chapter 2 - welcome to bugianens'...

Chapter 2 ndash Physics of semiconductor detectors

1

CHAPTER 2 PHYSICS OF SEMICONDUCTOR DETECTORS


21 ndash Interaction of electromagnetic radiation with matter

211 - Overview

Energy moving through space is identified with the name of electromagnetic radiation and it is characterized by quantity of energy E speed c frequency ν and wavelength λ with which is moving These quantities are all correlated together by the following equations (where h is Planck constant1 and c is the speed of light in vacuum2)

If one of these quantity is known it is so possible to reach for all the others (the factor hc occurs so often in atomic and nuclear physics that it can be considered as a separate constant3)

Different values of energy frequency and wavelength creates the flavours of electromagnetic radiation but difference between them is evident only after the interaction with matter when they show particle-like behaviour out of wave-light behaviour Hence in the definition of radiation the charged particles are included (such as alpha and beta radiation beams of charged particles created by accelerating machines electromagnetic radiation or photons and beams of neutral particles such as neutrons)

This chapter is meant to describe the basic physics that stands behind interaction of radiation and particles with matter what are its consequences and how these principles are applied in semiconductor silicon detectors technology

22 - Electromagnetic and particulate radiation

The principal types of radiation can be first divided into two main categories electromagnetic (X-rays produced outside the nucleus and γ-rays emanated from within nuclei) and particulate (α particles protons neutrons electrons β- positrons β+) This distinction as already mentioned belongs to the proper ldquohistoryrdquo of the radiation drawn by the history of the particle (subject connected to the concepts of energy loss of a particle range interactions) and by the history of the target atoms (that leads to displacements recombination ionization excitation radiation damage and build-up concepts) A beam of radiation that passes through matter can lead to the complete absorption (electronic transitions and vibration-rotational transitions) to some scattering (Rayleigh Rutherford Raman and Mie scattering) andor to the passage with no interaction These processes can be explained in terms of interactions between particles that are stopped or scattered The basic effect of the interaction can be the scattering absorption thermal emission refraction and reflection of the incoming radiation

With the absorption and emission spectra (of molecules) it is possible to outline characteristic structures and so to identified and quantified molecules by these lsquofingerprintsrsquo The spectra are determined by position (wavelength) of absorptionemission line knowing the difference of energy levels of the transition 1 413x10-18 keVsec2 3x108 msec3 124 eVmicrom = 1240 MeVfm

2


and by strength of absorptionemission line knowing the probability of the transition The most commonly used transition is the electron transition in the atoms and vibration-rotational modes in the molecules

Moreover a particle travelling through matter can lose energy gradually (losing energy nearly continuously through interactions with the surrounding material) or catastrophically (moving through with no interaction until losing all its energy in a single last collision) Gradual energy loss is typical of charged particles whereas photon interactions are of the all-or-nothing kind

23 - Photon interactions with matter

First the all-or nothing type interactions are considered

231 - Attenuation coefficients

The description of the attenuation of a beam of particles all with the same energy and all travelling in the same direction is given by an exponential law

that performs the exponential decrease of the number of particles N(x) at x given depth into the material

from the initial number where microL is the linear attenuation coefficient4

This law follows from the fact that over any short distance the probability of losing a particle from the beam is proportional to the number of particles left into it if particles are present in high number many are going to be lost but if the number left decreases the same does the rate of loss

The exponential attenuation law does not describe what happens to the energy carried by the photons removed from the beam and it is possible that some of that may be carried through the medium by other particles including some new photons

The average distance travelled by a photon before it is absorbed is given by λ the attenuation length or mean free path that is the reciprocal of the linear attenuation coefficient

4 It gives a measure of how fast the original photons are removed from the beam (if of high values the original photons are removed after travelling only small distances)

3


It follows an alternative way of expressing the exponential attenuation law

The distance over which one half the initial beam is absorbed is called the half thickness and is related

to the linear attenuation coefficient and to the mean free path by

The attenuation of photons depends on the total amount of material in the beam path and not on how it is distributed because the probability for a photon to interact somewhere within the matter depends on the total amount of atoms ahead of its path (since they interact only with single atoms)

Therefore it is useful to describe the attenuation process without the dependence on the density of material but only on the kind of material This is obtained by introducing the mass attenuation coefficient μm which relates the linear attenuation coefficient to the density of the material ρ

This means for example that the mass attenuation coefficient is the same for ice liquid water and steam whereas the linear attenuation coefficients differs greatly

It is so possible to have a ULTERIORE definition of the attenuation law

that states that the total attenuating effect of a slab of given type material can be described by quoting the mass attenuation coefficient which is characteristic of the materials chemical composition and the photon energy together with the materials density and thickness The product ρx the areal density5 of a thickness x of the attenuating material is also called the density-thickness and is often quoted instead of the geometrical thickness x Although the SI6 unit of density-thickness is kgm-2 the obsolete unit gcm-2 is still used in the literature

5 mass per area6 International System of measurements

4


If an absorber is made of a composite material the mass attenuation coefficient is readily calculated by adding together the products of the mass attenuation coefficient and the proportion (α) of the mass due to each element present in the material

The law of attenuation always describes the attenuation of the original radiation If the radiation changes degrades in energy it is not completely absorbed or if secondary particles are produced then the effective attenuation decreases and so the radiation penetrates more deeply into matter than predicted It is also possible to have an increasing number of particles with depth in the material this process is called build-up and has to be taken into account when evaluating the effect of radiation shielding

232 - Effects of photon interaction

Gamma rays x rays and light are photons with different energies depending on their energy and the nature of the material photons can interact in three main ways photoelectric effect (or photoelectric absorption) Compton scattering and pair production

