Microelectronic Circuits Study Guide (Chapter 1-3)

Chapter 1: Signals and Amplifiers Microelectronics generally refers to the field of integrated circuits, up to and including microprocessors - the devices reviewed in this text may be utilized individually as discrete circuits, or as components of an integrated circuit 1.1: Signals A signal is a carrier of information that can be processed in various ways - a real-world measurement is first converted by a transducer to an electrical signal - the signal can be equivalently represented in two ways: Thévenin form (a voltage source vs (t) having a source resistance Rs (t) in series), or Norton form (a current source is (t) having a source resistance Rs (t) in parallel) - these forms are interchangeable (since vs (t) = Rs is (t), but Thévenin is preferred when the source resistance is low; Norton is preferred when the source resistance is high (see Exercise 1.1: when oriented in series, most of the source voltage is delivered to the load when the load resistance is much larger than the source resistance; when oriented in parallel, most of the source voltage is delivered to the load when the load resistance is much smaller than the source resistance) Information is contained in a signal as a time-varying magnitude of the current or voltage - the analysis of the signal is easiest to conceptualize if it can be represented mathematically, which can be difficult for arbitrary signals 1.2: Frequency Spectrum of Signals A signal can be represented according to its frequency spectrum, as a set of individual sinusoids of the form va (t) = Va sin(ωt + φ), where Va is the peak amplitude, ω is the angular frequency, and φ is the phase shift - sinusoid amplitude is commonly discussed as its average, represented by the root-mean-square, √ rather than its peak value; the RMS value is simply peak - the time-domain and 2 frequency-domain representations of a signal are equivalent and interchangeable The frequency spectrum is determined as a Fourier series (for periodic signals) and Fourier transforms (for both periodic and aperiodic signals, including arbitrary signals) - the Fourier series produces a set of sinusoids with harmonically related frequencies, i.e., multiples of a fundamental frequency - the Fourier transform is a continuous function over all frequencies, but the most significant components of the signal are typically in a select range; e.g., an audio signal encodes information for all frequencies, but only those within the range audible to humans (20 Hz - 20 kHz) are significant; similarly, an analog video signal 2 only needs to be sampled between 0 and 4.5 MHz 1.3: Analog and Digital Signals Most physical signals are analog, featuring a magnitude that can take on any value within a certain range, and the amplitude is a continuous function - other signals are digital, where the amplitude at specific times is within a defined set of discrete values, such that the signal can be represented as a sequence of rational numbers - an analog signal can first be sampled, by measuring it at regular time intervals, and then quantized or digitized, by mapping each measurement to a discrete value (usually through rounding) While many number systems may be used to represent digital measurements, binary number systems are the simplest and most common representation - the quantization can either use only two values (low and high, represented as binary 0 and 1, or as a low voltage such as 0 V and a high voltage such as +5 V), or a set of values that are represented each represented by a sequence of N binary values from b0 , b1 , . . . , bN −1 , where the recorded value is represented as the sum D = b0 20 + b1 21 + . . . + bN −1 2N −1 - the mapping or rounding that is inherent in sampling creates a quantization error, which can be reduced by increasing the resolution of the quantization (i.e., increasing N ) - quantization can be achieved by an analog-to-digital (A/D) converter, which inputs an analog input signal and outputs a series of binary values that together represent the digital output - the reverse process can be achieved through a digital-to-analog (D/A) converter, which converts a digital input signal into an analog output signal (though lost resolution cannot be recaptured) 1.4: Amplifiers Signal amplification: Transducers typically sample a signal with too little amplitude for effective processing, so the signal needs to be amplified to a larger