor. Fermi level in an extrinsic semiconductor Case of n-type material: In an n-type material, there are electrons which are set free for the movement in the material by the donor atoms. They are relatively free compared to the rest of the electrons in the material, and hence possess more energy. As a result, the energy levels for those electrons will be elevated to a position much higher than the valence band and lie in the band gap very close to the conduction band in the band diagram as shown in Figure. They are called donor levels. 1 The energy difference between the top of the valence band and the donor levels is denoted as 𝐸𝑑. At a low temperature, the thermal energy will be less and hence cannot elevate electrons from valence band to conduction band. But the same energy will be enough to transfer electrons from the donor level to the conduction band, since 𝐸𝑔 − 𝐸𝑑 is small. These are the ones which are mainly responsible for conduction in an n-type material. Since the energy of these electrons is purely due to the thermal energy which is of the order of 𝐸𝑔 − 𝐸𝑑 , they occupy energy levels very close to the bottom of the conduction band after excitation. Therefore, the conduction electrons get distributed between the energy levels in the bottom of the conduction band, and the donor levels. Thus, Fermi level in an n-type material at low temperatures will be in the forbidden band, at the level 1 2 (𝐸𝑔 − 𝐸𝑑 ) below the bottom of the conduction band. Case of p-type material: In case of p-type material, the acceptor atoms give rise to holes that are relatively free compared to those which are in the conduction band. Hence the holes that are due to acceptor atoms possess higher energy than those in the conduction band since the energy for holes increases in the downward direction in the band diagram. They occupy energy levels in the band gap close to the valence band as shown in Figure. These energy levels are referred to as acceptor levels. The energy difference between the acceptor levels and the top of the valence band is denoted by 𝐸𝑎. Since, 𝐸𝑎 is small, very small amount of thermal energy suffices to transfer the holes from the acceptor levels into the valence band. At low temperatures the holes which participate in the conduction, undergo excitation and de- excitation between the acceptor levels and the energy levels in the top position of the valence band, i.e., through an energy difference of 𝐸𝑎. 2  1 As a result, the average energy of the holes participating in conduction becomes equal to 2 𝐸𝑎. Thus, the Fermi level in a p-type material at low temperature will be in the forbidden band at the level 1 𝐸 from the top of the valence band. 2 𝑎 Conductivity of semiconducting materials In a semiconductor, the current is due to free electrons as well as holes. Hence, current due to electrons is Ie = Ne e a ve …………….(1) where ve is the drift velocity of electrons, a is the area of cross section of the material, Ne is the number density of electrons, e is the charge due to electrons. Similarly, the current due to holes is: Ih = Nh e a vh ………………. (2) where vh is the drift velocity of holes and Nh is the number density of holes, e is the charge due to holes. The total current is I = Ie + Ih ………………(3) Therefore I = e a (Neve + Nhvh) …………… (4) The current density is J = I / a …………….. (5) Therefore J = e (Neve + Nhvh) ……………… (6) We Know that J = σE Therefore σE = e (Neve + Nhvh) ……………. (7) σ = e Ne (ve / E) + Nh (vh / E) ……………………... (8) Mobility of electrons is µe = (ve / E) and mobility of holes is µh = (vh /E) Therefore σ = e (Neµe + Nhµh) ……………… (9) For intrinsic semiconductors Ne = Nh = ni where ni is called the density of intrinsic charge carriers. Therefore σ = ni e (µe + µh) ……….. (10) For n type of semiconductors, Ne >> Nh, Therefore, from equation (9), σ ≈ Ne e µe ……. (11) As each donor atom contributes one free electron, Ne is also the density of donor impurity atoms. For p type semiconductors, Nh >> Ne Therefore σ ≈ Nh e µh ………………………(12) where Nh is the density of holes which is same as the density of acceptor impurity atom Hall effect When a material in which a current flow is subjected to a magnetic field at right angles to the direction of current flow, an electric field is induced across the material in a direction perpendicular to both the direction of the magnetic field, and the direction of current flow. This