233 - Photoelectric effect

In order to remove a bound electron from an isolated atom a threshold energy is needed itrsquos the ionization potential and it varies depending on what shell the electron occupies It has been given a letter name to the shells (K L M ) depending on the principal quantum number (n = 1 2 3 ) As example for hydrogen atom H the ionization potential from n=1 corresponds to an ultraviolet photon but for heavier elements the K-shell ionization shifts rapidly into the x-ray regime The following equation summarizes the dependence of the ionization potential from the atomic number Z of the atom (so from the dimension of the atom)

PLOT CURVE XE

5


The figure show that ionization cross section peaks just above threshold for each shell to then fall rapidly (asymp ν-3) at higher energy due to the difficulty in transferring the excess photon momentum to the nucleus For n gt 1 there is subshell structure (2s 2p12 2p32 ) The photoelectric effect will be important in the design of x-ray proportional counters

When other atoms are present as in molecules and solids the electronic energy levels will be very different as will the photoelectric cross sections For solids in vacuum the thresholds can be asymp 1 eV and it depends on the crystalline structure and on the nature of the surface The ionization potential in this case is usually called work function Photon absorption efficiencies approach 100 in the visible and ultraviolet but the overall device efficiencies are limited by the electron escape probabilities In a semiconductor a photon can be thought of as rdquoionizingrdquo an atom producing a rdquofreerdquo electron which remains in the conduction band of the lattice Thresholds are of order 01ndash1 eV for intrinsic semiconductors and of order to 001ndash01 eV for extrinsic semiconductors The latter photon energies correspond to infrared photons Photochemistry is somewhat similar in that photons produce localized ionization or electronic excitation

234 - Compton scattering

The Compton scattering takes place when a photon scatters off a free (or bound) electron yielding a scattered photon with a new lower frequency and a new direction as shown For an unbound electron initially at rest it is possible to have the following equations7

mmm

Low energy photons lose little energy while high energy photons called γ rays lose a lot of energy The wavelength increases by of order 00024 nm independently from the wavelength The Compton cross section is given by the following expresses Klein-Nishina formula[metto solo una referenza e non tutta la formulaccia o no]

The largest Compton scattering cross section is at small energy and it decreases monotonically with energy At low energies lots of scattering events take place but very little energy is lost It is a consequence that the energy absorption cross section is small at low energy because little energy is transferred to the electron and it rises to a peak for photon energies around 1 MeV that declines at higher energy

7 h (mec) has units of length and equals 00024 nm

6


235 - Pair production

Photons with energies in excess of 2mec2 produce electron-positron pairs and an interaction with a nucleus is needed in order to balance momentum The pair production cross section starts at 1022 MeV for then rising to an approximately constant value at high photon energy in the gamma ray region of the spectrum of electromagnetic radiation Cross sections scale with the square of the atomic number

24 - Interactions of charged particles with matter

The most common way in which charged particles (such as electrons protons and alpha and beta particles) can interact with matter is the electromagnetic interaction that involves collisions with electrons in the absorbing material and is the easiest mechanism to detect them They can also interact through one of the two kinds of nuclear interactions the weak interaction or the strong interaction

The main process of energy loss producing excitation and ionization is the inelastic collisions with an electron it can also happen an inelastic collisions with a nucleus that leads to Bremsstrahlung and coulombic excitation Eventually there could also be elastic collisions with a nucleus Rutherford diffusion and elastic collisions with an electron

7


241 - Electromagnetic interaction

Two main mechanisms characterize the electromagnetic interaction the first is the excitation and ionisation of atoms and the second is the so-called bremsstrahlung word meant to describe the emission of electromagnetic radiation (photons) when a charged particle is severely accelerated (usually by interaction with a nucleus) Moreover there exists a third kind of interaction producing Cherenkov radiation that absorbs only a small amount of energy (but it plays an important role in the detection of very high energy charged particles) Charge mass and speed of the incident particle as well as the atomic numbers of the elements of the absorbing material define the contribution of each mechanism

Individual interactions - scattering

Unlike photons each charged particle suffers many interactions along its path before finally coming to rest losing only a small fraction of its energy during every interaction (for example a typical alpha particle might make 50000 collisions before it stops) Hence the energy loss can usually be considered as a continuous process Although the amount of scattering at each collision may be small the cumulative effect may be quite a large change in the direction of travel Occasionally an incident particle passes very near a nucleus and then there is a single large deflection (this nuclear scattering effect is most pronounced for light incident particles interacting with heavy target nuclei)

Stopping power

The most important way to describe the net effects of charged particle interactions with matter and the

rate of energy loss along the particles path is with the linear stopping power S l also known as (where E

is the particles energy and x is the distance travelled)

commonly measured in MeV m-1 It depends on the charged particles energy on the density of electrons within the material and hence on the atomic numbers of the atoms So a more fundamental way of describing the rate of energy loss is to specify the rate in terms of the density thickness rather than the geometrical length of the path so energy loss rates are often given as the quantity called the mass stopping power

where ρ is the density of the material and ρx is the density-thickness

ADD BETHE BLOCK

P12 jorne knetter 20-24

27-35 silicon detectors

Fino a 47 interessa

8


P 46 strip e pixel

Excitation and ionization(riformula e connetti allrsquoaltro scritto)

Electromagnetic interaction between the moving charged particle and atoms within the absorbing material is the dominant mechanism of energy loss at low (non-relativistic) energies it extends over some distance keeping not necessary for the charged particle to make a direct collision with an atom Energy can be transfered simply by passing close by but only certain restricted values of energy can be transferred The incident particle can transfer energy to the atom raising it to a higher energy level (excitation) or it can transfer enough energy to remove an electron from the atom altogether ( ionisation) This is the fundamental mechanism operating for all kinds of charged particles but there are considerable differences in the overall patterns of energy loss and scattering between the passage of light particles (electrons and positrons) heavy particles (muons protons alpha particles and light nuclei) and heavy ions (partially or fully ionized atoms of high Z elements) Most of these differences arise from the dynamics of the collision process in general when a massive particle collides with a much lighter particle the laws of energy and momentum conservation predict that only a small fraction of the massive particles energy can be transferred to the less massive particle The actual amount of energy transferred will depend on how closely the particles approach and from restrictions imposed by quantization of energy levels The largest energy transfers occur in head-on collisions