amplitude - linear amplifiers scale the input value as a linear proportion, in order to avoid distortion that adds new information - thus, ideal amplifiers exhibit the relationship vo (t) = Avi (t), where A is the amplifier gain Voltage amplifiers operate by increasing the voltage of a signal, as above - power amplifiers operate by increasing the current with little or no voltage gain - an audio system may feature both a voltage amplifier (as a preamplifier), and a power amplifier (which feeds the voltage to the speakers with a higher current, thus amplifying the volume of both soft and loud music passages) The voltage gain of a linear voltage amplifier is defined as: 3 Av = vvoi The transfer characteristic of the linear amplifier is shown as a plot of vi against vo , and is a straight line with slope A for a linear voltage amplifier - similarly, a current amplifier has a current gain Ai = iioi - both types of amplifiers have a power gain Ap = Av Ai = PPLI = vvoi iiio Gain is often expressed in decibels on a logarithmic scale, such as voltage gain = 20 log(|Av |) dB; an increase of 6 dB indicates a doubling of the voltage, and an increase of 20 dB indicates a tenfold increase in voltage - similarly, current gain = 20 log(|Ai |) dB - however, power gain is expressed as 10 log(|Ap |) dB; on this scale, power doubles with an increase of 3 dB, and by a factor of 100 with a 20 dB increase Since amplifiers add power, they require connection to a power source - an example amplifier is connected to a positive power source VCC , which delivers a current ICC , and a negative power source VEE , which delivers a (negative) current IEE - both of the opposite terminals of the voltage sources (the negative terminal of VCC , and the positive terminal of VEE ) are connected to ground, along with a ground terminal of the amplifier - together, these power sources deliver DC power: Pdc = VCC ICC + VEE IEE The amplifier exhibits a power-balance equation: Pdc + PI = PL + Pdissipated A power efficiency for the amplifier can also be calculated as: L ν = PPdc · 100 Amplifier saturation: Amplifiers only generate a linear signal over a limited range of input voltages, where the power delivered by the power supplies can linearly scale to produce the proportional output - if the voltage is too high, amplifier saturation occurs, where the voltage increase is limited by a ceiling and distortion of the signal occurs Symbol conventions: iC (t): A time-varying current signal; i.e., a total instantaneous current IC : The time-invariant DC component of iC (t) ic (t): The time-varying alternating component of iC (t) Ic : The peak amplitude of ic (t) Thus, iC (t) = IC + ic (t), and ic (t) = Ic sin(ωt) 4 1.5: Circuit Models for Amplifiers Amplifier types: Voltage amplifier: Avo = vo vi Current amplifier: Ais = io is Transconductance amplifier (translates an input voltage to an output current): Gm = vioi Transresistance amplifier (translates an input current to an output voltage): Rm = vi o i 5 While each amplifier type may more natively fit a particular scenario, all four types are equivalent Voltage amplifier: According to this model, the amplifier is a voltage-controlled voltage source with an open-circuit voltage gain of Avo , an input resistance Ri , and an output resistance Ro - the transfer function is v0 = Avo vi RLR+LRo , and the L - in an ideal amplifier, Ro = 0, and Av = Avo ; voltage gain is Av = Avo RLR+R o thus, Avo is the open-circuit voltage gain, when 100% of the resistance is at the load - similarly, the source resistance diminishes the amount of voltage measured over the input: vi = vs R R+iR , so it is ideal to maximize the value of Ri - for i s any combination of parameters, the actual gain is: RL i Av = vvos = Avo RiR +Rs RL +Ro In some cases, the voltage is acceptable, but the source resistance is high, and raisin the input resistance to compensate would cause signal attenuation - a buffer amplifier can be used to drive the voltage with greater current Cascaded amplifiers: Sometimes multiple stages are needed to achieve a desired amplification - e.g., in the following scenario, the stage-1 amplifier reduces the input voltage from 100 k Ω to 1 k Ω; the stage-2 amplifier has a high voltage gain; and the stage-3 amplifier acts as a buffer amplifier to drive the amplified voltage with low output resistance 1.6: Frequency Response of Amplifiers Amplifier frequency response can be measured by providing a