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Chapters 1-3: Important concepts

The transmitted intensity It of a beam of photons of wavelength through a sample of. Absorption cutoff occurs at GaAs, Si, Ge, and InSb band gaps are such that c occurs beyond the visible region in c falling within the visible. When recombination occurs between a conduction band electron and a valence band hole, the energy released can be given off in the form of light luminescence.

Direct band-to-band recombination in direct band gap semiconductors have a much higher probability of light emission as compared to those in indirect materials. Broadly divided into three categories: Photoluminescence: if the recombining carriers were caused by optical excitation. Cathodoluminescence: if the recombining carriers were caused by high energy electron bombardment. Electroluminescence: if the recombining carriers were caused by injection of excess carriers by forward biasing a p-n junction, for example. For steady state excitation, the recombination rate and the generation rate for EHPs are equal, and one photon is emitted for each photon absorbed.

The fraction of energy converted to heat is given by 1. Thus, the amount of energy converted to heat per second. Thus, the number of photons emitter per second. Best example: cathode ray tube CRT basis of television sets, oscilloscopes, and other display systems. Electrons emitted from the heated cathode are accelerated towards the anode by high field, deflected by electric or magnetic fields by the horizontal and vertical plates, and made to hit the screen coated with a phosphor at desired locations.

When electrons hit these phosphors, the energy of the electrons gets transferred to the electrons of the phosphors, and they get excited to higher states, and eventually fall to the ground state, thus causing recombination and light emission. Three phosphor dots are used for each pixel, capable of transmitting three primary colors: red, green, and blue RGB thus by varying the intensity and position of the electron beam, a wide range of colors and picture can be attained. Photoconductivity: increase in the conductivity of a sample due to excess carriers created by optical excitation. With excitation turned off, the photoconductivity decreases to zero since all excess carriers eventually recombine.

Direct recombination occurs spontaneously, i. The net rate of change in the conduction band electron concentration at any time t. Similarly, excess holes in an n-type material recombine with a rate Note: for direct recombination, the excess majority carriers which is equal to the excess minority carriers decay at exactly the same rate as the minority carriers, however, there is a large percentage change in the minority carriers as compared to the majority carriers. A more general expression for carrier lifetime for near not sufficiently extrinsic samples is 1 2 3 4.

In indirect materials, the probability of direct band-to-band recombination is very small recombination in these materials proceed through the assistance of recombination or trapping centers located within the band gap, which trap carriers of one type followed by trapping carriers of the opposite type, thus annihilating the pairs.

The resulting energy loss is often in the form of heat given to the lattice instead of light emission , and, thus, these materials are not well suited for optoelectronic applications. There are four probabilities associated with a recombination center: a hole capture: when an electron from the recombination center falls to the valence band, b hole emission: when an electron makes a transition from the valence band to the recombination center, c electron capture: when an electron falls from the conduction band to the recombination center, and d electron emission: when an electron makes a transition from the recombination center to the conduction band.

Each of these processes has their own probabilities and time constants, and the resulting analysis is significantly complicated. If process a follows process c or vice versa, recombination takes place, whereas, if process b follows process a or vice versa or process c follows process d or vice versa , it is known as reemission, and the recombination center behaves like a trapping center.

Generally, centers that are located towards the middle of the band gap e. Alternate definition: in a center located within the band gap, if after capturing one type of carrier, the most probable next event is the capture of opposite type of carrier, then it is a recombination center, however, if the most probable next event is reexcitation, then it is a trap. The recombination can be slow or fast, depending on the amount of time the carrier spends in the center before the capture of the opposite type of carrier happens, thus, computation of lifetime for this kind of indirect recombination is sufficiently complicated.

The decay of excess carriers can be measured by a typical photoconductive decay measurement, where light shining on a sample is suddenly switched off, and the resulting decay of current passing through the sample is measured, the rate of decay of this current gives the excess carrier lifetime. Auger Recombination. Note: the lifetime is proportional to the inverse of the doping concentration. However, at relatively high doping levels, the lifetime decreases at a faster rate with an increase in the doping concentration.

This is because a different recombination mechanism, called the Auger recombination becomes dominant at high doping levels. In this recombination mechanism, the electron and hole recombine without involving trap levels, and the released energy of the order of the energy gap is transferred to another carrier a hole in p-type material and an electron in n-type material. This process is somewhat the inverse of the impact ionization process, in which an energetic carrier causes EHP generation.

Since two electrons in n-type material or two holes in p-type material are involved in this process, it is highly unlikely except in heavily doped material. The recombination lifetime associated with the Auger recombination process is inversely proportional to the square of the majority carrier concentration, i.

It is obvious that near the surface of any semiconductor device, the carrier recombination rate should be very high, due to extra defects and traps at the surface. Thus, the diffusion flux of minority carriers at the surface is determined by the surface recombination processes. The capture cross-section typically describes the effectiveness of the localized state in capturing a carrier. The product may be visualized as the volume swept out per unit time by a particle with cross-section If the surface state lies within this volume around the carrier, then the carrier gets captured by the surface state.

For any temperature T, there is a thermal generation rate g T balanced by a recombination rate r T. Now, if a steady light is shone on the sample, an optical generation rate will be added to g T , and the carrier concentrations n and p would increase to their new steady state values. Thus, the excess carrier concentrations can be given by In general, when trapping is present, , and. Note: when excess carriers are present,. When excess carriers are present, the equilibrium Fermi level is no more meaningful; instead, the carrier concentrations are defined in terms of quasi-Fermi levels also referred to as Imref, which is Fermi spelled backwards as.

Imref for the minority carriers deviates significantly from the equilibrium Fermi level, whereas, for majority carriers, the Imref stays very close to the equilibrium Fermi level, and the separation between these two Imrefs is a measure of how far the system is from equilibrium. With concentrations varying with position, the Imrefs would also vary with position, thus drawing Imrefs in band diagrams clearly shows the positional variations of the carrier concentrations.

Assume no trapping, and a Determine the electron and hole concentrations n and p respectively, and their percentage change from the equilibrium concentrations. Hence, the equilibrium electron ,and compare their locations with the. And the percentage change in the hole concentration Note: with optical excitation under the low-level injection approximation , there is a very large change in the minority carrier concentration, whereas the change in the majority carrier concentration is hardly noticeable! Thus, the majority carrier Imref almost coincides with the equilibrium Fermi level, whereas the minority carrier Imref shows a large departure from the equilibrium value.

Devices which change their resistance while exposed to light. Examples: automatic night light controllers, exposure meters in cameras, moving-object counters, burglar alarms, detectors in fiber optic communication systems, etc. Considerations in choosing a photoconductor for a given application: sensitive wavelength range, time response, and optical sensitivity responsivity of the material.

The photoconductivity change while exposed to light is. Obvious that for large changes in , the carrier mobility and lifetime should be high e. Time response is limited by recombination times, degree of carrier trapping, and time required for the carriers to drift through the device in an electric field. Dark resistance i. Generally, all these requirements cannot be satisfied simultaneously, and some kind of optimization is required.

Diffusion of Carriers. When excess carriers are created in a semiconductor and their concentrations vary with position, then there is a net carrier motion from regions of higher concentration to regions of lower concentration. This type of motion is called the diffusion, and it is an important charge transport mechanism in semiconductors. Diffusion and drift are the two main current transport mechanisms.

Diffusion processes. Natural result of the random motion of individual electrons. Electrons move randomly and experience collisions, on the average, after each mean free time. Since the motion is truly random, an electron has equal probability of moving into or out of a volume through a boundary. Consider any arbitrary distribution n x , with x divided into segments wide, with n x evaluated at the center of each segment.

In , half of the electrons in segment 1 to the left of would move into segment 2 , and in the same time, half of the electrons in segment 2 to the right of would move into segment 1. Therefore, the net number of electrons moving from segment 1 to segment 2 through within a mean free time , where A is the area perpendicular to x.

The minus sign in the expression for implies that the diffusion proceeds from higher electron concentration to lower electron concentration. Similarly, holes diffuse from a region of higher concentration to a region of lower concentration with a diffusion coefficient Thus,. Note: electrons and holes move together in a carrier gradient, however, the resulting currents are in opposite directions because of the opposite charges of the particles.

Diffusion and Drift of Carriers: Built-in Fields.

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The total current may be due primarily to one of these two components, depending upon the carrier concentrations, their gradients, and the electric field. Thus, minority carriers can contribute to current conduction significantly through diffusion, even though they contribute very little to the drift term due to their low concentrations. Since electrons drift opposite to the direction of the electric field due to their negative charge , their potential energy increases in the direction of the electric field.

Thus, the electric field can be given by. Note: electrons drift downhill in a band diagram, therefore, the electric field points uphill in a band diagram. Equating the hole current density equation to zero, we get. This is an extremely important equation valid for both carrier types, and is called the Einstein relation gives the relation between D and , which is a function only of temperature, and allows calculation of one if the other is known.

Thus, the electric field varies inversely with distance and has positive values throughout. Since E x varies inversely with x, hence Ei and consequently, both EC and EV will have a logarithmic ln dependence on x. In the description of conduction processes, the effects of recombination must be included, since they can cause a variation in the carrier distribution. Hole conservation equation:. When the current is carried entirely due to diffusion negligible drift , then we obtain the diffusion equation for electrons, given by.

In steady state, if a distribution of excess carriers is maintained, then the diffusion equations become. Case study: suppose excess holes are injected into a semi-infinite n-type bar, which maintains a constant concentration relevant problem in a forward biased diode. Obviously, the excess holes would diffuse into the n-type bar, recombine with the electrons. The steady state distribution of excess holes causes diffusion, and therefore, a hole current in the direction of decreasing concentration, given by. Counterpart of the Hall effect experiment.

Independently determines the minority carrier mobility and diffusion coefficient. Basic principle a pulse of excess holes is created in an n-type bar which has an applied electric field , as time progresses, the holes spread out by diffusion and move due to the electric field, and their motion is monitored somewhere down the bar, 1 2 3 4. For measurement of diffusion coefficient, assume the pulse spreads without drift and neglect recombination; then the diffusion equation can be rewritten as. Note: the peak values of the pulse decreases and the pulse spreads in time.

The length of the sample is 2 cm and the two measurement probes are separated by 1. The voltage applied across the two ends of the sample is 5 V. A pulse arrives at the collection point 0. Calculate the electron mobility and diffusion coefficient, and verify whether Einstein relation is satisfied.

