Semiconductor Physics Motivation Is it possible that there might be current flowing in a conductor (or a semiconductor) even when there is no potential difference supplied across its ends? Look at the image below: Now, in case 1, there is some electric current flowing through the wires as the switch is ON. But, what about case 2 where there is no switch and thus, no potential difference? Is there any current flowing through the wires in case 2? Actually, there are a lot more phenomena going on inside the conducting material which shall modify our answer. These very concepts cover the physics of the semiconductors used in the manufacturing of diodes and transistors and other important electronic devices and are thus presented in this module. Apart from this, the following module the anomalous effects of magnetic fields and electric fields and their interference and the relationship between material and their conductivity.
Learning Objectives This module uncovers the phenomenon underlying semiconductor physics and deals in the quantitative study of the nature of semiconductor devices. After this module, you should be able to: 1. Study principles articulating charge carriers and solve their numerical values. 2. Justify the need for the existence of extrinsic semiconductors. 3. Formulate concepts such as drift velocity, Hall Effect and diffusion in extensive detail. 4. Appraise the above concepts so as to refer to them while studying devices. Suggested time 5 hrs Prerequisite Basic physics Charge Carriers The conduction of electricity in solid substances takes place with the help of atomic particles, referred to as charge carriers. These charge carriers are basically free electrons which move through the bulk of solids and transfer electric charge from one region to another. As per the convention of potentials, the electrons move from a region of lower potential to a region of higher potential and hence, the direction of current is reverse of that of electron flow. Let us now look at the physics behind generation of free electrons and other charge carriers in a semiconductor. At 0 K, there is no thermal energy in molecules and hence electrons remain in the valence band.
As the temperature increases, these electrons start gaining kinetic energy. Some of the electrons gain sufficient kinetic energy to cross the energy gap between the valence band and conduction band, and get excited, so move into the conduction band. Once their energy reaches the level of conduction band, these electrons become free to move in the bulk of the substance and conduct charges. In metals, free electrons are the only charge carriers that are present. Their concentration is very high because of the overlapping that takes place between the valence and conduction bands. Because of this, metals are good conductors of electricity. In semiconductors however, this concentration depends upon the energy gap between the two bands. In insulators, this gap is too high for electrons to jump into the conduction band and so, insulators do not conduct electricity. When an electron leaves the valence band in semiconductors, it leaves behind a positively charged vacancy, referred to as a hole. These holes are mobile in the valence band. An electron and its corresponding vacancy in the valence band collectively form the electron-hole pair (EHP). Hence a semiconductor has two types of charge carriers- the electrons in conduction band and the holes in valence band. The flow of these particles sums up to give the current flowing through the semiconductor. The following figure gives a diagrammatic representation for the electron-hole pair formation in a p-n junction:
Figure 3.1 Electron-hole pair formation in a p-n junction Intrinsic and Extrinsic Semiconductors An intrinsic semiconductor is one which is purely composed of only one type of element and is free from any defects or impurities. Intrinsic semiconductors do not have any charge carriers at 0 K and at higher temperatures, EHPs are the only charge carriers in the substance. In a steady state, there is generation of EHPs by thermal agitation, as well as there is recombination of EHPs, which leads to a dynamic equilibrium in the concentration of electrons and holes. Since the electrons and holes are always created in pairs, the concentration of both of these is the same. That is, where np and nn are the electron and hole concentrations; and ni is the intrinsic carrier concentration which is a function of temperature. Apart from thermal agitation, it is also possible to generate electrons and holes by a process known as doping. This involves insertion of impurities into the substance, which create or disrupt the covalent bonds present in the material, thus creating new free electrons and holes. Such semiconductors which have impurities in them are known
as extrinsic semiconductors, and exhibit a greater conductivity that their intrinsic counterparts. Extrinsic semiconductors can be classified into two categories- n type and p type. The n type semiconductors are the ones which have been doped by infusing a small amount of donor element. Donor elements are the elements of group 15 with five valence electrons- four out of which form covalent bonds with the semiconductor atom and one is left free. Hence the n-type semiconductors have an excess of electrons in them. The p type semiconductors are the ones which have been doped by infusing a small amount of acceptor element. The acceptor elements are elements of group 13 with three valence electrons- all of which form covalent bonds with the semiconductor atom, leaving a hole. Hence the p-type semiconductors have an excess of holes. Figure 3.2 Examples of Pure and Impure (n-type and p-type) semiconductors