sinusoidal wave of a known amplitude and frequency, and measuring the amplitude and phase shift of the output (a linear amplifier will not change the frequency) - the transfer function, T (ω), has both a gain (|T (ω )| = VVio ) and a phase shift (∠T (ω) = φ) the complete frequency response can be evaluated as a Bode plot, which reveals both the gain (amplitude response) and the phase shift (phase response) at each input frequency 6 The frequency response of an amplifier can be determined from the circuit model, but where capacitors and inductors are replaced by their reactances 1 (for capacitors, reactance = jωC ; for inductors, reactance = jωL ) - the trans(s) , and can be fer function can be represented as a complex function T (s) = VVio(s) evaluated as T (jω) to determine the response to various physical frequencies Single-time-constant (STC) networks are circuits comprising only one reactive element and one resistance, which can be modeled according to one time conL - different types of STC networks have a characteristic stant τ = RC = R response to various inputs, such as sinusoidal, step, and pulse - most networks exhibit either low-pass-filter or high-pass-filter characteristics - tuned amplifiers feature both a low-pass filter and a high-pass filter, and therefore pass frequencies only within a desired range The components of the circuit also determine the shape of the responses at gain falloff points: internal capacitance can cause a loss of gain at high frequencies, while the inclusion of a coupling capacitor between amplifier stages can cause a loss of gain at low frequencies - integrated circuit amplifiers typically do not feature coupling capacitors, but are directly-coupled DC low-pass amplifiers 7 Chapter 3: Semiconductors Solid-state devices such as diodes and transistors are made from semiconducting materials- this chapter focuses on the physical properties and behavior of semiconductor materials Most semiconductors are made from silicon, through a specific fabrication process - an integrated circuit containing a vast number of transistors can be carved from a single silicon crystal, leading to a monolithic circuit - the conductivity of silicon is altered to create these devices by the controlled introduction of impurity atoms, in a process called doping 3.1: Intrinsic Semiconductors Semiconducting materials are defined as any material exhibiting conductivity between conductors (e.g., copper) and insulators (e.g., glass) - semiconducting materials may be made of one element, such as germanium or silicon, or compounds, typically (group III or V) + (group II or VI), such as gallium-arsenide Silicon is the most widely used semiconducting material in use today - silicon has four valence electrons, and can form four covalent bonds with neighboring atoms to form a regular lattice structure - at low temperatures, all covalent bonds are intact and all electrons are non-mobile, so silicon is an insulator; however, at room temperature, an electron can break free and drift apart and move according to an electric field (“thermal generation”) - the deficit of an electron from its original atom creates a positively charged “hole” that attracts an electron from a neighboring atom, so the “hole” moves in the opposite direction of the electric field - the number of free electrons and holes scales proportionally with temperature - the conductivity of a material is measured as a number of charge carriers per unit volume (cm3 ) - free electrons and holes also settle in other atoms and therefore stop acting as charge carriers (“recombination”) This process can be quantified: temperature affects the thermal generation rate, which in turn determines the number of charge carriers, which in turn determines the recombination rate: ni = n = p - also, p · n = (ni )2 , where: n = the current / average / typical number of free electrons per unit volume p = the current / average / typical number of holes per unit volume ni = the number of charge carriers of either type (holes or electrons) per unit volume For any material: 8 3 Eg ni = BT 2 e2kT , where: B = a material-dependent parameter for the material Eg = bandgap energy k = Boltzmann’s constant (8.62 · 10−5 eV ) K For silicon: 3 B = 7.3 · 15015 k − 2 Eg = 1.12eV At room temperature, ni = 1.5 · 1010 chargecmcarriers 3 3.2: Doped Semiconductors The value of ni for silicon at room temperature is too small to be useful, and also fluctuates too much with