Any combination of drift and diffusion implies a gradient in the steady state Imrefs. Under general case of nonequilibrium electron concentration with drift and diffusion, the total electron current can be written as. Using the expression for n x in terms of the electron Imref, and applying Einstein relation, it can be shown that. Therefore, any drift, diffusion, or a combination of the two in a semiconductor sets up currents proportional to the gradient in the Imrefs, or, in other words, no current implies constant Imrefs. Practice Problems 3. The absorption coefficient at this wavelength is.

What power is delivered to the sample as heat? Calculate the kinetic energies of the electron and the. Assume high-level injection condition State and justify whether would decay with the same profile till it reaches zero. What is the separation of the Imrefs Clearly draw the band diagram showing the Imrefs and the equilibrium Fermi level compute the change in the sample conductivity after illumination. The sample is illuminated uniformly to generate. The electron lifetime in the sample is Calculate the sample resistivity and the percent change in the conductivity after illumination due to the majority and the minority carriers.

Determine the optical generation rate injection and no trapping. Also, plot the potential V x as a function of position. Starting from the continuity equation and assuming low-level injection and no current flow, determine the expression for the build-up of excess holes as a function of time. If the excess conductivity at ; and after sufficiently long time, it is , determine the optical generation rate trapping. What should be the minimum values of the lifetime and the diffusion length in the original sample for authentic measurement results? However, by a simple modification, it can be made to include the effects of recombination.

Assume an n-type semiconductor, the peak voltage of the pulse displayed on the CRO screen is proportional to the peak value of the hole concentration under the collector terminal at time td, and that the displayed pulse can be approximated as a Gaussian, which decays due to recombination by , where is the excess hole lifetime. The electric field is varied and the following date taken: for , the peak.

PN junctions are important for the following reasons: i PN junction is an important semiconductor device in itself and used in a wide variety of applications such as rectifiers, Photodetectors, light emitting diodes and lasers etc ii PN junctions are an integral part of other important semiconductor devices such as BJTs, JFETS and MOSFETs iii PN junctions are used as test structures for measuring important semiconductor properties such as doping, defect density, lifetime etc.

The discussion associated with the PN junctions will proceed in the following order: i PN junction in equilibrium ii dc IV characteristics in forward bias iii characteristics in reverse bias iv dynamic characteristics v Circuit models vi Design perspective Device Structure : The Figure below shows a simplified structure of a PN junction:. The structure can be fabricated by diffusing P-type impurity in the n-epilayer grown over an substrate. While the doping in the n-epilayer can be uniform, the doping in the P-region is often either Gaussian or error-function in nature.

The doping profiles and the junction are schematically illustrated below:. Even though the doping in both N and P-regions may in general be nonuniform, for simplicity, we shall assume them to be uniform in the initial analysis because the basic device physics remains almost the same A simplified, one-dimensional abstracted view of a PN junction described by the region within the dotted lines of device schematic is shown below:.

We shall assume that the thicknesses of P and N-regions are large enough so that one can ignore the presence of Ohmic contacts and the heavily doped N-region and consider only the P and N regions for analysis. Such a diode with wide N and P-regions is called a widebase diode. The PN junction that we shall study will therefore be a 1-D structure with uniformly doped P and N regions with thicknesses sufficiently large to ignore effects of contacts and other layers.

It shall be represented simply as. As mentioned earlier, the characteristics of a semiconductor device is completely specified in equilibrium if the variation of potential as a function of position is specified. As a first step to obtaining this potential profile, we shall sketch the energy-band diagram of the device. The energy band diagram would provide us with i a qualitative variation of potential in the device.

As usual, the energy band diagram of the PN junction will be obtained by combining the energy band diagrams of N and P-type semiconductors separately. When the N and P-regions are brought into contact, the electrons would flow from regions of higher Fermi-energy to regions of lower Fermi energy and holes would flow in the opposite direction. Because of loss of electrons, the N-region would acquire a net positive charge due to the uncovered positively charged donor atoms and P-region would acquire a negative charge due to uncovered negatively charged acceptor atoms.

At equilibrium there is no net flow of either electrons or holes so that the PN junction has a single constant Fermi level. The transfer of charges will affect only the regions close to the junction so that regions which are far still have the same energy band diagram i. Similarly, as we approach the junction from the P-side, the conduction band must bend downwards towards the Fermi energy to indicate the fact that the region is getting depleted of holes Using these principles, the final energy band diagram can be sketched as. As a result of transfer of charges from N and P-regions, the region next to the junction is charged and is known as the space charge region.

The charge on the N-side is positive and on the P-side negative. As a result, the space charge region will have an electric field directed from the N to the Pregion with a maximum value at the junction and zero at the edges of the space charge region. As a result of the electric field, there will be a net voltage across the space charge region known as the built-in voltage.

The magnitude of the built-in voltage can be quickly estimated from the energy band diagram. We do this by performing an analog of Kirchoffs voltage law: We start from a point in the N-region away from the space charge region at the energy and then move to a point in the P-region away from the space charge region again at energy via any path other than the Fermi-energy and add up the energy gained or loss at each step of the path, then the net sum should be zero! An important result that can be deduced from Eq.

Using the relationship , the expression for built-in voltage for a PN junction having non-degenerate semiconductors can be written as. Example 1. Will the built-in voltage increase or decrease with increase in temperature? Substitution of the doping values in Eq. The built-in voltage decreases with increase in temperature due to exponential increase of intrinsic carrier concentration with temperature.

There is another method by which the magnitude of built-in voltage can be obtained. In this case we start with the fact that in equilibrium, the net electron current is zero:. Although there is a net voltage across P and N-regions, the built-in voltage does not appear across the external terminals.

If it did, then upon connection of a resistor across it, a current would begin to flow. This contradicts the fact that no current can flow in equilibrium. So how does the voltage across the external terminals become zero? Keeping in mind that contact potential between any two materials is simply the difference of their work-functions, we obtain.

For simplicity we asume that both anode and cathode metals are same say aluminium so that Using four equations given above, it is easy to see that. The energy band diagram gives only a qualitative variation of potential across the space charge region. The detailed nature of this potential can be obtained through the solution of Poisson equation:. Because of the exponential terms in the expression for charge density, the analytical solution of the Poisson's equation becomes difficult.

This difficulty is overcome through the assumption that the electron and hole density within the space charge region is negligible as compared to the ionized donor or acceptor atom density. This approximation, known as the depletion approximation, allows the Poisson equation to be simplified to:. The table below shows the charge density as a function of potential within the space charge region for a PN junction with same doping in N and P regions for simplicity. The data in the table shows that over a large range of potential, the depletion approximation is valid.

Only for regions close to the space charge edge, does the approximation become weak. Simplified Charge density With the depletion approximation , the charge density can be expressed as. This implies that electric field outside the depletion region is constant. However, to be consistent, this electric field must be zero, otherwise it would imply a non-zero current, some applied bias etc. The Poisson equation with these boundary conditions can be easily solved to obtain the following results.

Solution: Electric field: It is max. The variation of potential across the depletion region is parabolic. Using the boundary condition that potential must be continuous at the junction:. The depletion width increases with decrease in doping but the magnitude of maximum electric field decreases even though the space charge region gets wider.

This is because while the width of the space charge region increases as , the charge density with in the. This results in a net decrease in charge and therefore the electric field at the junction. The difficulty in this problem is that while it is clear that in Eq. The answer depends on where the depletion edge in N-region lies. This gives a of 0. We have to check whether our assumption is correct or not. Use of Eq. As the PN junction is reverse biased, the depletion width increases so that eventually the depletion edge would lie in the higher doped N-region.

In that case also a new value of builtin voltage would have to calculated and used in the expressions for depletion width, electric field etc. Calculate the depletion width at equilibrium. Using the previous example, we know that the depletion edge will lie in the higher doped Nregion so that. To find the depletion widths , we can adopt the methodology used for uniformly doped PN junctions except that solution of Poisson's equation is carried out in three regions, with region I being P-type , region II being N-type with doping and region III with N-type doping of The boundary conditions are similar except that two new boundary conditions describing continuity of potential and electric field will have to be used at the boundary of regions II and III.

An alternative to working out the solution by beginning from Poisson's equation is to use some of the results already obtained with uniformly doped PN junctions. For example, we know that the electric field will vary linearly and can be sketched as. Using the concept of charge neutrality, meaning that net charge on the P-side must be balanced by net charge on the N-side, we can write. The slopes of electric field in each region can be written straight from Poisson equation. For example, in region II, so that.

In these equation refers only to the magnitude of the maximum electric field. The area under the curve is simply the total voltage across the junction so that. The Figure below shows a comparison of an actual charge profile computed using a 1-D device simulator and charge profile under depletion approximation for a doping of. The Figure above shows that the transition region is about , almost same as the depletion width predicted by the depletion approximation! The depletion approximation therefore appears to be a poor assumption. However, a careful look shows that the depletion assumption overestimates the charge in region I but underestimates the charge in region II.

Since, the electric field and potential are determined by the integral of charge density, the error in electric field and potential profile is not large! Obtain expressions for electric field and potential Integration of Poisson's equation in regions 1 and 2 and matching the electric field at the boundary gives. Integration of electric field with the condition that the net voltage across the space charge region is , gives. Let us consider next an heterojunction and sketch its band diagram at equilibrium and find its barrier height.

There exists a discontinuity in conduction band and valence band at the junction. Their magnitudes can be expressed as. The barrier height can be determined by performing an analog of Kirchoffs law. We start from a point at Fermi energy in the P-type GaAs far from the junction and arrive again at the Fermi energy but on the side of N-AlGaAs, again far from the junction and add up all the energy increments along the way:.

The first term is the usual term that is present in the expression for built-in voltages of homojunctions also. The second term is the additional term that results from the presence of conduction-band discontinuity. Let us consider the Forward bias first and examine qualitatively the mode of operation. There are plenty of holes in P-type region and would like to move to N-region via diffusion but are prevented by the electric field or the energy barrier at equilibrium. The drift and diffusion currents cancel each other.

Similarly, there are plenty of electrons in N-type region and would like to move to P-region via diffusion but are prevented by the electric field or the energy barrier at equilibrium. The drift and diffusion currents again cancel each other. The application of forward bias reduces the barrier and the electric field allowing significant electron and hole current to flow:. The fraction of electrons that are able to cross over to the P-side or the fraction of holes that are able to cross over to the N-side and contribute to current goes exponentially with the barrier height remember,.