Speaking in terms of relative carrier concentrations, we say that in the n type semiconductor, electrons are the majority carriers and holes are the minority carriers. Similarly p type semiconductors have holes as their majority carriers and electrons as their minority carriers. Two reference videos for the same are provided here: https://www.youtube.com/watch?v=hk1e7g-nukm (For your convenience you can get them inside Learn More quadrant) https://www.youtube.com/watch?v=ut5qtayrg6c (For your convenience you can get them inside Learn More quadrant) Now, let us quantify the phenomena underlying the carrier concentrations. Speaking in terms of relative carrier concentrations, we say that in n type semiconductor, electrons are the majority carriers and holes are the minority carriers. Similarly p type semiconductors have holes as their majority carriers and electrons as their minority carriers. Carrier concentrations The concentration of electrons at any energy level (E) at a given temperature (T) follows Fermi-Dirac distribution. That is, where E f is known as Fermi-level energy.the Fermi Dirac distribution is only valid if the number of fermions is large enough so that adding one more fermion to the system has negligible effect on E f This probability distribution can be used to find the average number of electrons in an energy state E
In the above equation, if we replace N(E) by the effective density at energy level Ec (conduction band), then we get Assuming that Ec>>Ef, we get Similarly, the concentration of holes can be given by where Nv is the effective hole density at energy level E v (valence band). Assuming that Ev<<Ef, we get The product of electron and hole concentration is constant at equilibrium (even if doping is varied), and the product is written as
Hence we can write and Drift Velocity and Current Density At room temperature, these conduction electrons move randomly inside the conductor more or less like a gas molecule. During motion, these conduction electrons collide with ions (remaining positive charged atom after the valence electrons move away) again and again and their direction of motion changes after each and every collision. As a result of these collisions atoms move in a zig-zag path. Then, shouldn t there be current flow? No! Since in a conductor there are a large number of electrons moving randomly inside the conductor. Hence, they have no net motion in any particular direction. Since the number of electrons crossing an imaginary area ΔA from left to right inside the conductor very nearly equals the number of electron crossing the same area element from right to left in a given interval of time leaving flow of electric current through that area nearly equals to zero. Figure 3.3 There is random movement of electrons in an idle conductor
Now we apply some potential difference using a battery across the two ends of the conductor. Thus, an electric field is set up inside the conductor. As a result of this electric field setup inside the conductor and conduction electron experience a force in direction opposite to electric field and this force accelerates the motions of the electrons. Figure 3.4 Electrons move opposite to the direction of applied current As a result of this accelerated motion electrons drift slowly along the length of the conductor towards the end at higher potential. Due to this acceleration, the velocity of the electron increases only for a short interval of time. This extra velocity gained by the electrons is lost in subsequent collision and the process is continued till the electron reaches the positive end of the conductor. If τ is the average time between two successive collisions and E is the strength of applied electric field then the force on the electron due to applied electric field is F=eE where, e is the amount of charge on electron If m is the mass of electron, then acceleration produced is given by a=ee/m
Since the electron is accelerated for an average time interval τ, an additional velocity acquired by the electron is vd=aτ or vd=(ee/m)τ This small velocity imposed on the random motion of electrons in a conductor on the application of an electric field is known as drift velocity. Thus, this drift velocity is defined as the velocity with which free electrons drift towards the positive end of the conductor under the influence of externally applied electric field. Relation between drift velocity and electric current Consider a conducting wire of length L having a uniform cross-section area A in which an electric field is present: Figure 3.5 Cross-section of the semiconductor under study Consider in the wire that there are n free electrons per unit volume moving with the drift velocity vd. In the time interval Δt each electron advances by a distance vdδt and the volume of this portion is AvdΔt and the number of free electrons in this portion is navdδt and all these electrons cross the area A in time Δt. Hence the charge crossing the area in time Δt is ΔQ=neAvdΔt Or, I = ΔQ/Δt = neavd
This is the relation between the electric current and drift velocity. If the moving charge carriers are positive rather than negative then the electric field force on charge carriers would be in the direction of electric fields and drift velocity would be left to right. Current density: (Total flow of charge per time over a cross section area A). In terms of drift velocity current density is given as J = I/A = nevd High field effects While deriving at the relation between drift velocity and current density, we had assumed that drift current is proportional to the electric field, and that the conductivity is not a function of the electric field. However, large electric fields can cause drift velocity and thus the current to have sub-linear dependence on the electric field. Figure 3.6 Drift Velocity vs Electric Field