Energy loss by heavy particles

A massive particle that collides with an electron loses relatively small quantity of energy at each collision For example a slow alpha particle hitting an electron transfers a maximum of only 005 of its energy to the electron Since head-on collisions are rare usually the energy loss is much lower In order to significantly reduce the incident particles energy many collisions are needed so the energy loss can be considered as a continuous process Although the energy given to an electron may be a small fraction of the incident energy it may be sufficient to ionize the atom and for making the ejected electron travel some distance away from the interaction point leaving a trail of excited and ionized atoms of its own These knock-on electrons can leave tracks called delta rays Mostly however the knock-on electrons lose their energy within a very short distance of the interaction point

The energy dependence of the rate of energy loss (stopping power) by excitation and ionization of heavy particles for some typical materials is shown in figure 416 This graph is a plot of the energy-loss rate as a function of the kinetic energy of the incident particle Note that the stopping power is expressed using density-thickness units To obtain the energy loss per path length you would need to multiply the energy loss per density-thickness (shown on the graph) by the density of the material As for photon interactions it is found that when expressed as loss rate per density-thickness the graph is nearly the same for most materials There is however a small systematic variation the energy loss is slightly lower in materials with larger atomic numbers The diagram shows the rate of energy loss for the extreme cases of carbon (Z = 6) and lead (Z = 82) At high incident energies there is also some variation with density of the same material because a higher density of atomic electrons protects the more distant electrons from interactions with the incident particle This results in lower energy loss rates for higher densities

9


10


FIGURE DA COMMENTARE COL PHYSICS REVIEW E DATA BOOKLET p267

For low energies the stopping power varies approximately as the reciprocal of the particles kinetic energy The rate of energy loss reaches a minimum called minimum ionization point (MIP) to then start to increase slowly with further grow in kinetic energy Minimum ionization occurs when the particles kinetic energy is about 25 times its rest energy and its speed is about 96 of the speed of light in vacuum Although the energy loss rate depends only on the charge and speed of the incident particle but not on its mass it is convenient to use kinetic energy and mass rather than the speed At minimum ionization the energy loss is about 02 MeV (kg m-2)-1 (= 3 times 10-12 Jm2kg-1 in SI units) and it slightly decreases with the increasing atomic number of the absorbing material

Before losing all its kinetic energy into the material a penetrating particle percurs some distance called range of that particle Energy loss along the path is shown in figure 417 The rise near the end of the path is due to the increased energy loss rate at low incident energies At very low speeds the incident particle picks up charge from the material becomes neutral and is then entirely absorbed by the material

Particles of the same kind with the same initial energy have nearly the same range for a given material The number of particles as a function of distance along the path is shown in figure 418 The final small variation in the range is called straggling and is due to the statistical nature of the energy loss process which consists

11


of a large number of individual collisions subjected to some fluctuation In spite of that the average range can be used to determine the average energy of the incident particles

Energy loss by electrons and positrons

Concerning the electrons and positrons loss of energy they also ionize but with several differences with heavy particles (for example they have lower loss rates at high energies than heavier particles travelling at the same speed) There is also a slight difference between the interactions of positrons and of electrons resulting in a slightly higher energy loss for the positrons

An electron is easily scattered in collisions with other electrons because of its light mass as a result the final erratic path is longer than the linear penetration (range) into the material with greater straggling

Bremsstrahlung effect

Literally translated from German into braking radiation bremsstrahlung is an effect that occurs whenever the speed or direction of a charged particle motion changes (when it is accelerated) and consist in the emission of electromagnetic energy (photons) when the acceleration takes place It is most noticeable when the incident particle is accelerated strongly by the electric field of a nucleus in the absorbing material Since the effect is much stronger for lighter particles it is much more important for beta particles (electrons and positrons) than for protons alpha particles and heavier nuclei (but it happens also for them) Radiation loss starts to become important only at particle energies well above the minimum ionisation energy (at particle energies below about 1 MeV the energy loss due to radiation is very small and can be neglected) At relativistic energies the ratio of loss rate by radiation to loss rate by ionization is approximately proportional to the product of the particles kinetic energy and the atomic number of the absorber So the ratio of stopping powers is

where E is the particles kinetic energy Z is the mean atomic number of the absorber and E is a proportionality constant E asymp 800 MeV

12


Electron-photon cascades

A high energy electron performing Bremsstrahlung results in a high energy photon as well as a high energy electron and a high energy photons performing pair production results in a high energy electron as well as a high energy positron in both cases two high energy particles are produced from a single incident particle It follows that the products of one of these processes can be the incident particles for the other with the result of a cascade of particles which increases in number while decreasing in energy per particle until the average kinetic energy of the electrons falls below the critical energy The cascade is then absorbed by ionization losses Such cascades or showers can penetrate large depths of material

13


INTERACTIONS OF NEUTRAL PARTICLES WITH MATTER (cenni)

Neutral particles such as the neutron can interact with matter only through the nuclear interactions that produce charged particles allowing to detect their presence Uncharged particles are not subject to coulomb interaction8 but loose the energy through strong interaction with the nuclei of the material In case of semiconductors the particle knocks and then shift the nuclei from the original lattice position creating vacancies or interstitial defects

8 At the first order

14


22 ndash Physics and behavior of semiconductors

This chapter is meant as an exhaustive introduction about the physical principles that stand behind the behaviour of devices made from semiconductor materials Such semiconductor devices are widely used in the electronics (power-switching devices) because of their specific electrical conductivity σ which is between that of good conductors (gt1020 free electron density) and that of good insulators (lt103 free electron density)