Although the electric field favors the flow of holes to the P-region, there are very few holes in N-region to begin with! The number of holes in N-region is , a very small number. Further, the number of holes is fixed and unaffected by the bias. Similarly, the number of available electrons in P-region for current flow is very small and unaffected by the applied bias.

The only thing that the applied reverse bias does is to increase the junction electric field or the barrier height as shown below. The increased electric field does not alter the current flow because the bottleneck is the small number of carriers available for current conduction. Current in Reverse bias is very small and almost constant Static I-V Characteristics: The dc current-voltage characteristics of the PN junction diode will be obtained using the semiconductor equations listed below:. Since the current is the same everywhere, one can choose the region within the device for calculation of current-voltage characteristics.

Big Question : Where in the device should the current be calculated such that its computation besides being easy is also accurate? To appreciate the ease or difficulty of carrying out the computation in this case, let us consider a symmetric junction with. For a forward bias of 0. As we shall see later, the net electron current flowing through the junction for this device at a forward bias of 0.

Because the drift current is five orders of magnitude larger than the net current, the drift and diffusion currents would have to be calculated to an accuracy of. This makes the estimation of total current via an analysis at the junction virtually impossible! Let us consider a region for estimation of current which is far from the junction in say N-type semiconductor.

Far from the junction, on the N-side, the current is expected to be primarily an electron current. Any holes which are injected from the P-side would recombine and disappear away from the junction. The electron density being constant, the electron current would be primarily a drift current so that. It might appear that this is a very good place for estimation of current because we have just one component and only one unknown , the electric field.

However, this electric field is extremely difficult to estimate because of its very small value. The voltage applied across the diode gets dropped partially across the junction and partially outside it. The bottleneck for current flow in a PN junction is the space charge region where the potential barrier exists.

As a result, is almost equal to the applied voltage. The two examples discussed earlier illustrate that the choice of position in the PN junction for computation of its I-V characteristics is very important. As first demonstrated by Shockley, the computation of currents in PN junction diode is best done at the edges of depletion region as explained below:. During the course of the analysis, several assumptions will be made. There are two ways of justifying these assumptions. One of them is: i Make the assumption ii Solve the resulting simplified equations to obtain the current-voltage characteristics iii Check that the assumptions made are consistent with the results obtained.

The assumptions made will be consistent only for certain range of currents, so that the range of validity of the model will be obtained. The other approach is to justify the assumptions in the beginning of the analysis, based on available device characteristics. These assumptions would define the range of validity of the obtained model. We shall follow a mix of these two approaches. Assumption 1 : Negligible recombination within the Junction We shall justify this assumption using the first approach, namely that the assumption would be shown to be consistent with the results obtained within certain limits.

All the ho;es that are injected at reach the point so that Similarly all the electrons that are injected at This allows the total current to be expressed as : The total current can be computed by computing the minority carrier currents at the edges of depletion region in N and P-regions reach the point , so that. Assumption 2 : Minority carrier current is largely diffusive We shall justify this assumption using the second approach, namely that the validity of this assumption will be demonstrated prior to analysis.

This is described in Appendix A. The task of computing the currents boils down to the computation of minority carrier profiles: p x in N-region and n x in P-region. The minority carrier profile can be determined by solving the continuity equation with appropriate boundary conditions For hole density in N-region:. In Silicon, the dominant recombination mechanism is the Shockley-Hall-Read recombination which can be described by the relation. There are two extreme cases: i Wide Base diode: For this case, the minority carrier densities can be simplified to:.

The minority carrier densities decay exponentially with the distance from the junction, with a characteristics decay length of. It can be shown that the average distance a hole diffuses before recombining is equal to so that it is called the diffusion length. The other extreme case is : ii Narrow Base diode: The minority carrier profile can be simplified to.

The task of determining the I-V Characteristics now reduces to finding a relationship between the minority carrier densities at the edges of depletion region and the applied voltage. We start with the relation:. The low level injection assumption invoked earlier can be used here also for simplification.

The first obvious consequence is that. The quasi-Fermi level on the N-side must coincide with the Fermi level of the metal forming the ohmic contact to the N-side if an ideal contact with no voltage drop across it is assumed. Similarly, the quasi-Fermi level on the P-side must coincide with the Fermi level of the metal forming the ohmic contact to the P-side if an ideal contact with no voltage drop across it is assumed. Since a voltage V is applied between the two ohmic contacts:. The total current density for the diode at a bias of V volts can now be expressed as Wide base diode:.

Example 2. Calculate also the net current flowing through the device. Solution : The wide-base diode is model valid here. Using the expressions derived earlier:. Confirm that they are much smaller than minority carrier diffusion currents calculated in example 2. The minority hole current in N-region can be written using the results of previous example as:. The hole current is primarily diffusion current and the sum of hole and electron currents is equal to the total current.

The electron current on the N-side is therefore simply:. The term in the bracket is simply the hole diffusion current which has already been obtained earlier:. An electron mobility of was assumed. Let us calculate the hole drift current at the depletion edge where there is an electric field of The hole drift current is which is much smaller than the diffusion current component.

Calculate the total current flowing through the diode. Solution : This is an example of a diode that can neither be considered a fully wide-base diode nor a fully narrow-base diode. On the P-side, the diode is very thick so that we can use the expression for electron current valid for wide base diodes. Therefore as before. On the N-side so that the narrow-base model can be used. The net current will be 0. The current is predominantly determined by the narrow base side of the junction. Solution: This diode can be modeled as a narrow-base diode.

We have already calculated the hole current in example 2. The electron can similarly be calculated as. The net current will be This illustrates that for comparable doping values, narrow-base diodes provide higher current for the same bias or equivalently have a smaller turn-on voltage. The expression for current was derived on the basis of two assumptions: i negligible recombination within the depletion region.

The first assumption determines the lower limit, while the second assumption determines the upper limit. Lower limit: As stated earlier, this is determined by neglect of space charge recombination. If the hole continuity equation is integrated across the depletion region, we obtain the relation.

So what we need to do first is to get an estimate for the SCR recombination current: We shall use a simple model for the Shockley-Hall-Read recombination:. The recombination is assumed to take place via a single deep level at the midgap with equal hole and electron recombination lifetimes Within the depletion region:. Because of the exponential dependence of p and n on the voltage which varies quadratically with x , the function is a rapidly varying function of the form shown.

The recombination rate would have a peak value where the factor maximum value. The sharp variation of U implies that most of the recombination current comes from a small region around the peak value. This allows the following simplification to be performed:. If we compare this recombination current with ideal diode current, we can see two major differences: i The ideal diode current increases as while the recombination current increases.

It is for this reason that the SCR current is considered as an index of material quality because the recombination lifetime is very sensitive to fabrication conditions. The upper limit for the validity of ideal diode equation is determined by the assumption of low level injection condition. This low level injection condition will first break down for the region which has the smaller doping level.

We shall assume, for the sake of discussion, that N-region is the lightly doped region. The low level injection assumption had allowed the following simplifications to be made: i Minority carrier current is diffusive. The major departure in I-V Characteristics is caused by the breakdown of ii and iii relations because they are associated with an exponential factor.

The iii simplification amounted to neglect of the IR drop in the N-region. This drop is negligible when. The upper limit for the validity of the ideal equation is then: for wide base diode 95 for wide base diode 96 , the ideal diode equation. For this example, the ideal diode equation is valid over five orders of magnitude variation of current. It is because of the wide range of validity of the final equation, that the assumptions of negligible SCR recombination and low level injection are such good assumptions! Thus the current includes an additional component due to light which represents the current due to flow of carriers generated effectively within a distance of one diffusion length of the depletion edge.

There would be an optical generation current due to generation within the depletion region as well which can be written as , where W is the total depletion width. Since depletion width is often much smaller than diffusion length, this component can be neglected.

However, in some especially designed PIN diode structures, this component is the dominant current. This however is true only if the contact can be assumed to be ideal. For practical contacts, the excess carrier density may be small but is nonzero. These contacts are characterized by a parameter called surface recombination velocity, which for holes can be defined as.

Assume a diode with. The equation above shows that as , the expression becomes identical with that derived for ideal contacts. Thus an ideal contact is one with an infinite recombination velocity. More practically when the factor , then the contact could be considered almost. The assumption that minority carrier current is largely diffusive can be shown to be true provided low level injection conditions prevail within the device:. Consequences of Low Level Injection: In the N-region: In the P-region: We will need another result before we can demonstrate the soundness of our assumption: The regions outside the space charge region are quasi-neutral so that: In the N-region:.

We will now show that the minority carrier currents can be assumed to be diffusive provided low level injection condition prevails. Although this result is general, we shall assume that the N and P regions are of comparable doping. This implies that the electron and hole currents close to the depletion edge will also be comparable. We have already shown that electron and hole diffusion currents are comparable and that for low level injection electron drift current is much larger than the hole drift current in the Nregion so that.

We shall first consider the neutral P-region and show that for low level injection conditions, the hole quasi Fermi level can be considered to be almost flat. Therefore, as long as the IR drop is sufficiently small, the hole quasi-Fermi level can be assumed to be constant. How much is sufficiently small? As shown in the main text, the expression which results from making the assumption is.

That hole and electron quasi-Fermi levels can be assumed to be flat within the depletion region can be demonstrated as follows:. Substitution of the expressions for electron and hole densities in the expression for current results in. Since most of the recombination occurs within a very narrow spatial region and electric field is a slowly varying function, it can be taken out of the integral with a value at the position of maximum recombination rate.

The required expression for current can now be obtained by substituting this expression in Eq. C5 I - V characteristics in Forward Bias. This expression was derived under the assumptions: i Low Level Injection ii Negligible recombination within the SCR Although the equation was derived in the context of forward bias, much of the derivation remains valid in reverse bias also.

In reverse bias, instead of injection of minority carriers in P and N-regions, there is extraction of minority carriers from them. Holes now flow from and electrons from. These are the conditions for low level injection if "injection" is interpreted as having a negative value in this case. The electrons diffuse from the bulk of the P-region to the edge of the depletion region after which they are swept away by the junction field. Similarly, the holes diffuse from the bulk of. The source of electrons in P-region and holes in N-region is thermal generation of carriers.