phenomenon is called Hall effect. Charge density & Hall coefficient 3 Consider a rectangular slab of a semiconductor material in which current I is flowing in the positive X direction. Let the semiconducting material be of n type which means that the charge carriers are electrons. Let a magnetic field B be applied along Z direction and hence the electron experiences the Lorentz force FL given by FL = −Bev …………………………………………………... (1) where e is the magnitude of charge on the electron, v is the drift velocity. Applying Fleming’s left-hand rule, we see that the force is exerted on the electrons in the negative y direction and hence the electrons are deflected downwards. Thus, the density of electrons increases in the lower end of the material and the bottom edge becomes more negatively charged. On the other hand, the loss of electrons from the upper end causes the top edge of the material to become positively charged. Hence, a potential VH called Hall voltage appears between the upper and lower surfaces of the semiconductor material which establishes an electric field EH called as Hall field across the conductor in the negative Y direction. The field EH exerts an upward force FH on the electrons given by FH = −e EH …………………………………. (2) The deflection of electrons continues in the downward direction due to the Lorentz force F L and thus contributes to the growth of Hall field. As a result, the force FH which acts on the electron in the upward direction also increases. These two opposing forces reach an equilibrium at which stage. FL = FH Therefore using equations (1) and (2), the above equation becomes −Bev = −eEH Therefore EH = Bv ……………………… (3) 𝑉𝐻 If d is the distance between the upper and lower surfaces of the slab, then 𝐸𝐻 = 𝑑 𝑉𝐻 = 𝐸𝐻 𝑑 = 𝐵𝑣𝑑 ………………….. (4) Let ω be the thickness of the material in the Z direction. Therefore, its area of cross section normal to the direction of I is = ωd. 𝐼 Therefore, the current density 𝐽 = ωd W.K.T. J = nev = ρv …………………. (5) 4 where n is the charge carrier concentration, and ρ is the charge density. 𝐼 Therefore 𝜌𝑣 = 𝜔𝑑 𝐼 Hence 𝑣 = 𝜌𝜔𝑑 ………………… (6) Substituting for v from equation (6), then equation (4) becomes 𝐵𝐼 𝑉𝐻 = 𝜌𝜔 ……………………….. (7) 𝐵𝐼 𝜌 =𝑉 …………………………… (8) 𝐻𝜔 Thus, by measuring VH, I and ω and by knowing B, the charge density ρ can be determined. The polarity of Hall voltage developed at the top and bottom edges of the specimen can be identified using probes. For the set up described in the figure, if the top edge acquires positive polarity for Hall voltage, then the charge carriers must be electrons which means the semiconducting is of n-type. On the same reasoning, for a material of p-type, the polarity at the tope edge will be negative. Hall coefficient RH For a given semiconductor, the Hall field EH depends upon the current density J and the applied field B. EH ∝ JB or EH = RH J B where RH is the Hall coefficient From the above equation, 𝐸𝐻 𝑅𝐻 = …………………. (9) 𝐽𝐵 Substituting for EH and J, we get 𝐵𝑣 𝑅𝐻 = 𝜌𝑣𝐵 1 Thus 𝑅𝐻 = 𝜌 ………………………… (10) Thus, Hall coefficient can be evaluated once the charge density ρ is known. Expression for Hall voltage in terms of Hall coefficient From equations (7) & (10), we have 𝐵𝐼 1 𝐵𝐼 1 𝑉𝐻 = = ( ) and 𝑅𝐻 = 𝜌𝜔 𝜌 𝜔 𝜌 𝐵𝐼 Using the above two equations, we get 𝑉𝐻 = 𝑅𝐻 ( 𝜔 ) Light Emitting Diode: Types of Light Emitting Diodes There are different types of light-emitting diodes present and some of them are mentioned below. Gallium Arsenide (GaAs) – infra-red 5  Gallium Arsenide Phosphide (GaAsP) – red to infra-red, orange Aluminium Gallium Arsenide Phosphide (AlGaAsP) – high-brightness red, orange-red, orange, and yellow Gallium Phosphide (GaP) – red, yellow and green Aluminium Gallium Phosphide (AlGaP) – green Gallium Nitride (GaN) – green, emerald green Gallium Indium Nitride (GaInN) – near-ultraviolet, bluish-green and blue Silicon Carbide (SiC) – blue as a substrate Zinc Selenide (ZnSe) – blue Aluminium Gallium Nitride (AlGaN) – ultraviolet Working Principle of LED The working principle of the Light-emitting diode is based on the quantum theory. The quantum theory says that when the electron comes down from the higher energy level to the lower energy level then, the energy emits from the photon. The photon energy is equal to the energy gap between these two energy levels. If the PN-junction diode is in the forward biased, then the current flows through