221 - Conduction in a solid

SemiconductorsBasicspdf

After Quantum Mechanics discoveries a theory about solid state materials that includes semiconductors has been commonly approved by the scientific community The structure of an isolated atom shows numerable states of the electrons surrounding the nucleus characterized univocally by a definite energy En

9 In a solid it is to be taken into account the entire number of the atoms that constitutes the lattice the interactions among the atoms and their high existing number10 make the electron states so dense to make them forming a continuous band of allowed energy These bands can be separated by gaps that electrons cannot occupy the forbidden gaps Because of their fermionic nature11 electrons fill the states starting from the lowest energy level available filling up the energy bands to a maximum energy E0 (see figure 101)

Qualitatively there are two possible configurations one with the last band partially filled and the other with the last band completely filled The partially filled (or empty) band is called conduction band while the band below it is referred to as valence band Because of the thermal energy available at the absolute

9 n is a set of integer numbers10 ~ 1022 atomscm3

11 In Quantum Mechanics the fermions belongs to one of the two fundamental particle classes (fermions and bosons) Fermions distinguish from bosons for the fact that they obey to Paulirsquos Exclusion Pinciple that states that a single quantic estate cannot be occupied by more than one fermion (while the bosons are free to largely crowd the same quantic state)

15


temperature T some higher energy levels are populated In the case of a partially filled band the solid is a conductor because when an electric field is applied the electrons can freely change states in the conduction band In the case of completely filled bands the gap width between the valence and the conduction band can make the solid an insulator (Φ ~10eV) or a semiconductor (Φ ~1eV) In fact the

thermal energy available at T 300K is sufficient to bring some electrons into the conduction band if the

gap is of the order of 1eV To calculate the number of electrons with an energy above a given value E 0 one must applies Boltzmann statistics [] which gives the density of electrons having energy greater than E0 (ie n(E gt E0) = e1048576 E0 kBT kB = 13807 _ 10104857623JK where kB is the Boltzmann constant)

Classification of Semiconductors

Although there is a large variety of semiconductor materials today available there is one of them that stands out from the group and dominates the scene it is the silicon Its properties are entirely well known it is quite easy to find and to manage practically and ndash last but not least for the productive processes ndash not expensive Nonetheless according to their chemical composition each different kind of semiconductor can have different properties and so used for different specified duty in the applications

Elementary semiconductors are located within the IV group of the Periodic Table of Elements []and they are the Silicon (Si) the Germanium (Ge) the grey tin (α-Sn) and Carbon (C) that can solidify in two different structures (graphite and diamond that is an insulator but with the same crystal structure as Si Ge and α-Sn)

TABLE GROUP IV ELEMENTS (nella didascalia commenta anche gli altri elementi)

The main characteristic of the IV elements mentioned is that they all have the outer shell of the individual atoms is exactly half filled and so by sharing one of the four electrons of the outer shell with another Si atom it is possible to obtain a three-dimensional crystal structure with no preferencial direction (except for graphite) and it is also possible to combine two of IV group semiconductors in order to form useful compounds (such as SiC or SiGe) with new peculiarity (for example the SiC is a borderline compounds between semiconductor and insulator and can be useful for high temperature electronics)

By completing the outer shell by sharing electrons with other atoms can be obtained also with other compounding12 so obtaining compounds that are semiconductors too Elements of group III (II) can so be combined with elements of group V (VI) with covalent bonds (but in contrast with IV group ones they show also a certain degree -~30- of ionic bonds) Most of the III-V semiconductors exist in the so-called zincblende structure (cubic lattice) and some in the wurtzite structure (hexagonal lattice) GaAs and GaN are the most known and most utilized of them (optical application because they are direct semiconductors)

TABELLA 14 httpwwwpdf-search-enginecomsemiconductor-an-introduction-pdfhtml

It also exists the II-IV class of semiconductors characterized by an higher ionic bond degree total percentage -~60- since the respective elements differ more in the electron affinity due to their location in the Periodic Table of Elements Also I-VII compounds can form semiconductors with larger energy gap There are other elementary semiconductors such as selenium and tellurium from group VI the

12 8N atomic rule

16


chalcogenes but only with two missing valence electrons to be shared with the neighboring atoms so they have the tendency to form chain structures

TABELLA 5

There are also some spare compounds that works as semiconductors they are the IV-VI compounds (PbS PbSePbTe) V-VI (B2Te3) II-V (Cd3As2 CdSb) and a number of amorphous semiconductors (the a-SIH amorphous hydrogenate silicon for example is a mixture of Si and H) Moreover it is still possible to cite the chalcogenide glasses (As2Te3 As2Se3 that can be used in xerography)

Silicon

Silicon is the most widely used material in radiation detectors mostly because it is the only semiconductor material having a native oxide with good interface properties that fit for a high integration technology and the most chosen for devices involving semiconductors It has four valence electrons so it can form covalent bonds with four of neighbors atoms When the temperature increases the electron in the covalent bond can become free generating holes that can afterwards be filled by absorbing other free electrons so effectively there is a flow of charge carriers The effort needed to break off an electron from its covalent bond is given by Eg (bandgap energy) There exists an exponential relation between the free-electron density and Eg given by the formula

For example at T=300 K ni = 108 x 1010 electronscm2 and at T=600 K ni = 154 x 1015 electronscm2

In pure silicon at equilibrium the number of electrons is equal to the number of holes The silicon is called intrinsic and the electrons are considered as negative charge-carriers Holes and electrons both contribute to conduction although holes have less mobility due to the covalent bonding Electron-hole pairs are continually being generated by thermal ionization and in order to preserve equilibrium previously generated pairs recombine The intrinsic carrier concentrations ni are equal small and highly dependent on temperature