It was shown earlier, in the context of forward bias, that :. This expression is equally valid in reverse bias also, with the difference that the last term now represents generation of carriers within the space charge region, instead of recombination. In Forward bias, we had neglected this term but as we shall see, this term is the dominant term under reverse bias for Silicon PN junction diodes.

The negative sign indicates generation! Over a large fraction of the depletion width, the electron and hole densities are much smaller than the intrinsic carrier density so that. The first tem represents the current due to minority carrier diffusion and the second due to generation within the space charge region.

Example 3. Other things remaining the same, will it still be true that the reverse leakage current is dominated by generation current within the depletion region? Solution : The generation current would increase by a factor while the ideal diode saturation current would increase by a factor The two currents are now comparable.

For even smaller bandgaps, the reverse leakage current will be determined entirely by the ideal diode saturation current. The reverse current increases slowly with increase in reverse bias till impact ionization induced breakdown begins to occur within the space charge region. Impact Ionization: An electron or a hole travelling through a region of high electric field can acquire enough energy to create another electron-hole pair. It is natural to expect that the ionization coefficients would a function of carrier energy and therefore the electric field. There are a variety of models for impact ionization coefficient, simplest of which is : for Silicon As the reverse bias increases, the electric field within the junction also increases thereby increasing the probability of impact ionization.

An electron or hole generated due to impact ionization within the depletion region can acquire enough energy again to cause another impact ionization. The new electron-hole pairs generated can in turn generate further electron-hole pairs. As a result of this process, a single carrier entering the depletion region can get multiplied many times over. This process of multiplication is known as Avalanche Multiplication.

The normal reverse current gets multiplied by the avalanche multiplication process. When avalanche multiplication becomes large, very large reverse current begins to flow and breakdown is said to occur. To obtain an expression for breakdown voltage, it has to be precisely defined. This is explained using the Figure below:. Since no holes are assumed to enter the depletion region, p x must be due to impact ionization in the region.

An equal number of electrons also must have been generated also so that, the number of electrons that would come out of the depletion region must be:. For this case:. It can therefore be said that whenever the maximum electric field at the junction acquires a critical value of , breakdown would occur. Taking at breakdown allows an estimate of the breakdown voltage to be determined rapidly for any PN junction diode. Solution : We first perform a check whether at breakdown, the depletion width still lies in the lightly doped region or not.

If it does then, This shows that depletion width will extend into the higher doped N-region as well resulting in the following diagram. Solution : The question can be answered by examining the electron and hole density profiles within the depletion region generated due to impact ionization.

These are shown below:. In the junction, the electron density is maximum near the high field region at the junction and hole density is minimum. As a result most of impact ionization is done by. This is of interest because junctions have a curvature near the periphery which can be considered as cylindrical. Integration of Poisson's equation with the boundary condition that electric field at the depletion edge is zero we obtain. The expressions above can be used to find the breakdown voltage by using the fact that at breakdown, the electric field is equal to the critical field.

The table below shows the breakdown voltages computed for a doping of and different radii of curvature. The expression for multiplication factor derived earlier suggests that multiplication can be empirically modeled as. The avalanche breakdown is the most common mechanism of breakdown in PN junction diodes. There is another mechanism called Zener breakdown that comes into play in diodes with heavily doped P and N regions.

As noted earlier, in reverse bias, the holes are required to flow from the P-side to the N-side and electrons from P-side to the N-side. The reverse current is normally small because there are so few holes in N-region and electrons in P-region. However, there are plenty of electrons in valence band of P-side and plenty of empty states in the conduction band of N-side.

Except via tunneling, the electrons from the valence band of P-region cannot flow to empty states in the conduction band of N-side due to presence of a potential barrier When the probability of tunneling becomes significant, large reverse current begins to flow and Zener breakdown is said to occur. The Figure below depicts the tunneling process:. The barrier that the electron sees while tunneling, can be approximated as a triangular barrier as shown below:.

As expected, the transmission probability increases exponentially with the thickness of the barrier which can be expressed as. As doping increases, the electric field increases causing barrier to become narrower and tunneling probability to increase. To achieve significant tunneling, the barrier width should be only a few tens of Angstroms. The field calculated for Avalanche breakdown was , which is lower than that required for Zener breakdown.

It appears, therefore, that avalanche breakdown would always precede Zener breakdown! However, it is not the electric field but the carrier energy that is really important for impact ionization.


A very high electric field in a very narrow region may not allow a carrier to gain enough energy so that impact ionization becomes significant. As a result, Zener breakdown occurs in very heavily doped junctions only with small depletion widths. Because of the small depletion widths, the breakdown voltage, despite the high electric field, is often Volts. Diodes which have breakdown voltages larger than Volts break down due to Avalanche multiplication process.

In the intermediate range both the processes may be active. It is possible to determine the breakdown mechanism by measuring the temperature sensitivity of the breakdown voltage. Diodes which break down via avalanche multiplication have a positive temperature coefficient, while those that breakdown via tunneling have a negative temperature coefficient. The increase in avalanche breakdown voltage with temperature occurs due to increased scattering which makes it more difficult for carriers to acquire energy from the electric field.

The decrease of Zener breakdown voltage with increase in temperature occurs because of increased carrier velocity which increases the flux of carriers attempting to cross the barrier. Since transmission probability remains unchanged, the tunneling current increases with temperature.

Most of the diodes that go under the name Zener diodes have a breakdown via avalanche multiplication rather than tunneling. To determine the behavior of the PN junction under time varying excitation, we start from the continuity equation where time explicitly comes into picture:. By virtue of depletion approximation, the last term is zero. The second term can also be neglected because it is much smaller than the first term while ;.

The first three are familiar terms. The last term can be rewritten by noting that the junction depletion charge can be expressed as:. The last term represents the current due to time variation of the junction depletion charge The current due to variation of depletion charge can be expressed as:. An expression for junction dV capacitance can be easily obtained using the depletion approximation and will be discussed later. The net current can now be expressed as:. Let us next look at he minority carrier currents. Under low level injection approximation, they can be assumed to be diffusive so that:.

As for the static case, the computation of these currents requires determination of minority carrier profiles, which in turn requires solution of continuity equation. For the computation of p x , the hole continuity equation in N-region has to be solved:. This is where the other major difference between static and time varying characteristics comes. Unlike the static case, is no longer zero. For low level injection condition: so that integration of hole continuity equation across the N-region for a long base diode gives:.

The first term in the expression above is also present under static conditions and represents the current due to recombination of injected holes. The second term represents the current due to time variation of stored excess hole charge For the N-side also a similar expression can be written:. With these expressions for minority carrier currents, the total current after neglecting the SCR recombination current, can be expressed as:. This representation of the diode's behavior is known as the Charge Control model.

The operation of many semiconductor devices can be conceptualized as a two step process where the applied voltage modulates the charge within the device, which then modulates the current. The change in diffusion charge with the applied bias can be represented by a diffusion capacitance defined as.

To use the above equation, a relationship between the diffusion charges and capacitances and the applied bias is needed. This again requires solution of the continuity equation with the boundary condition that. This is clearly unacceptable for we know from experience that the electrical properties of a macroscopic sample do not depend on its dimensions. Much better results are obtained using the Born-von Karman boundary conditions, referred to as cyclic boundary conditions.

From the newly obtained geometry it becomes evident that for any value of x, we have the cyclical boundary conditions: Using the free-electron wave function Expression 1. In the case of a three-dimensional crystal with dimensions Born-von Karman boundary conditions can be written as follows: the 6 where Chapter 1 are integer numbers.

Energy bands of a crystal intuitive approach In a single atom, electrons occupy discrete energy levels.

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What happens when a large number of atoms are brought together to form a crystal? In a lithium atom, two electrons of opposite spin occupy the lowest energy level 1s level , and the remaining third electron occupies the second energy level 2s level. The electronic configuration is thus All lithium atoms have exactly the same electronic configuration with identical energy levels. If an hypothetical molecule containing two lithium atoms is formed, we are now in the presence of a system in which four electrons "wish" to have an energy equal to that of the 1s level.

But because of the Pauli exclusion principle, which states that only two electrons of opposite spins can occupy the same energy level, only two of the four 1s electrons can occupy the 1s level. This clearly poses a problem for the molecule. The problem is solved by splitting the 1s level into two levels having very close, but nevertheless different energies Figure 1.

Energy Band Theory 7 If a crystal of lithium containing N number of atoms is now formed, the system will contain N number of 1s energy levels. The same consideration is valid for the 2s level. The number of atoms in a cubic centimeter of a crystal is on the order of As a result, each energy level is split into distinct energy levels which extend throughout the crystal.

Each of these levels can be occupied by two electrons by virtue of the Pauli exclusion principle. In practice, the energy difference between the highest and the lowest energy value resulting from this process of splitting an energy level is on the order of a few electron-volts; therefore, the energy difference between two neighboring energy levels is on the order of eV. This value is so small that one can consider that the energy levels are no longer discrete, but form a continuum of permitted energy values for the electron.

This introduces the concept of energy bands in a crystal. Between the energy bands between the 1s and the 2s energy bands in Figure 1. In that case, a forbidden energy gap is produced between permitted energy bands. The energy levels and the energy bands extend throughout the entire crystal. Because of the potential wells generated by the atom nuclei, however, some electrons those occupying the 1s levels are confined to the immediate neighborhood of the nucleus they are bound to. The electrons of the 2s band, on the other hand, can overcome nucleus attraction and move throughout the crystal.

This implies that atoms are placed in an orderly and periodic manner in the material see Annex A4. While most usual crystalline materials are polycrystalline, semiconductor materials used in the 8 Chapter 1 electronics industry are single-crystal. These single crystals are almost perfect and defect-free, and their size is much greater than any of the microscopic physical dimensions which we are going to deal with in this chapter. In a crystal each atom of the crystal creates a local potential well which attracts electrons, just like in the lithium crystal described in Figure 1.

The potential energy of the electron depends on its distance from the atom nucleus. Equation 1. These electrons actually induce a screening effect between the nucleus and outer shell electrons, which reduces the attraction between the nucleus and higherenergy electrons. The energy of the electron as a function of its distance from the nucleus is sketched in Figure 1. How will an electron behave in a crystal? In order to simplify the problem, we will suppose that the crystal is merely an infinite, one- 1. Energy Band Theory 9 dimensional chain of atoms.