the diode. The flow of current in the semiconductors is caused by the flow of holes in the direction of current and the flow of electrons in the direction opposite of the current. Hence there will be recombination due to the flow of these charge carriers. The recombination indicates that the electrons in the conduction band jump down to the valence band. When the electrons jump from one band to another band the electrons will emit the electromagnetic energy in the form of photons and the photon energy is equal to the forbidden energy gap. For example, let us consider the quantum theory, the energy of the photon is the product of both the Planck constant and frequency of electromagnetic radiation. The mathematical equation is shown E = hf 6 where h is known as a Planck constant, and the velocity of electromagnetic radiation is equal to the speed of light ‘c’. The frequency radiation is related to the velocity of light as an f= c / λ. λ is denoted as a wavelength of electromagnetic radiation and the above equation will become as a E = hc / λ From the above equation, we can say that the wavelength of electromagnetic radiation is inversely proportional to the forbidden gap. In general silicon, germanium semiconductors this forbidden energy gap is between the conduction and valence bands are such that the total radiation of electromagnetic wave during recombination is in the form of infrared radiation. Applications of LED : Picture phones and digital watches. Camera flashes and automotive heat lamps. Aviation lighting. Digital computers and calculators. Traffic signals and Burglar alarms systems. Microprocessors and multiplexers. Optical Communication. Indicator lamps in electric equipment. Photodiode : A photodiode is a PN-junction diode that consumes light energy to produce an electric current. These diodes are particularly designed to work in reverse bias conditions. when a photon of ample energy strikes the diode, it makes a couple of an electron-hole. This mechanism is also called the inner photoelectric effect. If the absorption arises in the depletion region junction, then the carriers are removed from the junction by the inbuilt electric field of the depletion region. Holes in the region move towards the anode, and electrons move toward the cathode, and a photocurrent will be generated. When the diode is connected in reverse bias, then the depletion layer width can be increased. This biasing will cause quicker response time for the diode. So, the relation between photocurrent & illuminance is linearly proportional. Power Responsivity It is the ratio of the photocurrent which is generated to the absorbed optical power. 7 It is normally maximum in a wavelength area wherever the photon energy is higher than the bandgap energy & declining within the bandgap region wherever the absorption reduces. Applications of Photodiode These diodes are used in smoke detectors, compact disc players, and televisions and remote controls. Used for exact measurement of the intensity of light in science & industry. Generally, they have an enhanced, more linear response than photoconductors. Used in numerous medical applications like instruments to analyse samples, detectors for computed tomography, and also used in blood gas monitors. Dielectric materials A dielectric is an electrically non-conducting material which provides electrical insulation between two conductors which are at different potentials, and also serve as an electrical charge storage aid under certain circumstances. If its main purpose is to provide just the insulation, then it is referred to as insulating material, but if it is employed for charge storage, then its name dielectric remains. Materials such as glass, ceramics, polymers, paper, porcelain, wood, waxed paper, vegetable oils, natural rubber, etc., are very popular as electrical insulation. A dielectric may be defined in terms of the energy band structure. The forbidden gap Eg is very large in dielectrics and excitation of electrons from the normally full valence band to the empty conduction band is not possible under ordinary conditions. Therefore, conduction cannot occur in dielectrics. A dipole is an entity in which equal positive and negative charges are separated by a small distance. Dielectrics are classified as (i) polar (dipole) dielectrics and (ii) non polar (neutral) dielectrics based on dipole