Doping

DEF INTRINSIC ESTRINSIC SEMICOND

In order to fabricate a power-switching device it is necessary to increase greatly the free hole or electron population This is achieved by deliberately doping the silicon by adding specific impurities called dopants The doped silicon is subsequently called extrinsic and as the concentration of dopant Nc increases the resistivity ρ decreases

Pure silicon can be doped with other elements in order to change its electrical properties it can be doped with P (phosphourous) and so it will have more electrons (type N doping with free negative charges) or with B (boron) and so it will have more holes (type P doping with free positive hole charges)

A group V dopant is called a donor having donated an electron for conduction The resultant electron impurity concentration is denoted by ND - the donor concentration

17


If silicon is doped with atoms from group III such as B Al Ga or In which have three valence electrons the covalent bonds in the silicon involving the dopant will have one covalent-bonded electron missing The impurity atom can accept an electron because of the available thermal energy The dopant is thus called an acceptor which is ionised with a net positive charge Silicon doped with acceptors is rich in holes and is therefore called p-type The resultant hole impurity concentration is denoted by NA - the acceptor concentration

To be pointed out that for manufactury industries it is not easy to grow large area silicon crystals doped with a rate of less than 10 around the resistivity wanted Final device electrical properties will therefore vary widely in all lattice directions Tolerances better than plusmn1 per cent in resistivity and homogeneous distribution of phosphorus can be attained by neutron radiation commonly called neutron transmutation doping NTD The neutron irradiation flux transmutes silicon atoms first into a silicon isotope with a short 262-hour half-lifetime which then decays into phosphorus Subsequent annealing removes any crystal damage caused by the irradiation Neutrons can penetrate over 100mm into silicon thus large silicon crystals can be processed using the NTD technique

Charge Carriers

Electrons in n-type silicon and holes in p-type are called majority carriers while holes in n-type and electrons in p-type are called minority carriers The carrier concentration equilibrium can be significantly changed by irradiation by photons the application of an electric field or by heat Such carrier injection mechanisms create excess carriers

The product of electron and holes densities is always equal to the square of intrinsic electron density regardless of doping levels

It follows that for n-type doped semiconductors

18


and for p-type doped semiconductors

Charge transportation

A first mechanism of charge transportation into semiconductor can be focused in the so called drift mechanism it is simply the application of an electric field at the extremity of the semiconductor and the charge particles will move at a velocity proportional to the electric field given

The current is calculated as shown in the following formula

In general the drift current is expressed as

while the total current is the sum of the current given by holes and electrons drifts

It is important to underline that in reality the velocity does not increase linearly with electric field but it saturates at a critical value The following equation expresses the velocity saturation

19


A second charge transportation mechanism is the diffusion that is given by the fact that charged particles move into the semiconductors from a region of high concentration to a region of low concentration

Diffusion current is proportional to the gradient of charge (dndx) along the direction of current flow as shown in the following equation

It is important to say that a linear charge density profile means constant diffusion current whereas nonlinear charge density profile means varying diffusion current as shown in the plot

20


Surprisingly there exists a relation between the drift and diffusion currents NONOSTANTE they are totally different it is the Einsteinrsquos relation

PN junctions

A pn junction is the location in a semiconductor where the impurity changes from p to n while the monocrystalline lattice continues undisturbed A bipolar diode is thus created which forms the basis of any bipolar semiconductor device

When N-type and P-type dopants are introduced side-by-side in a semiconductor a PN junction (or diode) is formed

In order to understand the operation of a diode it is necessary to study its three operation regions equilibrium that introduces the depletion zone and the built-in potential reverse bias that introduces the junction capacitance and forward bias that introduces the IV characteristics

Processes forming p-n junctions

CHP1 BARRY

Diffusion across the junction

Each side of the junction contains an excess of holes or electrons compared to the other side and this situation induces a large concentration gradient Therefore a diffusion current flows across the junction from each side

21


As free electrons and holes diffuse across the junction a region of fixed ions is left behind This region is known as the depletion region

The fixed ions in depletion region create an electric field that results in a drift current

At equilibrium the drift current flowing in one direction cancels out the diffusion current flowing in the opposite direction creating a net current

of zero The figure shows the charge profile of the PN junction

It exist a built-in potential because of the junction

22


Reverse biasing

There are two ways for biasing the junction one the is direct the other is the reverse way When the N-type region of a diode is connected to a higher potential than the P-type region the diode is under reverse bias which results in wider depletion region and larger built-in electric field across the junction

Varying the value of VR it is consequently possible to vary also the width of the depletion zone changing also the capacitance value this leads to identify the PN junction as with the same behavior of a voltage dependent capacitor Its capacitance is described by the following equation

with

A useful application of this statement is to use the junction to form a LC oscillator circuit that varies the frequency by changing the VR (and so changing the capacitance)

23


Forward bias

When the N-type region of a diode is at a lower potential than the P-type region the diode is in forward bias This situation leads to shorten the depletion width and decrease the built-in electric field

Under forward bias minority carriers in each region increase due to the lowering of built-in fieldpotential Therefore diffusion currents increase to supply these minority carriers

TANTE EQ

Minority charge profile should not be constant along the x-axis in order to have a concentration gradient and so diffusion current recombination of the minority carriers with the majority carriers accounts for the dropping of minority carriers as they go deep into the P or N region

IV characteristics of PN junction

The current and voltage relationship of a PN junction is exponential in forward bias region and relatively constant in reverse bias region

Junction currents are proportional to the junctionrsquos cross-section area so two PN junctions put in parallel are effectively one PN junction with twice the cross-section area and hence twice the current

24


Constant-voltage diode model and reverse breakdown

Diode operates as an open circuit if VDlt VDon and a constant voltage source of VDon if VD tends to exceed VDon

When a large reverse bias voltage is applied breakdown occurs and an enormous current flows through the diode

There exist two kinds of reverse breakdown Zener and Avalanche breakdown The first is a result of the large electric field inside the depletion region that breaks electrons or holes off their covalent bonds whilethe second is a result of electrons or holes colliding with the fixed ions inside the depletion region