This assumption may seem rather coarse, but it preserves a key feature of the crystal: the periodic nature of the position of the atoms in the crystal. The periodic nature of the potential has a profound influence on the wave function of the electron. We will study the behavior of an electron with an energy E lying between and. This case is similar to a 1s electron previously shown for lithium. The periodic nature of the crystal lattice suggests that the wave function satisfies the Bloch theorem 1.

This system can be written in a matrix form: In order to obtain a non-trivial solution for A, B, C and D, i. Let us call this term P E and rewrite Expression 1. Because the argument in the exponential term of 1. Therefore, simultaneous solution of both left- and right-hand side of Equation 1. This defines permitted values of energy forming the energy bands, and forbidden values of energy constituting forbidden energy bands.

This important result is the same to that intuitively unveiled in Section 1. Note: In the case when the electron energy is greater than value and Equation 1. Energy Band Theory 13 Using Expression 1. Figure 1. The E k diagram for a free electron is also shown. It can be observed that the energy of the electron in a crystal coarsely represents the same dependence on k as that of a free electron.

The main differences reside in the existence of forbidden energy values and curvatures of each segment of the E k curves. The E k curves shown in Figure 1. This particular region of the k-space is called the first Brillouin zone. The second Brillouin zone extends from third zone extends from to to and from and from to to the etc.

Applying the Born-von Karman boundary conditions Expression 1. Since we limit our study to the first Brillouin zone, the kvalues which have to be considered are given by the following relationship: duplicate of the the value is excluded because it is a wave number. Therefore, the values of k to consider are: There are thus N wave numbers in the first Brillouin zone, which corresponds to the number of elementary lattice cells. For every wave number there is a permitted energy value in each energy band.

By virtue of the Pauli exclusion principle, each energy band can thus contain a maximum of 2N electrons. The one-dimensional volume of the first Brillouin zone is equal to Since it contains N k-values, the density of k-values in the first Brillouin zone is given by: 1. Energy Band Theory 15 In the case of a three-dimensional crystal, energy band calculations are, of course, much more complicated, but the essential results obtained from the one-dimensional calculation still hold. In particular, there exist permitted energy bands separated by forbidden energy gaps.

The 3-D volume of the first Brillouin zone is where V is the volume of the crystal, the number of wave vectors is equal to the number of elementary crystal lattice cells, N. The density of wave vectors is given by: 1. Valence band and conduction band Chemical reactions originate from the exchange of electrons from the outer electronic shell of atoms. Electrons from the most inner shells do not participate in chemical reactions because of the high electrostatic attraction to the nucleus. Likewise, the bonds between atoms in a crystal, as well as electric transport phenomena, are due to electrons from the outermost shell.

In terms of energy bands, the electrons responsible for forming bonds between atoms are found in the last occupied band, where electrons have the highest energy levels for the ground-state atoms. However, there is an infinite number of energy bands. The first lowest bands contain core electrons such as the 1s electrons which are tightly bound to the atoms. The highest bands contain no electrons. The last ground-state band which contains electrons is called the valence band, because it contains the electrons that form the -often covalent- bonds between atoms.

The permitted energy band directly above the valence band is called the conduction band. At higher temperatures, some electrons have enough thermal energy to quit their function of forming a bond between atoms and circulate in the crystal. These electrons "jump" from the valence band into the conduction band, where they are free to move. The energy difference between the bottom of the conduction band and the top of the valence band is called "forbidden gap" or "bandgap" and is noted In a more general sense, the following situations can occur depending on the location of the atom in the periodic table Figure 1.

In other words, electrons can leave the atom and move in the crystal without receiving any energy. A material with such a property is a metal. In case C, a significant amount of energy equal to or higher has to be transferred to an electron in order for it to "jump" from the valence band into a permitted energy level of the conduction band.

This means that an electron must receive a significant amount of energy before leaving an atom and moving "freely" in the crystal. A material with such properties is either an insulator or a semiconductor. The distinction between an insulator and a semiconductor is purely quantitative and is based on the value of the energy gap. In a semiconductor is typically smaller than 2 eV and room-temperature thermal energy or excitation from visible-light photons can give electrons enough energy for "jumping" from the valence into the conduction band.

The energy gap of the most common semiconductors are: 1. Insulators have significantly wider energy bandgaps: 9. Diamond exhibits semiconducting properties at high temperature, and tin right below germanium in column IV of the periodic table becomes a semiconductor at low temperatures. It is worthwhile mentioning that it is possible for non-crystalline materials to exhibit semiconducting properties. Some materials, such as amorphous silicon, where the distance between atoms varies in a random fashion, can behave as semiconductors. The mechanisms for the transport of electric charges in these materials are, however, quite different from those in crystalline semiconductors.

It is convenient to represent energy bands in real space instead of k-space. By doing so one obtains a diagram such as that of Figure 1. The maximum energy of the valence band is noted the minimum energy of the conduction band is noted and the width of the energy bandgap is It is also appropriate to introduce the concept of a Fermi level.

The Fermi level, represents the maximum energy of an electron in the 18 Chapter 1 material at zero degree Kelvin 0 K. At that temperature, all the allowed energy levels below the Fermi level are occupied, and all the energy levels above it are empty. In an insulator or a semiconductor, we know that the valence band is full of electrons, and the conduction band is empty at 0 K. Therefore, the Fermi level lies somewhere in the bandgap, between and In a metal, the Fermi level lies within an energy band. One can, however, represent E k along main crystal directions in k-space and place them on a single graph.

For 1. Energy Band Theory 19 example, Figure 1. Crystal A is an insulator or a semiconductor crystal B is a metal The energy band diagrams, plotted along the main crystal directions, allow us to analyze some properties of semiconductors. For instance, in Figure 1. A semiconductor exhibiting this property is called a direct-band semiconductor. Examples of direct-bandgap semiconductors include most compound elements such as gallium arsenide GaAs.

In such a semiconductor, an electron can "fall" from the conduction band into the valence band without violating the conservation of momentum law, i. This process has a high probability of occurrence and the energy lost in that "jump" can be emitted in the form of a photon with an energy In Figure 1. A, the minimum energy in the conduction band and the maximum energy in the valence band occur at different k-values. A semiconductor exhibiting this property is called an indirect bandgap semiconductor.

Silicon and germanium are indirectbandgap semiconductors. In such a semiconductor, an electron cannot "fall" from the conduction band into the valence band without a change in momentum. This tremendously reduces the probability of a direct "fall" of an electron from the conduction band into the valence band, as will be discussed in Chapter 3. Parabolic band approximation For electrical phenomena, only the electrons located near the maximum of the valence band and the minimum of the conduction band 20 Chapter 1 are of interest.

These are the energy levels where free moving electrons and missing valence electrons are found. In that case, as can be seen in Figure 1. Near the minimum of the conduction band one can thus write: Near the maximum of the valence band one can write: with A and B being constants. This approximation is called the "parabolic band approximation" and resembles the E k relationship found for the free electron model.

Concept of a hole To facilitate the understanding of electrical conduction in a solid one can make a comparison between the flow of electrical charge in the energy bands and the movement of water drops in a pipe. Let us consider Figure 1. A two pipes which are sealed at both ends. The bottom pipe is completely filled with water and the top pipe contains no water it is filled with air.

In our analogy between electricity and water, each drop of water corresponds to an electron, and the bottom and top pipes correspond to the valence and the conduction band, respectively. When the filled or empty pipes are tilted, no movement or flow of water is observed, i. Thus the semiconductor behaves as an insulator Figure 1. Let us now remove a drop of water from the bottom pipe and place it in the top pipe, which corresponds to "moving" an electron from the valence to the conduction band. If the pipes are now tilted, a net flow of liquid will be observed, which correspond to an electrical current flow in the semiconductor Figure 1.

The water flow in the top pipe conduction band is due to the movement of the water drop electron. In addition, there is also water flow in the bottom pipe valence band since drops of water can occupy the space left behind as the air bubble moves. It is, however, easier to visualize the motion of the bubble itself instead of the movement of the "valence" water.

Energy Band Theory 21 If, in this water analogy, an electron is represented by a drop of water, a bubble or absence of water in the "valence" pipe represents what is called a hole. Hence, a hole is equivalent to a missing electron in the crystal valence band. A hole is not a particle and it does not exist by itself. It draws its existence from the absence of an electron in the crystal, just like a bubble in a pipe exists only because of a lack of water.

Holes can move in the crystal through successive "filling" of the empty space left by a missing electron. The fact that the electron is in a crystal will influence its response to an applied force. As a result, the apparent, "effective" mass of the electron in a crystal will be different from that of an electron in a vacuum.

In the case of a free electron Relationship 1. The mass is a constant since E is a square function of k. Using the rightmost term of 1. Unlike the case of a free electron the effective mass of the electron in a crystal is not constant, but it varies as a function of k Figure 1. Additionally, the mass in the crystal will be different for differing energy bands. The following general observations can be made: if the electron is in the upper half of an energy band, its effective mass is negative if the electron is in the lower half of an energy band, its effective mass is positive 1.

Energy Band Theory 23 if the electron is near the middle of an energy band, its effective mass tends to be infinite The negative mass of electrons located in the top part of an energy band may come as a surprise, but can easily be explained using the concept of a hole. In the case of silicon the mass of electrons near the minimum of the conduction band along the direction is equal to and in the orthogonal directions it is is called the longitudinal mass and the transversal mass, while is the mass of a free electron in a vacuum.

These masses are related to the energy by the following relationship called "parabolic energy band approximation": where is the lowest energy state in the conduction band along the [] or [] Figure 1. In most practical cases, for the sake of simplicity, the effective mass is considered to be constant.

A There are two vectors and which correspond to a same energy value In a two-dimensional crystal Figure 1. B the locus of values corresponding to the energy level is an ellipse in the plane. The three-dimensional case cannot be drawn on a sheet of paper, but extrapolating from the 1D and 2D cases it is easy to conceive that the k values corresponding to the energy level form ellipsoids in the space Figure 1. In a three-dimensional crystal such as silicon there are 6 equivalent crystal directions [], [], [], [], [] and [] which present an energy minimum conduction band minimum.