moment. Polar and non-polar dielectrics If the centres of gravity of the positive nuclei and the electron cloud coincide the molecule is called non polar molecule. Hence the electric dipole moment is zero and are called non polar molecules and the materials made up of non polar molecules are called non polar dielectrics. If the points of action of the resultant charges of a molecule do not coincide in space even in the absence of external electric field the molecule will possess an intrinsic dipole moment or permanent or rigid dipole moment. Such molecules are called polar molecules and the materials made up of polar molecules are called polar dielectric. Matter comprising different molecules - both non-polar and polar should be regarded as polar matter. Whether a molecule is a polar or a non-polar molecule can be judged from its structure. Monoatomic molecules such as He, Ne, Ar and molecules consisting of two identical atoms linked by a non-polar bond, such as H2, N2, 02, Cl2 are non-polar. The magnitude of the electric moment of a molecule is  = ql where q is the electric charge of a molecule and l is the arm of the dipole. 8 Dielectric polarization The displacement of charges in the molecules of a dielectric under the action of an applied electric field, leading to the development of dipole moment is known as dielectric polarization. Linear dielectrics show a direct proportionality between the electric moment p acquired by the particle during the process of polarization and the intensity E of the electric field acting on the particle as p = E The coefficient of proportionality  is known as the polarizability of a given particle. The polarization P is given by P = NE Where N is the number of molecules per unit volume Types of polarization The four important types of polarizations are (i)electronic polarization (ii)ionic polarization (iii)orientation polarization and (iv) space charge polarization (i) Electronic Polarization This occurs due to displacement of the positive and negative charge in a dielectric material owing to the application of an external electric field. The separation created between the charges leads to development of the dipole moment. This process occurs throughout the material. Thus the material as a whole will be polarized. The electronic polarizability ,  0 ( r − 1) e =  Where N is the number of atoms/unit volume, o is the dielectric constant of vacuum is 8.854x10-12 F/m &r is the relative dielectric constant or relative permittivity for the material. (ii) Ionic Polarization Ionic polarization occurs in materials which are ionic ex: NaCl. In the absence of the electric field, the ions are arranged in a regular manner. When a field is applied to the molecule such as sodium chloride, the sodium and chlorine ions are displaced in opposite directions until ionic binding forces stop the process, thus increasing the dipole moment. Again it is found that this induced dipole moment is proportional to the applied field and an ionic polarizabilityi is introduced to account for the increase, giving i = iE 9 (iii) Orientational Polarization This type of polarization occurs only in polar substances. The existence of a permanent moment is purely a matter of molecular geometry. The permanent molecular dipoles in such materials can rotate about their axis of symmetry to align with an applied field which exert a torque in them. This additional polarization effect is accounted by an orientationalpolarizability term αo. 2 0 = where is the permanent dipole moment, k is the Boltzmann constant, and T is the 3 temperature (K). In thermal equilibrium with no field applied the permanent dipoles contribute no net polarization since they are randomly oriented. When electric field is applied, dipole alignment largely offset by thermal agitation. However, since it is observed that the orientational polarization is of the same order as the other forms of polarization, it is only necessary for one dipole in 10-5 to be completely aligned with the field to account for the effect. Whereas the orientationalpolarizability αo is temperature dependent, higher the temperature the greater is the thermal agitation and the lower is αo,. (iv) Space Charge Polarization Space charge polarization occurs in multiphase dielectric substances in which there is a change of resistivity between different phases. When an electric field is applied at