temperature - carrier concentration can be improved by injecting a dopant, which introduces an impurity that substantially increases either holes or free electrons n-type doped silicon can be created with a dopant having five valence electrons, such as phosphorus; the extra electron becomes a donor, without creating a hole: the positive charge remains with the dopant (“bound charge”) - the degree of doping determines (ND ) determines the concentration of donor atoms, which is usually much greater than the number of intrinsic free electrons ni ; thus, the number of free electrons in n-type doped silicon nn ≈ ND ; the number of holes, pn , is initially unchanged, but diminishes due to recombination with free electrons - accordingly, in n-type material, electrons are the “majority charge carriers” and holes are the “minority charge carriers”) - when the rates of thermal generation and recombination are equal (“thermal equilibrium”), 2 pn nn = ni2 , where pn ≈ NnDi p-type doped silicon can be created with a dopant having three valence electrons, such as boron - as a consequence, boron accepts an extra electron from a neighboring silicon atom, creating a bound negative charge at the boron atom (“acceptor” atom) and a hole in the neighboring atom that can act as a charge carrier - the degree of doping with acceptor atoms (NA ) creates a number of holes (pp = NA ) that is much greater than the number of intrinsic charge charge carriers - as above, at thermal equilibrium, pp np = ni2, where np ≈ n2i NA Note that both n-type and p-type silicon are electrically neutral; the type of material just determines whether an electric current results in a propagation of electrons or a propagation of holes 9 3.3: Current Flow in Semiconductors Drift current: When an electric field is applied to a semiconductor crystal, holes move in the direction of the field, and free electrons move in the opposite direction The velocity of holes is: vp−drif t = µp E cm , where: s V E = strength of electric field, measured in cm 2 2 - for intrinsic silicon, µp = 480 cm , µp = hole mobility, measured in cm V·s V·s Similarly, the velocity of free electrons is: , where: vn−drif t = −µn E cm s E = strength of electric field 2 µn = electron mobility - for intrinsic silicon, µn = 1, 350 cm V·s The hole drift current density (i.e., current of holes per cross-sectional unit area) is: Jp = IAp = qpvp−drif t For a particular section of material, the hole drift current is: Ip = Aqpvp−drif t = Aqpµp E = Jp A, where: A = cross-sectional area q = charge magnitude (for a single electron, q = 1.6 · 10−19 coulombs) p = concentration of holes Similarly, the electron drift current density (i.e., current of electrons per crosssectional unit area) is: Jn = − IAn = −qpvn−drif t And the electron drift current is: In = −Aqnvn−drif t = −Aqnµn E = Jn A The total drift current density is: J = Jp + Jn = q(pµp + nµn )E, or J = σE, where: σ = the conductivity of the material measured by σ = q(pµp + nµn ) Alternatively, total drift current can be expressed in terms of resistivity, where: J = Eρ Conversely, resistivity can be defined as: , where ρ= E J ρ = resistivity measured in terms of Ω · cm Diffusion current: When the concentration of charge carriers is higher in one part of a material than another, the charge carriers tend to diffuse, thus creating 10 a diffusion current from the high-concentration area to the low concentration area For an area of material with an abundance of holes: dp(x) Jp = −qDp dx , where Dp = the diffusivity or diffusion constant of holes, i.e., the rate at which holes can diffuse through the material dpx = rate of change of concentration of holes (note: this is negative, since dx diffusion causes concentration to fall) Similarly, for an area of material with an abundance of free electrons: dn(x) Jn = qDn dx , where Dn is the diffusivity of electrons dpx = rate of change of concentration of free dx electrons Thermal voltage: For any material: Dp n = , or VT = D µn µp VT = kT q This is known as the thermal voltage; at room temperature, T ≈ 300 and VT = 25.9 mV 3.4: The pn Junction with Open-Circuit Terminals A pn junction can be created by creating a region of p-type silicon adjacent to a region of n-type silicon, with metal contacts at each end (the metal contact at the p-type silicon end forms the anode, and the metal contact at the n-type silicon end forms the cathode): In an open circuit (when the terminals are