The locus of k-values corresponding to a particular energy value is 6 ellipsoids Figure 1. The center of these ellipsoids are the six k-values corresponding to the conduction band energy minima. For simplification the ellipsoids can be approximated by spheres Figure 1. D , which is equivalent to equating the transverse and the longitudinal mass The energy in the vicinity of the maximum of the valence band is given by: 1. Energy Band Theory 25 1. Density of states in energy bands The density of permitted states in a three-dimensional crystal is given by 1.

Its value is: per crystal unit volume. If we define f k as the probability that these states are occupied, then the electron density, n, in an energy band can be calculated by integrating the product of the density of states by the occupation probability over the first Brillouin zone: Similarly, the density of holes within an energy band is given by: The function n k represents the density of permitted states in an energy band.

The function f k is a statistical distribution function which is a 26 Chapter 1 function of the energy, Under thermodynamic equilibrium conditions, f k is the Fermi-Dirac distribution function defined as:[11] where is an energy value called the "Fermi level", k is the Boltzmann constant, and T is the temperature in Kelvin. The Fermi-Dirac function is plotted in Figure 1. In order to integrate Expressions 1. To do this, let us consider a unit cell of the reciprocal crystal lattice where and are given by Relationship 1.

D : The number of unit cells in that volume is given by the volume of the shell divided by the unit volume of the cell: The number of k vectors and thus the number of energy levels, since there is an energy level for each k vector is equal to the number of unit cells. Using the Pauli exclusion principle which states that there can be only 2 electrons for each k vector , the number of electrons is given by: Using the parabolic band approximation, constant effective mass, one obtains: and using a 1.

In the case of electrons with a mass located near the bottom of the conduction band, the energy is referenced to the minimum of the conduction band which yields: In the case of holes with a mass located near the top of the valence band, the energy is referenced to the maximum of the valence band and one obtains: Integration of Equations 1. To integrate, a change of variables can be used where which yields: is called the "effective density of states in the conduction band". It represents the number of states having an energy equal to which, when multiplied by the occupation probability at yields the number of electrons in the conduction band.

Likewise the total number of holes in 1. Energy Band Theory 29 the valence band can be calculated using this technique, based on Equation 1. The effective density of states for holes in the valence band is: The density of holes and electrons in the conduction and valence bands is shown in Figure 1. C for a Fermi level at midpoint of and 1. Intrinsic semiconductor By virtue of Expressions 1. A semiconductor is said to be "intrinsic" if the vast majority of its free carriers electrons and holes originate from the semiconductor atoms themselves.

In that case if an electron receives enough thermal energy to "jump" from the valence band to the conduction band, it leaves a hole behind in the valence band. Thus, every hole in the valence band corresponds to an electron in the conduction band, and the number of conduction electrons is exactly equal to the number of valence holes: and or, if simplifying approximation : where 30 Chapter 1 where is called the intrinsic energy level.

It is the energy of the Fermi level in an intrinsic semiconductor. One can generally consider that it lies right in the middle of the energy bandgap Expression 1. However, the variation of with temperature is illustrated in Figure 1. When temperature is raised an increasing number of electron gather sufficient thermal energy to leave the semiconductor atoms and become free to move in the conduction band. These electrons are called "free electrons". Since they can move in the crystal they can contribute to an electrical current.

An equal number of "free holes" can move in the crystal and contribute to an electrical current as well. The conductivity of a material directly depends on the number of free carriers it contains free electrons and free holes : the larger the number of carriers, the higher the conductivity. Thus, the conductivity of an intrinsic semiconductor increases with temperature Figure 1.

Using equations 1. Energy Band Theory 31 1. Extrinsic semiconductor The silicon used in the semiconductor industry has a purity level of One can, however, intentionally introduce in silicon trace amounts of elements which are close to silicon in the periodic table, such as those located in columns III boron or V phosphorus, arsenic. If, for instance, an atom of arsenic is substituted for a silicon atom, it will form four bonds by sharing four electrons with the neighboring silicon atoms Figure 1.

The thermal energy of the crystal at room temperature is large enough to remove the loosely held fifth electron from the arsenic's outer electronic shell, such that this electron will now reside in the conduction band where it is free to move in the crystal. Arsenic atoms in silicon are called donor atoms because each of these atoms "donates" an electron to the crystal. The free electron can contribute to electrical conduction.

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Similarly, substituting a silicon atom with an atom from the third column of the periodic table, such as boron, will result in a missing electron Figure 1. The boron atom can easily capture an electron to form a fourth bond with silicon atoms, thereby creating an immobile negatively charged boron atom. This releases a hole in the crystal, located in the valence band. This hole can move about in the crystal, thereby participating in electrical conduction. Because in silicon group III atoms create a hole which can be "filled" with an electron, these atoms are called acceptor atoms.

Such atoms are usually introduced into the semiconductor 32 Chapter 1 in very small amounts 1 atom of boron per atoms of silicon, for instance. We will see later that the introduction of even minute amounts of these impurities dramatically modify the electrical properties of a semiconductor. Atoms possessing the property of releasing or capturing electrons in a semiconductor are indiscriminately called doping impurities, doping atoms, or dopants.

The introduction of a donor atom such as phosphorus P or arsenic As in silicon gives rise to a permitted energy level in the bandgap in Figure 1. This level is located a few meV below the bottom of the conduction band, and at very low temperature contains the electrons which can be given by the impurity atoms to the crystal. Similarly, the introduction of an acceptor atom such as boron B in silicon gives rise to a permitted energy level in the bandgap. This level is located a few meV above the top of the valence band.

At room temperature electrons in the top of the valence band possess enough thermal energy to "jump" into the energy levels created by the impurity atoms or: valence electrons are "captured" by acceptor atoms , which 1. Energy Band Theory 33 gives rise to holes in the valence band. These holes are free to move in the crystal. When an electron is captured by an acceptor atom, a hole is thus released in the crystal, and the acceptor atom boron becomes ionized and carries a negative charge, -q, as shown in Figure 1.

Donor and acceptor impurities are commonly introduced into semiconductors to increase electron or hole concentrations, which modifies the electrical properties of the material. The energy levels created in the bandgap by the presence of such impurities are situated close to the top of the valence band or the bottom of the conduction band. Other elements, such as gold, iron, copper and zinc introduce one or several energy levels in the bandgap of silicon. These levels are located closer to the center of the bandgap and are called "deep levels". The latter usually have a detrimental effect on semiconductors, which is why the semiconductor industry uses crystals having a very high degree of purity.

The influence of deep levels on the properties of semiconductors will be discussed in Section 3. A semiconductor containing donor impurities is called an N-type semiconductor, since most of the carriers have a negative charge, and a semiconductor containing acceptor impurities is called a P-type semiconductor, since most of the carriers have a positive charge. The concentration of donor and acceptor atoms in the semiconductor are labeled and respectively, and are expressed in atoms per cubic centimeters Thus, an N-type semiconductor has more free electrons than holes, and vice-versa.

However, the material itself is charge neutral due to the ionized impurities which carry a charge equal and opposite to that of the free carriers. If a doping atom is not ionized, it does not release a free carrier in the crystal, and therefore, does not contribute to electrical conduction. Consider a donor impurity, such as arsenic in silicon. The ionization of the arsenic atom is a reversible process which can be written as: where represents a non-ionized arsenic atom, and an ionized atom.

Quite naturally the total impurity concentration is equal to the sum of the ionized and non-ionized impurity concentrations: The probability of occupancy of the donor level, can be obtained by substituting for E in the Fermi-Dirac distribution function. Previously Equation 1. In other words, each energy level could be populated with two electrons. In this case, however, an ionized arsenic atom can receive only one electron.

Example: Consider the following numerical example in silicon: - 50 meV assuming the doping concentration is very low 1. Electron-hole equilibrium Consider a semiconductor crystal containing both N-type and P-type impurities. Because the crystal is charge neutral one can write: Chapter 1 36 As we have seen in the previous Section all doping impurities are ionized at room temperature, therefore, and can thus be re-written in the following form: Relationship 1. Energy Band Theory 37 released by doping impurities. In that case, and the semiconductor is intrinsic even though it is doped.

The influence of high and low temperatures on carrier concentration is illustrated by Problem 1. In an N-type semiconductor the Fermi level is located in the upper half of the bandgap, above the intrinsic energy level, The Fermi level increases logarithmically with the donor atom concentration, It is now possible to introduce a new variable, the Fermi potential, unit: volt.

It is defined by the following relationship: Using Equation 1. Combining Relationships 1. In a P-type semiconductor the Fermi level is located in the lower half of the bandgap, below the intrinsic energy level, The Fermi level decreases with increasing acceptor atom concentration, 1. Energy Band Theory 39 Using Equation 1. A graphical representation of electron and hole concentrations for both N- and P-type semiconductors is shown in figure 1.

Note the position of the Fermi level, and the asymmetry of carrier densities for both types. Degenerate semiconductor We have hitherto assumed that the introduction of doping impurities in a semiconductor does not affect certain intrinsic parameters of the crystal, such as the width of the energy bandgap. As we have seen before the presence of donor doping atoms such as phosphorus or arsenic introduces a permitted energy level, in the bandgap.

Typical doping concentrations are in the to range, which is small compared to the actual number of semiconductor atoms in silicon. As a result the width of the bandgap is reduced from to in Figure 1. Such a semiconductor is called a "degenerate" semiconductor or a "degenerately doped" semiconductor. A degenerate semiconductor exhibits electrical properties similar to those of a metal. Alignment of Fermi levels Often, the doping concentration in a semiconductor is not one constant value throughout the material.

Consider a piece of N-type semiconductor in which the doping concentration varies along one direction of space, x. The concentration of doping atoms is described by the function shown in Figure 1. Consider now that leftmost and rightmost parts of the sample are separated.

According to Relationship 1. Imagine a test energy level in the bandgap having an energy, located between and In the left part of the sample the test level has a low probability of being populated with an electron, because In the right part of the sample, on the other hand, the test level has a high probability of being populated with an electron, because Let us now consider the entire sample, and in particular, focus on the middle region where the doping concentration changes abruptly.

If the energy bands near stay as they are in the leftmost and rightmost parts of the sample, the test level will have both a high and a low probability of being occupied by an electron, which is a contradiction in itself. The test level must have a single occupation probability. This condition can be satisfied only if at the immediate left of is equal to at the immediate right of And since this condition must be true for any arbitrary position along the x-axis, the Fermi level must be unique and constant throughout the sample.