Text Book Fundamentals of Semiconductor Physics and Devices First Edition Author Donald A Neamen

Publishing McGraw-Hill Company

Dear Fabio The materials are from the text book Fundamentals of Microelectronics The author is Behzad Razavi I just used the teaching slides provided by the author I think you had better refer to the text book The details of the text book are as followsBook title Fundamentals of MicroelectronicsAuthor Behzad RazaviPublication company John Wiley amp Sons IncYear 2008 Hopefully the information will be useful for you Sincerely

25


Jen-Shiun ChiangProfessorDepartment of Electrical EngineeringTamkang UniversityTamsui Taipei Taiwan----- Original Message ----- From Fabio Rivero To chiangeetkuedutw Sent Wednesday August 12 2009 618 PMSubject reference request

Goodmorning Dr Changsorry if I disturb you I am an italian student of Biomedical Physics working on my master theses about 3D Silicon Detectors at Cern Looking for references on the subject physics of semiconductors in the internet I went through your slides on this web-pagehttpwwweetkuedutw~chiangcourseselectronics-1electronicshtmch02pdf I found that some pictures you put into are really good at explaining the semiconductors behaviour and characteristics so I want to ask you if I can borrow some of them for my thesis (citing your slides in my theses references) or knowing the reference from which you took them I would also be glad if you have other interesting links or references on the subjectThank you very much

26


23 ndash Semiconductor silicon detectors

In principle a semiconductor detector behaves like a ionization chamber with a simple configuration made by an absorbing medium in the case the semiconductor connected to two electrodes The electrodes are themselves connected to an external bias supply that creates the electric field through the pn junction and when a particle passes to the material and generates charges particles this electric field drifts the generated charges to the respective electrodes producing the outgoing signal

The study of physical laws that stand behind interaction between radiation and matter leads to the fact that to study electromagnetic radiation it is necessary for the radiation to interact in some fashion with some physical rdquodetectorrdquo Electromagnetic radiation can be thought of in terms of waves or of particles but there is no strict dividing line between these views since electromagnetic radiation always retains both particle and wave characteristics The particle viewpoint is more useful when at high frequencies where the photons are energetic whereas the wave viewpoint is more useful when at low frequencies To determine which viewpoint may be more useful it is to be considered the average photon occupation number of the modes of the radiation field

Thinking of the radiation in terms of photons as for first example shows that the basis of photon detectors have to be found through the variations on the photoelectric effect the Compton scattering and the pair production

Solid State Detectors

Silicon is characterized by a unit cell of crystalline with face-centered cubic structure (like the one of diamond) in which each atom has the four nearest neighbors joined together in a tetrahedral configuration by a single covalent bond containing two electrons one and one given by both of each atom

It is useful to represent this three-dimensional construction in two dimensions with the Si atoms forming a regular square grid It is limited by the fact that it does not accurately represent the atomic arrangement far beyond that of nearest neighbors and any particular atom is not simultaneously the nearest neighbors of another atom

FIGURE

Highly purified silicon is known as an intrinsic semiconductor and is a poor conductor of electricity since the electrons are tied up in valence bonds However if one of those bonds is broken by absorption of a photon

27


or by thermal excitation then an electron is raised in energy into the conduction band leaving behind a hole Both the electron and hole are mobile charge carriers although they may have very different mobilities

An extrinsic semiconductor is formed if one of the silicon atoms is replaced by an impurity atom with a different number of valence electrons For example arsenic atoms have five valence electrons So when an arsenic atom is placed in a silicon crystal there is an extra electron not tied up in the valence bonding which therefore is a free carrier This is known as N-type doping since it provides excess negative charge carriers Boron has three valence electrons so a boron impurity leaves a hole as a free carrier This is P-type doping since the hole carries positive charge Consider what would happen if a piece of P-type Si and a piece of N-type Si were joined together At the boundary there would be a high concentration of free electrons on the N side and some of these electrons would diffuse across the boundary into the P side There would likewise be a high concentration of holes on the P side and some of these would diffuse across the boundary to the N side The donor and acceptor atoms would be left behind with static positive and negative charges which would set up an electrostatic field After enough carriers had diffused across this field would prevent further diffusion The region around the boundary would be depleted of the majority charge carriers (electrons on the N side and holes on the P side) and is appropriately called the depletion region By applying reverse bias (positive voltage to the N side and negative voltage to the P side) one can increase the size of the depletion region

Silicon Diode Detectors

By applying reverse bias to a PN junction one is effectively storing charge on the equivalent of a parallel plate capacitor (the depletion region is an insulator and the P and N regions are conductors) Imagine disconnecting the bias If photons are absorbed letrsquos say within the depletion region electron-hole pairs are produced The electrostatic field within the depletion region will sweep the electrons to the N side and the holes to the P side decreasing the amount of stored charge After some time one could reapply the bias restoring the original charge and the current would reveal how many photons had been absorbed in the depletion region The sensitivity of this technique is limited by thermodynamic fluctuations In thermodynamic equilibrium at a temperature T the uncertainty in the stored charge (the charge fluctuations at fixed voltage) is given by

(Q)2 = kTC (5-11)

This is known as kTC noise For a temperature of 150 K and a capacitance of 1 pF

Q 280 eminus (5-12)

which is relatively high if one wants to detect individual photons In order to be limited by photon statistics rather than kTC noise one would need of order 105 photons ( 105 300) There is also dark noise from thermally activated leakage currents (which depend exponentially on temperature)

Charge-Coupled Devices (CCD)

Instead of a PN junction another way of obtaining a silicon detector is to start with a P-type substrate of silicon growing on the surface an insulating oxide (SiO2) and then depositing small and thin (semi-transparent) metallic electrodes on top of it Each electrode defines a MOS (metal-oxide-Si) capacitor