high temperature, the electric charges get accumulated at the interface due to sudden change in conductivity. There is accumulation of charges with opposite polarity at opposite parts in the resistivity phase. Total polarization The space charge polarization is not an important factor in most dielectrics. Therefore the total polarizability of a dielectric material is equal to the sum of electronic, ionic and orientation polarizabilities. =e + i+o The total polarization is given by P = Pe + Pi + Po Internal fields or local field in solids and liquids Internal field is defined as the sum of the electric field applied and the field due to the neighbouring dipoles. Therefore the internal field is given as Ei ( A) = E + E' 10 Consider a dielectric material, either liquid or solid, kept in an external uniform electric field of strength E. In the material let us consider an array of equidistant atomic dipoles arranged parallel to the direction of the field as shown. Let the inter-atomic distance be d, and the dipole moment of each of the atomic dipole be μ. In the linear array let us consider an atom X. At X the total field due to all the other dipoles in the array can be E'= E1 + E2 + E3 +……..   = + +....   d 3   (2d )3   1 E' =   d 3 n n =1 3 Therefore the internal field is given as 1.2 Ei ( A) = E +   d 3 If e is the electronic polarizability for the dipoles, then  = eE 1.2 e E Therefore Ei ( A) = E +   d 3 In three dimensional cases, the general equation for internal field is expressed as   Ei ( A) = E +   P  o  1 if it is cubic lattice, then  = , then the internal field is named as Lorentz field given as 3 P ELorentz = E + 3 o Dielectric constant of a dielectric material Consider a parallel plate capacitor with vacuum in between the two plates. Let it be connected to a battery with potential V. Let the upper plate be connected to the positive of the battery and lower terminal to the negative of the battery. 11 Due to this, the positive charges get accumulated on the upper plate and negative charges get accumulated at the lower plate. The electric lines of force move from positive charges available at the upper plate to the negative charges near the lower plate. Let Q1 be the charge accumulated at the plates such that the potential across the plates become equal to the potential V that is applied across the battery. W.K.T charge Q1 = Cvac V, where Cvac is the capacitance developed across the capacitor with vacuum in between the plates. 𝑄1 Hence, 𝐶𝑉𝑎𝑐 = 𝑉 …………(1) If a dielectric material is introduced between the two plates of a parallel plate capacitor instead of vacuum, electric polarization takes place and negative charges get accumulated inside the dielectric near the upper plate and positive charges get accumulated inside the dielectric near the lower plate. Due to the presence of negative and positive charges, the electric lines of force get truncated and more charges get accumulated from the battery towards the plate such that the potential across the plate become equal to the potential applied across the battery. Let Q2 be the charge accumulated across the plates. Now Q2 > Q1 W.K.T Q2 = Cmat V Where Cmat is the capacitance developed across the capacitor with dielectric material in between the plates. 𝑄2 Hence, 𝐶𝑚𝑎𝑡 = 𝑉 …………(2) Since, Q2 > Q1, hence from equations (1) and (2) , 𝐶𝑚𝑎𝑡 > 𝐶𝑉𝑎𝑐 𝐶 The ratio 𝐶𝑚𝑎𝑡 is called static dielectric constant. 𝑉𝑎𝑐 Hence, dielectric constant is defined as the factor by which the capacitance of a capacitor increases if vacuum is replaced by a dielectric material between the two plates of a parallel plate capacitor. 12 Applications of dielectrics They are used for energy storage in capacitors To enhance the performance of a semiconductor device, high permittivity dielectric materials are used. Ceramic dielectric is used in dielectric resonator oscillator. Barium strontium titanate thin films are dielectric which are used in microwave tunable devices providing high tunability and low leakage current. Parylene is used in industrial coatings act as a barrier between the substrate and external environment. In electric transformers, mineral oils are used as liquid dielectric & they assist in cooling process. Castor oil is used in high voltage capacitors to increase its capacitance value. Electrets, a specially processed dielectric material acts as a electrostatic equivalent to magnets. 13