disconnected), an interaction occurs at the interface - the edge of the p-type material, featuring an excess of holes, binds extra electrons from the edge of the n-type material - both charges become bound, creating a “depletion region” - the depletion region exhibits a diffusion current ID from the p-type region to the n-type region, which is the sum of the hole diffusion current from the p-type region to the n-type region, and the free electron diffusion current from the n-type region to the p-type region - the 11 depletion of free electrons in the n-type region causes some of the bound positive charges to remain “uncovered,” and the depletion of holes in the p-type region creates uncovered bound negative charges The creation of uncovered bound charges in the depletion region creates a voltage difference across the border, and a static electric field from the n-type material to the p-type material - this electric field serves creates a “barrier voltage” V0 , which is measurable over the depletion region (from the n-type region to the p-type region) - this voltage prevents the further migration of holes and electrons, and reduces the diffusion current Drift current: In addition to blocking diffusion of majority carriers, the electric field promotes the transport of minority carriers - e.g., a thermally generated hole that appears in the n-type region may randomly migrate toward the edge of the barrier region, and the electric field sweeps it over the border into the p type region - this transport of minority charge carriers creates a drift current IS in the opposite direction of the diffusion current (i.e., from the n-type material to the p-type material) - the magnitude of this current is dependent upon the availability of charge carriers (and hence temperature), but independent of the actual voltage Equilibrium: In an open circuit, when ID > IS , the depletion region widens until the diffusion current falls; similarly, when ID < IS , the depletion region narrows until diffusion rises - thus, an equilibrium is established where ID = IS , and is maintained by the barrier voltage V0 - with no external voltage applied, equilibrium is established with a “junction built-in voltage” across the pn junction: D ) V0 = VT ln( NAn·N 2 i For silicon at room temperature, 0.6V < V0 < 0.9V This barrier voltage is strictly internal to the pn material, and is not observable between the cathode and anode, because contact voltages are created that balance the junction built-in voltage 12 Depletion region width: When equilibrium is reached, the width of each half of the depletion region is proportional to the concentration of majority charge carriers on the opposite side - the ratio of the widths is defined as: NA xn , where =N xp D xn = the width of the n-type portion of the junction (adjacent to the n-type region but positively charged) xp = the width of the p-type portion of the junction (adjacent to the o-type region but negatively charged) The total width q of the depletion region is: W = xn + xp = 2ǫq s ( N1A + N1D )V0 , where F ) ǫs = electrical permittivity of material (for silicon, ǫs = 1.04 · 10−12 cm The width of each side can be calculated as: xn = W NAN+AND D xp = W NAN+N D The charge stored in each half of the depletion region is defined as: |Q+ | = qAxn ND , and |Q− | = qAxp NA , where A = the cross-sectional area of the material Note that the charges are balanced: |Q+ | = |Q− | 3.5: The pn Junction with an Applied Voltage When a DC voltage is applied across the terminals of a pn junction, with the higher potential at the p terminal, the device is forward-biased; when the voltage is higher at the n terminal, the device is reverse-based In reverse bias, when a voltage VR is applied, the barrier voltage increases to V0 + VR , which significantly reduces ID - however, IS is not affected by the barrier voltage, as it is rate-limited by thermal generation, so a small reverse current q appears - the width of the depletion region also expands: W = 2ǫs ( N1 + N1 )(V0 + VR ) A D The amount q of charge stored in the junction is: QJ = A ND )(V0 + VR ) 2ǫs q(NNAA+N D In forward bias, the voltage VF reduces the build-in voltage V0 , which dramatically raises the diffusion current ID , and diminishes the width of the depletion region - the total observed current is I = ID − IS Carrier injection: When forward-biased, the pn junction allows many more of 13 the majority carriers to cross the barrier into the opposite