This is a very important property of the Fermi level, which can be enunciated the following way: a t thermodynamic equilibrium the Fermi level in a structure is unique and constant. This property not only applies to non-homogeneously doped semiconductors, but to metal-semiconductor structures and contacts between different semiconductors. Because is constant the conduction, valence, and intrinsic levels bend within a transition region around Figure 1. Energy Band Theory 41 Under thermodynamic equilibrium conditions electrons are transferred from the electron-rich right part of the sample where the Fermi level is highest into the electron-poor left part of the sample where the Fermi level is lowest , through a diffusion process which will be discussed in Chapter 2.

To make a comparison with fluid mechanics the alignment of the Fermi levels in the sample is similar to the alignment of the water levels in glasses of water connected together Figure 1. The diffusion process electron transfer or water transfer ceases when an equilibrium state is reached.

Since Relationships 1. The magnitude of this energy level bending reflects the presence of an internal potential, noted which, once multiplied by -q, is equal to the variation of the energy levels and between the left and the right of the sample Figure 1. The internal potential is a real electrical potential variation due to the 42 Chapter 1 appearance of an electric field in the semiconductor caused by the charge imbalance resulting from the diffusion of electrons from one part of the semiconductor to the other when thermodynamic equilibrium is established. Since the electron concentration is related to by Relationship 1.

It is easy to show that an equivalent relationship can be derived for holes: Relationships 1. They will play an important role in the theory of the PN junction Chapter 4. Problem 1. The unit for energy in the plot must be electron-volts eV. The unit for the wave function is Hint: it is possible to plot different units eV and wave function unit on the same y-axis, but if you do it as such, the wave functions and energy levels will have magnitudes with such difference that the wave functions will be much larger compared to the energy levels.

Therefore, the amplitude of the wave functions must be divided by , in order to get a "nice-looking" graph. The first of those is the classical particle-in-a-box problem. In its discrete form, the second-derivative operator can be written: where in the right-hand side of the latter expression is the constant distance between two successive mesh points. Then the wave functions must be sorted by ascending energy values, and the wave function corresponding to the lowest energy value is finally plotted.

In Figures 1. The answer should have the form: Problem 1. Arsenic atoms introduce a donor energy level at 0. Using the results of Problem 1. Energy Band Theory 49 References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 J. Moll, Physics of semiconductors, McGraw-Hill, pp. McKelvey, Solid-state and semiconductor physics, Harper International, p. A, p. Ashcroft, N. Mermin, Solid-state physics, Holt, Rinehart and Winston, p. Kazmerski, Polycrystalline and amorphous thin films and devices, Academic Press, pp.

Wolf, Semiconductors, J. Wiley and Sons, p. Muller and T. Kamins, Device electronics for integrated circuits, 2nd edition, J. McKelvey, Solid-state and semiconductor physics, Harper International, pp. IV, Addison-Wesley, p. Electrons and holes are no longer treated separately, but are considered as macroscopic carrier populations or carrier concentrations. As a result the use of quantum mechanics is no longer required. Rather, Maxwell's equations and concepts such as the conservation of charge and the diffusion resulting from concentration gradients will be used.

Drift of electrons in an electric field The electrons we have considered so far were found in ideal crystals with perfectly periodic potential variations. Actual crystals contain defects such as interstitials and vacancies due to displaced or missing atoms, and trace impurities.

Furthermore, the atoms vibrate around their equilibrium position. The amplitude of these vibrations depends, among others, on temperature. These vibrations can be studied formally using quantum mechanics. From the study of these vibrations emerges the concept of a phonon. The phonon is a quasi-particle representing the propagation of vibration -or heat- through the crystal.

The interaction between a free electron and phonons or crystal defects can be viewed as a series of collisions obeying the principles of conservation of energy and momentum. The trajectory of electrons is thus a series of random velocity vectors. In the absence of an applied external force all these small movements average out and the net displacement of the electron is zero, as shown in Figure 52 Chapter 2 2.

When an electric field is applied, on the other hand, a net drift of the electron in the opposite direction of the electric field is observed Figure 2. It is worthwhile noting that the random thermal velocity of electrons is much larger than the velocity produced by imposing an electric field. To obtain the current flow resulting from this process one must calculate the average drift velocity of the electrons caused by the electric field.

The analogy with Brownian motion in a liquid allows us to write two hypotheses concerning the motion of an electron: Each electron in the conduction band moves freely in the crystal between each collision. The average time between two collisions is called "relaxation time", and is noted The relaxation time for electrons in a semiconductor is on the order of a tenth of a picosecond at room temperature, during which the electron can travel on the order of 10 nanometers. The direction of the electron motion after a collision is random.

Collision events, are therefore, isotropic. Among all the electrons in the conduction band there are electrons which, at the instant undergo a collision event. Let us follow the evolution of this electron population. At some of these electrons will already have undergone new collisions. Therefore, at there is a smaller number of electrons, n t , which have not yet undergone a collision event.

Theory of Electrical Conduction 53 Integrating this equation between and t the evolution of the number of electrons that have not undergone a collision since can be obtained: Let us now describe the influence of a time-independent electric field, on an electron. The equation for the movement of a quasi-free electron with an effective mass is: Using Expression 2. This relationship is valid for the -dn dn 2. Mobility Using Relationship 2. The unit for the mobility velocity divided by an electric field is Using Relationship 2.

Since mobility is proportional to the relaxation time it decreases with temperature because thermal lattice vibrations -or phonons- increase with increasing temperature. Similarly, impurities and defects cause electron scattering collisions , and therefore, mobility decreases with increasing impurity or defect concentration.

A similar derivation can made for holes in the valence band and yields: where is the hole mobility which is defined by: The actual effective mass of electrons and holes is anisotropic see Relationship 1. Because of the cubic symmetry in Si, Ge or GaAs crystals, one can, however, use a scalar expression for the effective mass, defined by: where is called "conductivity effective mass".

In silicon the conductivity effective mass of electrons is equal to and 2. Theory of Electrical Conduction that of holes is vacuum. A more thorough analysis of the scattering of electrons by phonons yields the following dependence of mobility on temperature: and the dependence of mobility on impurity concentration, N: When the dependence on both temperature and impurities is taken into account, the mobility, is given by: Equation 2.

The mobility of electrons and holes in different semiconductors is shown in Figure 2. The resistivity can range from to By comparison, the resistivity of metals is on the order of and that of typical insulators is around 2. Theory of Electrical Conduction 57 2.

Hall effect According to Relationship 2. The conductivity of the sample can readily be measured, using an ohmmeter, for instance. The carrier concentration and the mobility can be separated by performing an additional measurement based on the Hall effect. When a magnetic field, is applied perpendicular to the direction of the carrier flow in a semiconductor sample a potential difference appears in the direction perpendicular to both the current flow direction and the direction of the magnetic field Hall effect, Let us examine the motion of electrons in a piece of N-type semiconductor under the combined effect of a longitudinal electric field, and of a magnetic field, perpendicular to it Figure 2.

The current density in the y-direction, is given by Equation 2. Each electron in motion is submitted to a Lorentz force having a magnitude equal to in a direction, x, perpendicular to both the electron velocity, thus also to and to Since no current can flow in the x-direction a transverse electric field which exactly counteracts the Lorentz force, is created, such that : 58 Chapter 2 If the width of the sample is W, a potential difference which can be measured, called "Hall voltage" will appear at the sides of the sample: If the thickness of the sample is h the current flowing in the y-direction is equal to: One defines the "Hall coefficient", which characterizes the combined effect of an electric field and a magnetic field on electrons by the following relationship: [5] Since the magnetic field is perpendicular to the direction of current flow the latter Equation can be rewritten in the following form using 2.

Once the mobility is known, Relationship gives access to the electron concentration. In the case of a P-type semiconductor, one finds: In conclusion the Hall effect allows the determination of the polarity of a semiconductor N- or P-type through the sign of the Hall coefficient.

In addition, when combined with a conductivity measurement it allows for the extraction of the majority carrier density and the majority carrier mobility. Theory of Electrical Conduction 59 2. Diffusion current In semiconductors current can be produced due to a concentration gradient of carriers. The current in this case is called diffusion current and is derived below. Consider a piece of semiconductor in which, for whatever reason, there is an electron concentration gradient. By analogy with the laws of diffusion in gases or liquids one can easily conceive that electrons will diffuse from the region where their concentration is highest to the region where it is lowest.

This flux, when multiplied by -q, is equal to the diffusion current density of the electrons: In a similar way a hole concentration gradient gives rise to a hole diffusion current. They represent the ease or the "fluidity" with which the carriers can move and diffuse in the semiconductor material. Drift-diffusion equations Based on the concepts derived in the previous sections we can now establish the drift-diffusion equations. The total hole current density in a semiconductor is composed of the sum of the drift and the diffusion components of current.

Similarly, the total electron current density in a semiconductor is composed of the sum of the drift and the diffusion components of current. Using 2. Einstein relationships The mobility and diffusion coefficient in a semiconductor are related to each other. This relationship is derived in the following section. Consider a piece of semiconductor material with a non-uniform doping concentration. Let the doping atoms be arsenic in silicon and for the sake of simplicity we will consider a one-dimensional case.

The doping impurities are N-type and their concentration is as shown in Figure 2. Assuming all doping impurities are ionized, we have that The presence of an electron concentration gradient gives rise to an electron diffusion current. The electrons diffusing to the left "leave behind" positively charged arsenic atoms. These atoms occupy substitutional sites in the crystal lattice, and unlike electrons, cannot move. Because of the increased number of electrons in the left-hand part of the sample and the presence of positive charges in the right-hand part an internal electric field develops locally.

This electric field tends to "recall" the electrons towards their place of origin. This electric field and the associated potential drop are noted where the 2. Theory of Electrical Conduction 61 subscript zero implies an internal or "built-in" field under thermal equilibrium. With no external bias applied to the sample there is no current flow and the force of the internal electric field exactly balances the diffusion force.

Using the drift-diffusion equation 2. It is equal to Transport equations The transport equations are a set of five equations that govern the behavior of semiconductor materials and devices. In the previous section we have related the flow of current to drift and diffusion mechanisms. The first two transport equations are the drift-diffusion equations given by Relationships 2.