The Hall effect Consider a semiconductor p-type bar with current in the positive x- direction. When a magnetic field is applied to it in a direction perpendicular to the flow of electrons (say positive direction), the holes will experience a magnetic force in addition to the electric force. Mathematically, the net force on a hole in this condition would be The y-component of this force is Hence in order to ensure that the holes move steadily even in the presence of an external magnetic field, one must try that the y components of electric and magnetic fields cancel each other. That is, This electric field starts to set up when the holes get unevenly distributed due to the presence of magnetic force. Once the electric field is sufficiently strong to overcome the magnetic force, the steady flow of holes in x- direction is restored. This establishment of opposing electric field is known as the Hall effect, and the resulting voltage Vh= EyW is called the Hall voltage. Writing as
for holes, we see that the hall field is directly proportional to the current density and the magnetic field intensity, the constant of proportionality being Diffusion. This constant is called Hall coefficient. Doping refers to the process of infusing impurities into a semiconductor, in order to increase its charge-carrier concentration and therefore, increase its conductivity. When a semiconductor is doped with an impurity, there is a chance that the impurity concentrations are different at different places. Because of this, electrons start moving from regions of high concentration to lower concentration. This is called diffusion and plays an important role in semiconductor physics. Diffusion Processes Figure 3.6 Diffusion in semiconductors When a semiconductor is doped with an impurity, there is a chance that the impurity concentrations are different at different places. Because of this, electrons start moving from regions of high
concentration to lower concentration. This is called diffusion and the current that results is called diffusion current. Suppose electrons are injected into a semiconductor bar from one end. Then the electron flux density is given by Where Dn is known as the electron diffusion coefficient and dn/dx is the electron concentration gradient along x-axis. Similarly for holes, we have The current density due to diffusion current is then given by Measuring the Band-Gap of a Semiconductor According to the band theory of solids, insulators and semiconductors are materials that possess a band-gap (i.e., a range of forbidden energy values) at the Fermi level. Thus, these materials have a completely filled energy band below the gap and an empty band above the gap. The width of this band-gap is what distinguishes insulators from semiconductors. In semiconductors the band-gap is small enough (<2 ev) that at finite temperatures thermal excitation of electrons across the gap, into the empty "conduction" band, is possible leading to a small but measurable conductivity. The temperature dependence of the resistivity of a pure (i.e., intrinsic) semiconductor is given by ρ(t) = B(T).e Eg / 2kBT
where, Eg is the width of the gap, and, B(T) is only very weakly dependent on temperature. To a good approximation we can take B(T) constant. Thus, we can easily measure the gap energy of a semiconductor material by measuring the resistance of a sample over a range of temperatures. One additional complication here is that very small amounts of impurities in a semiconductor can have large effects on the conductivity. This is desirable in designing many semiconductor devices but can complicate the measurement of the intrinsic gap energy. These impurity effects can be minimized by working at high enough temperatures that all dopant-type impurities are fully ionized (and thus the impurity contribution to the conductivity will not change with temperature). According to the equation given above, the resistivity of an intrinsic semiconductor is very sensitive to changes in temperature. This property is the basis for solid-state temperature measurement devices known as thermistors. The animation model provided in this module will show how the resistance of a thermistor varies as a function of temperature and thereby determine the gap energy for the semiconductor material used in the device. Apparatus used: (i) a thermistor (resistance 100 Ω ) (ii) an electrically controlled thermostat (room temperature to 200 C) (iii) a constant current dc source (0-1A) (iv) a thermometer (v) a dc milli-ammeter (1-100A) (vi) a dc voltmeter (1-10V)
Setup: Procedure: 1. Set up the circuit arrangement as shown in the above figure. 2. Switch on the constant current source and pass a small current through the thermistor. Allow the devices to set to constant values. 3. Now keep the current constant. Switch on the oven and adjust its temperature to around 20 C. 4. Calculate resistance at this temperature. 5. Keeping the current constant, increase the temperature of the oven in suitable steps and at each step calculate the resistance and note the temperature. 6. Decrease the temperature of the oven and obtain the mean of the two values of R at each temperature.
7. Repeat the experiment for a constant value of the current. Find the value of (1/T) for each temperature. 8. Draw a curve by plotting (1/T) in K-1 along x-axis and log10r along the y-axis. Using the slope of the curve you can now easily find Eg.