region - at the border of the depletion region in the n-type region, an excess number of holes are deposited, in the amount of: V pn (xn ) = pn0 e VT , where xn refers to the boundary between the depletion region and the n-type region 2 pn0 = the equilibrium concentration of n ni n This amount is the sum of the equilibrium concentration, pn0 , and the excess holes that are deposited by the voltage; the excess is measured as: V Excess = pn0 e VT − pn0 The concentration continues deeper into the diffusion region, and diminishes at an exponential rate: −x−xn V pn (x) = pn0 + pn0 eVT e Lp , where x − xn = the distance from the depletion region border Lp = the diffusion length of the material (smaller value = faster recombination of excess charge carriers with majority carriers in the region = more rapid dropoff in concentration) The diffusion current density under an applied voltage is: V −x−xn p Jp (x) = q D p (e VT − 1)e LP Lp n0 The recombination of holes with electrons in the n-type region is strongest at the border of the depletion region, and diminishes further from the border - the applied voltage causes a current that injects electrons into the n-type region to replenish those that participate in recombination - the total current in the n-type material is therefore the same as the current at the border: V p Jp (xn ) = q D pn0 (eVT − 1) L p The same process occurs with the diffusion of electrons into the p-type region, 14 and recombination with holes - the current density created by this process is: V Dn np0 (e VT − 1) Jn (−xp ) = q L n The total current, as the sum of both currents, is determined as: V D n )(e VT − 1) I = Aqn2i ( Lp NpD + LD n NA In reverse-bias, when the voltage is negative and significantly greater than the thermal voltage, the exponential term is essentially zero, so: V I = Is (e VT − 1) ≈ −Is , where Is = Aqn2i (LpDNpD + L DNn ) n A However, at a large negative voltage, breakdown occurs at a negative voltage V = −VZ Zener breakdown: When VZ < 5V, breakdown occurs due to the zener effect, where the voltage is strong enough to break some covalent bonds, thus generating a significant number of mobile charge carriers - only a small change in voltage in the neighborhood of −VZ is sufficient to create a large current Avalanche breakdown: When VZ > 7V, breakdown occurs due to the avalanche effect - in this model, the minority carriers crossing the depletion zone have enough kinetic energy to break covalent bonds - the liberated charge carriers also have a large kinetic energy, and thereby break covalent bonds between other toms - the resulting “avalanche” of freed charge carrier creates the breakdown current through the junction 3.6: Capacitative Effects in the pn Junction The pn junction stores charge in two areas: the depletion region (predominant during reverse bias), and the n-type and p-type regions (predominant during forward bias) - different mechanisms are involved in each area Junction√capacitance: The reverse-biased junction stores charge according to: QJ = α q V0 + VR , where α = A 2ǫs q NNAA+NNDD This can also be expressed as: dQj Cj = dV |VR =VQ = 2√Vα0 +Vr R At zero reverse-bias, the capacitance of the junction is Cj0 = 2√αV o The effects described above result in a pn junction that changes abruptly at the junction boundary - other devices have a graded junction with a gradient from n-type material to p-type material - the junction capacitance of these pn junctions changes as a function of the grading coefficient 15 Diffusion capacitance: In the forward-biased junction, carrier injection causes recombination with the majority carriers in each region, causing an excess of minority carriers - this excess manifests as a stored charge in each region - for the p-type region, the stored charge is determined as: Qp = Aq(pn (xn ) − pn0 )Lp = τp Ip , where τp = L2p Dp = hole lifetime in the n-type region Similarly, in the n-type region, the stored charge is: Q n = τn I n The time constants τp , τn represent the average time for an injected charge carrier to recombine with a majority charge carrier in the region - the charge carriers lost to recombination at this rate are replenished by the current Ip , In - the total stored charge during forward bias is: Q = τp I p + τn I n this can be expressed as a function of the total current: Q = τT I, where τT = the mean transit time of the junction - the capacitative effect is therefore determined as: Cd = ( VτTT )I 16