FFFN30/FYST15 - Semiconductor Physics

If all the doping atoms are ionized, which is the case at room temperature, one obtains: The permittivity of a material is given by the product of its relative multiplied by the permittivity of permittivity or dielectric constant, 2. Theory of Electrical Conduction 63 vacuum where the permittivity of vacuum is equal to For example, silicon, which has a dielectric constant of Another set of equations which describe the evolution of carrier concentration with time can be derived.

However, the local carrier concentration may vary for the following reasons: External forces can be applied to a region of the semiconductor material such that carriers are either added to or removed from that region i.

The width of the bandgap in a semiconductor is small enough to allow for electrons to "jump" from the valence band into the conduction band and reciprocally. In addition, electrons can also "jump" from the conduction or valence band into permitted energy levels located inside the bandgap. These levels arise from the presence of trace impurity elements or crystalline defects. If, for instance, an electron jumps from the valence band into the conduction band, it becomes free to move in the crystal. At the same time, a free hole is created in the valence band, which is free to move as well.

Such an event is called "carrier pair generation" or, more simply, "generation". An electron can also "fall" from the conduction band into the valence band. In this process called "recombination" both a free electron and a free hole are lost. An external source energy can increase the hole and electron concentration. If enough energy is transferred to an electron in the valence band, it can "jump" into the conduction band, a process by which a free electron-hole pair is created. The external generation rates for electrons and holes are noted and respectively unit: A typical example where external generation is useful is the conversion of sun light into electrical energy in a solar cell.

If the rates of spontaneous generation and recombination are equal, both and are equal to zero. The extrinsic generation rates express the rate at which free electrons and holes are created by an outside source of energy, such as light illumination. Extrinsic generation involves only generation i. The cross-sectional area of the volume under consideration is A with length dx. An electron current density unit: enters the volume and a current density flows out of it. For one-dimensional current flow in the x-direction the variation of the number of free electrons in the volume Adx as a function of time is given by the number of electrons entering the volume, minus the number of electrons flowing out of the volume, plus the number of electrons generated minus the number of electrons recombined: 2.

Theory of Electrical Conduction 65 can be developed in series, which yields: Using the latter result Equation 2. Quasi-Fermi levels At thermodynamic equilibrium, and in the absence of applied external forces, the equilibrium carrier concentrations are a function of the internal potential in the semiconductor. The carrier concentrations are related to the internal potential by the Boltzmann relationships 1. These can be rewritten in the following form: 66 Chapter 2 and the pn product is given by: Under thermodynamic equilibrium conditions the Fermi level, unique for both electrons and holes.

For instance when excess carriers are continuously injected into the semiconductor material or if light is continuously shone on it, the relationship between the internal potential and the electron and hole concentrations, n x,y,z and p x,y,z becomes more complicated. The Boltzmann relationships, however, are still valid if one introduces the notion of "quasi-Fermi levels".

Quasi-Fermi levels are also called "imref", which means "imaginary reference", and quite conveniently, corresponds to the word "Fermi" spelled backwards. Instead of a single Fermi level common to both types of carriers let us define an electron quasi-Fermi level, and a hole quasi-Fermi level, The Boltzmann relationships can be rewritten in the following form: and the pn product is equal to: From Equation 2. Theory of Electrical Conduction Using the Einstein Relationship 67 we finally obtain: A similar calculation, made for holes, would yield: The two last relationships show that, in the most general case, the current is not linked to the gradient of the internal potential, but to the gradient of the quasi-Fermi levels.

Under thermodynamic equilibrium conditions and in the absence of external forces, however, constant, and therefore, Important Equations and 68 Chapter 2 Problems Problem 2. The intrinsic carrier concentration, is equal to Calculate the electron and hole concentration as well as the position of the Fermi level. Problem 2. Theory of Electrical Conduction 69 1: Assume that no external bias is applied to the sample. At time an arbitrary distribution of charge is injected into the sample: Show that excess charge will vanish exponentially as a function of time, and that the time constant is: where is the conductivity of the silicon.

On its left side there is a metal electrode which is separated from the silicon by a thin layer of air air is an insulator! Hint: This is similar to a parallel-plate capacitor. Since it is very difficult to solve Poisson's equation analytically for such a charge density, the so-called "depletion approximation" where the charge density is assumed to be equal to over a given distance, w, can be used.

Beyond w, the silicon remains neutral. In other words, we have: for 0 w Problem Figure 2. Theory of Electrical Conduction a: Plot where 71 n x and p x for 0 curves. From Relationship 1. For the y-axis choose either a linear or a logarithmic scale, whichever is most appropriate. Explain your results.

Since the left and right terms of the latter equation are both functions of the potential, iterations must be used until acceptable accuracy is reached see Annex 5. Chose an appropriate criterion for convergence. Between a and b the charge is given by the following expression: References 1 2 3 4 5 6 Ch.

Kittel, Introduction to solid-state physics, 6th Edition, J. Sze, Physics of semiconductor devices, J. Kamins, Device electronics for integrated circuits, J. Wiley and Sons, pp. Introduction As mentioned earlier there are electrons in the conduction band and holes in the valence band of a semiconductor, as long as the temperature is above zero Kelvin.

An electron in the conduction band is free to move in the crystal. It can also "jump" into a "vacant seat" in the covalent bond network Figure 3. This "vacant seat" is, of course, nothing but a hole. By doing this the electron releases energy. Such a phenomenon in which a free electron and a free hole both disappear is called a recombination event. Conversely, an electron can free itself from a covalent bond if enough energy is made available.

By doing this it "jumps" from the valence band into the conduction band and becomes free to move in the crystal. A free hole is also created in that process, which is called "generation of an electron-hole pair" Figure 3. Using an external source of energy such as illumination with light, one can, however, increase the carrier concentration and reach a state of non-equilibrium. Direct and indirect transitions In a semiconductor such as gallium arsenide GaAs the conduction band minimum where free electrons are located occurs at the same kvalue k is the wave vector as the valence band maximum.

The wave vector represents the momentum of the carriers. As shown in Figure 3. Therefore, when an electron from the conduction band recombines with a hole in the valence band the law of conservation of momentum is obeyed. A semiconductor where the minimum of the conduction band and the maximum of the valence band occur at the same k-value is called a direct-bandgap semiconductor, and the "jump" of an electron from the conduction band into the valence band is called "band-to-band recombination".

When a recombination event takes place the law of conservation of energy also implies that a quantum of energy is released in the form of a photon. The energy of that photon is such that where h is Planck's constant, v is the frequency of the photon, and is the bandgap energy. In most direct-bandgap semiconductors the photons emitted by recombination events have an energy corresponding to visible or near-infrared light. A recombination event where photons are emitted is called "radiative recombination" and is exploited in devices such as light-emitting diodes. In silicon and germanium the minimum of the conduction band and the maximum of the valence band do not occur at a same k-value.

A semiconductor where this is the case is called an "indirect-bandgap semiconductor". This can occur only if an appropriate momentum is transferred to the electron or the hole such that conservation of momentum is observed. This can happen through collision with a phonon or with several phonons. Since a precise value of momentum in Figure 3.

As a result there is no radiative recombination in silicon and germanium, and these materials cannot emit light. Rather recombination takes place via trap levels at various k-values within the band gap. Gallium arsenide emits photons with a wavelength of which corresponds to near-infrared, almost visible light. To fabricate semiconductor devices producing visible light more complex semiconductor materials are used, usually based on a combination of the 76 Chapter 3 elements of columns III and V of the periodic table, such as Ga, Al, P, As, and N.

Such semiconductors are called "III-V semiconductors". The main parameter that governs the electrical and optical properties of semiconductors is the bandgap energy, shown in Figure 3. The use of ternary compound semiconductors, such as or that of quaternary compounds, such as allows one to tailor the bandgap energy in order to produce a desired light wavelength.

The fabrication of a semiconductor material with an "engineered" bandgap energy is obtained, for example, by adjusting the x and y coefficients during the growth of a crystal. Semiconductors are transparent to photons that carry an energy, hv, smaller than the bandgap energy. Germanium, for instance, is used instead of glass to make infrared IR lenses for wavelengths larger than since its bandgap energy is larger then the energy of IR photons.

Photons with an energy equal or greater than the semiconductor bandgap energy, on the other hand, can be absorbed to generate electron-hole pairs. Figure 3. The absorption coefficient is a measure of the distance a light wave travels into the material before it is absorbed. In addition to band-to-band recombination mechanisms, a free electron can recombine with a free hole through "recombination centers" located within the energy bandgap.

These are permitted energy levels introduced 3. A recombination center acts as a catalyst that enables an electron to recombine at k values differing from the of the conduction band. This is especially true in indirect-bandgap semiconductors such as silicon or germanium, where band-to-band recombination events are very unlikely to occur. They contain some crystal defects such as interstitials excess semiconductor atoms in the crystal lattice , vacancies missing semiconductor atoms in the crystal lattice and dislocations imperfections in the crystal structure , as well as traces of impurity elements such as metallic atoms or oxygen.

These defects and impurities give rise to permitted levels within the energy bandgap. Let us consider one of these levels, having an energy within the bandgap. This permitted level can receive an electron from the conduction band case A in Figure 3. A level that is neutral if filled by an electron and positive if empty is called a "donor level", and a level that is neutral if empty and negative if filled by an electron is called an "acceptor level".

Permitted levels inside the bandgap are called generation-recombination centers, or, in short, "recombination centers". In Figure 3. Since these transitions involve energies smaller than that of the bandgap they are much more likely to occur than band-to-band transitions, especially in indirect-bandgap semiconductors like silicon or germanium. It is important to note that the terms and in the continuity equations 2. Natural, intrinsic generation in a semiconductor arising at any temperature above zero Kelvin, is encompassed in the intrinsic recombination-generation rate terms of the continuity equations, and Using the notations of Figure 3.

If it is negative, a net generation of carriers is observed. The energy released by a recombination event can give rise to different phenomena: In a band-to-band radiative recombination event, the energy is released in the form of a photon. In an Auger recombination event the energy released is transferred to another electron or hole , which becomes excited to a higher energy level.

In an indirect recombination event via an energy level within the bandgap, energy is transferred to the crystal lattice in the form of heat or phonons. Recombination of carriers takes place not only within the bulk of a semiconductor crystal, but at its surface as well. The surface is indeed a place where the periodicity of the crystal lattice is interrupted, and where contact with another substance air, metal, Within the bulk of the crystal a recombination-generation rate, or, in short, a recombination rate, is defined.

The recombination rate for electrons is noted 3.