28


Positive bias applied to the electrodes creates depletion regions13 serving as storage regions for electrons14 A charge-coupled device (CCD) is so obtained consisting of a 2 dimensional array of such pixels

The detection characteristics of CCD arrays depends on the method of illumination and certain physical characteristics of the manufacturing In all cases photons enter the semiconductor and are absorbed in or near the depletion region If the absorption occurs inside the depletion region the electron is drawn towards the positively charged electrode and trapped by the oxide and the hole is expelled from the depletion region while if the absorption occurs outside the depletion region it is necessary for the electron to diffuse to the boundary of the depletion region before the electron recombines15

Some CCDs are front illuminated meaning that the radiation passes through the semi-transparent electrode These are typically thick CCDs which have enhanced response in the red portion of the spectrum since the thick device is able to contain several optical absorption lengths even at long wavelengths where the absorption length is typically longest Quantum efficiencies are typically of order 70 Thinned CCDrsquos have been etched away from the underside the back and are typically back illuminated They have poorer red response because the thickness of the remaining material is only of the same order as the absorption length in the red while they have better blue response and higher peak quantum efficiencies typically nearly 90

The name CCDs is given by their method of signal readout CCDrsquos are essentially shift registers which preserve the integrity of the trapped charge bundles with charge-transfer efficiencies of order )CT ( 099999 per shift as the packets are shifted across the device in a rdquobucket brigaderdquo technique to a readout amplifier The readout generally takes place after the exposure and can be in the form of for example sequential readout of the final column of the CCD followed by a single step of all rows over by one column to repopulate the final column This is iterated until the entire device is read out An alternative technique is used in the Sloan Digital Sky Survey (SDSS) in which the telescope is stationary and the stars drift across the focal plane and the CDD (drift scan) in synchronism with the rate of charge packet shift across the CCD16

Advantages of CCD detectors for astronomy include high quantum efficiency good linearity low readout noise (( 3 eminus RMS) and large numbers of pixels (the SLOAN CCD camera has a total of 120 Megapixels enough to do simultaneous multi-color photometry over wide fields) Electron storage capacity can be of order 105 minus 106 electrons per pixel which gives a large but limited dynamic range Image defects can be caused by cosmic ray hits and by spillover from overfull packets due to bright sources One does not typically use CCDrsquos for high-speed photometry since the readout takes time Careful data reduction techniques for CCDrsquos include measurement of dark frames (which need to be subtracted) and rdquoflat fieldsrdquo (images under uniform illumination) by which the images are divided to obtain gain-corrected images

The above discussion is somewhat oversimplified There are varieties of semiconductor manufacturing processes and varieties of readout techniques for CCDrsquos

As an example many Hubble Space Telescope instruments have used CCDrsquos these include the STIS (Space Telescope Imaging Spectrograph) scheduled for repair in 2009 during Servicing Mission SM4 the ACS

13 depleted of holes the majority carriers in P-type silicon14 minority carriers15 it would reduce the detection efficiency16

29


(Advanced Camera for Surveys) also scheduled for repair and the ultraviolet-visible channel of WFC3 (Wide Field Camera 3) to be newly installed during SM4

Photon detection (zoboli questi due)

A photon can interact with a semiconductor and create charge when its energy is higher than the energy gap of the material For Silicon this value is 112eV and corresponds to a λ=112μm in the infrared region If a photon has a wavelength longer than 102 μm it will cross the silicon sensor without being attenuated

For indirect bandgap semiconductor such as germanium and silicon the absorption of a photon is made possible only with the involving of a phonon that gives the additional momentum necessary to the electron to jump to the conduction band Indirect bandgap semiconductors have the absorption coefficient growing gradually with the photon energy when the photon energy is high enough to allow the direct transition from the valence to the conduction band phonons are no longer required for the excitation and the absorption coefficient saturates For direct bandgap semiconductors such as GaAs the coefficient grows for energies nearby the energy gap value since the transition does not require an extra particle like the phonons in order to conserve momentum In all cases the incident electromagnetic radiation is attenuated in the matter by an exponential decay described by this equation [giarsquo scritta metti il num]

I(x) = I0 eminus_x (12)

where I0 is the initial beam intensity x is the depth and α the absorption coefficient

Particle detection (zoboli)

A charged particle passing through matter interacts with the atomic electrons through Coulomb forces The energy lost by the particle known also as Ionizing Energy Loss (IEL) depends on its energy and on material properties (such as density and atomic number) Electrons for instance due to their low mass can be easily deflected whereas protons or heavy ions proceed straight in the matter

Protons pions muons and electrons primary loose energy due to inelastic collision with the atomic electrons of the material in small amount at every collision and part of this energy creates free charge in the material The mean energy transferred per unit path length follows the Bethe-Bloch formula depending on parameters related to both the incident particle and to the physical properties of the absorbed material [8]

For high energy particles the mean energy transferred to the matter reaches a minimum which is nearly the same for protons electrons and pions and remain constant for higher energies as shown in Fig 16a Particles having energies high enough to reach this minimum are knows as Minimum Ionizing Particle (MIP) Fig 16b plots the energy deposition of a MIP particle in a silicon detector

30


Figure 16 RIFO a) Energy loss as a function of energy for pions[3] b) Typical Landau distribution energy from a semiconductor detector due to a MIP particle

The typical energy spectrum of a MIP particle crossing a semiconductor material follows a Landau distribution characterized by evident asymmetric shape given by the long tail for high energies losses due to high energy recoil electrons (δ rays) Due to the asymmetry the Most Probable Value (MPV) of energy loss corresponding to the peak differs from the mean energy lost which is shifted at higher energies In Silicon the most probable energy loss of a MIP corresponds to a most probable value of 80 electronhole pairs generated per micrometer For a 300μm thick detector the most probable collected charge is therefore 24000el (35fC) The energy range involved in the ATLAS experiment are well above this limit so particles are always supposed to release charge in a MIP like fashion

31


21 ndash Interaction of electromagnetic radiation with matter

211 - Overview

Energy moving through space is identified with the name of electromagnetic radiation and it is characterized by quantity of energy E speed c frequency ν and wavelength λ with which is moving These quantities are all correlated together by the following equations (where h is Planck constant1 and c is the speed of light in vacuum2)

If one of these quantity is known it is so possible to reach for all the others (the factor hc occurs so often in atomic and nuclear physics that it can be considered as a separate constant3)

Different values of energy frequency and wavelength creates the flavours of electromagnetic radiation but difference between them is evident only after the interaction with matter when they show particle-like behaviour out of wave-light behaviour Hence in the definition of radiation the charged particles are included (such as alpha and beta radiation beams of charged particles created by accelerating machines electromagnetic radiation or photons and beams of neutral particles such as neutrons)

This chapter is meant to describe the basic physics that stands behind interaction of radiation and particles with matter what are its consequences and how these principles are applied in semiconductor silicon detectors technology

22 - Electromagnetic and particulate radiation

The principal types of radiation can be first divided into two main categories electromagnetic (X-rays produced outside the nucleus and γ-rays emanated from within nuclei) and particulate (α particles protons neutrons electrons β- positrons β+) This distinction as already mentioned belongs to the proper ldquohistoryrdquo of the radiation drawn by the history of the particle (subject connected to the concepts of energy loss of a particle range interactions) and by the history of the target atoms (that leads to displacements recombination ionization excitation radiation damage and build-up concepts) A beam of radiation that passes through matter can lead to the complete absorption (electronic transitions and vibration-rotational transitions) to some scattering (Rayleigh Rutherford Raman and Mie scattering) andor to the passage with no interaction These processes can be explained in terms of interactions between particles that are stopped or scattered The basic effect of the interaction can be the scattering absorption thermal emission refraction and reflection of the incoming radiation

With the absorption and emission spectra (of molecules) it is possible to outline characteristic structures and so to identified and quantified molecules by these lsquofingerprintsrsquo The spectra are determined by position (wavelength) of absorptionemission line knowing the difference of energy levels of the transition 1 413x10-18 keVsec2 3x108 msec3 124 eVmicrom = 1240 MeVfm

2














3











4








PLOT CURVE XE

5






mmm




6







7






Stopping power






ADD BETHE BLOCK



Fino a 47 interessa

8


P 46 strip e pixel






9


10






11









12




13





14











15













12 8N atomic rule

16



TABELLA 5


Silicon




Doping





17




Charge Carriers




18









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31














3











4








PLOT CURVE XE

5






mmm




6







7






Stopping power






ADD BETHE BLOCK



Fino a 47 interessa

8


P 46 strip e pixel






9


10






11









12




13





14











15













12 8N atomic rule

16



TABELLA 5


Silicon




Doping





17




Charge Carriers




18









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31











4








PLOT CURVE XE

5






mmm




6







7






Stopping power






ADD BETHE BLOCK



Fino a 47 interessa

8


P 46 strip e pixel






9


10






11









12




13





14











15













12 8N atomic rule

16



TABELLA 5


Silicon




Doping





17




Charge Carriers




18









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31








PLOT CURVE XE

5






mmm




6







7






Stopping power






ADD BETHE BLOCK



Fino a 47 interessa

8


P 46 strip e pixel






9


10






11









12




13





14











15













12 8N atomic rule

16



TABELLA 5


Silicon




Doping





17




Charge Carriers




18









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31






mmm




6







7






Stopping power






ADD BETHE BLOCK



Fino a 47 interessa

8


P 46 strip e pixel






9


10






11









12




13





14











15













12 8N atomic rule

16



TABELLA 5


Silicon




Doping





17




Charge Carriers




18









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31







7






Stopping power






ADD BETHE BLOCK



Fino a 47 interessa

8


P 46 strip e pixel






9


10






11









12




13





14











15













12 8N atomic rule

16



TABELLA 5


Silicon




Doping





17




Charge Carriers




18









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31






Stopping power






ADD BETHE BLOCK



Fino a 47 interessa

8


P 46 strip e pixel






9


10






11









12




13





14











15













12 8N atomic rule

16



TABELLA 5


Silicon




Doping





17




Charge Carriers




18









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31


P 46 strip e pixel






9


10






11









12




13





14











15













12 8N atomic rule

16



TABELLA 5


Silicon




Doping





17




Charge Carriers




18









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31


10






11









12




13





14











15













12 8N atomic rule

16



TABELLA 5


Silicon




Doping





17




Charge Carriers




18









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31






11









12




13





14











15













12 8N atomic rule

16



TABELLA 5


Silicon




Doping





17




Charge Carriers




18









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31









12




13





14











15













12 8N atomic rule

16



TABELLA 5


Silicon




Doping





17




Charge Carriers




18









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31




13





14











15













12 8N atomic rule

16



TABELLA 5


Silicon




Doping





17




Charge Carriers




18









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31





14











15













12 8N atomic rule

16



TABELLA 5


Silicon




Doping





17




Charge Carriers




18









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31











15













12 8N atomic rule

16



TABELLA 5


Silicon




Doping





17




Charge Carriers




18









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31













12 8N atomic rule

16



TABELLA 5


Silicon




Doping





17




Charge Carriers




18









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31



TABELLA 5


Silicon




Doping





17




Charge Carriers




18









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31




Charge Carriers




18









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31









19





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31





20



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31



PN junctions





CHP1 BARRY



21







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31







22


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31


Reverse biasing



with


23


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31


Forward bias



TANTE EQ





24









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31









25




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31




26









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31









FIGURE


27






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31






(Q)2 = kTC (5-11)


Q 280 eminus (5-12)




28










29












30




31










29












30




31












30




31




31

chapter 2 - welcome to bugianens'...

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