电子工程师手册（英文版二）：ch022

Gildenblat, G.S., Gelmont, B., Milkovic, M., Elshabini-Riad, A., Stephenson, F.W., Bhutta, I.A., Look, D.C. “Semiconductors” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 22 Semiconductors 22.1 Physical Properties Energy Bands ? Electrons and Holes ? Transport Properties ? Hall Effect ? Electrical Breakdown ? Optical Properties and Recombination Processes ? Nanostructure Engineering ? Disordered Semiconductors 22.2 Diodes pn-Junction Diode ? pn-Junction with Applied Voltage ? Forward- Biased Diode ? I D -V D Characteristic ? DC and Large-Signal Model ? High Forward Current Effects ? Large-Signal Piecewise Linear Model ? Small-Signal Incremental Model ? Large-Signal Switching Behavior of a pn-Diode ? Diode Reverse Breakdown ? Zener and Avalanche Diodes ? Varactor Diodes ? Tunnel Diodes ? Photodiodes and Solar Cells ? Schottky Barrier Diode 22.3 Electrical Equivalent Circuit Models and Device Simulators for Semiconductor Devices Overview of Equivalent Circuit Models ? Overview of Semiconductor Device Simulators 22.4 Electrical Characterization of Semiconductors Theory ? Determination of Resistivity and Hall Coefficient ? Data Analysis ? Sources of Error 22.1 Physical Properties Gennady Sh. Gildenblat and Boris Gelmont Electronic applications of semiconductors are based on our ability to vary their properties on a very small scale. In conventional semiconductor devices, one can easily alter charge carrier concentrations, fields, and current densities over distances of 0.1–10 μm. Even smaller characteristic lengths of 10–100 nm are feasible in materials with an engineered band structure. This section reviews the essential physics underlying modern semiconductor technology. Energy Bands In crystalline semiconductors atoms are arranged in periodic arrays known as crystalline lattices. The lattice structure of silicon is shown in Fig. 22.1. Germanium and diamond have the same structure but with different interatomic distances. As a consequence of this periodic arrangement, the allowed energy levels of electrons are grouped into energy bands, as shown in Fig. 22.2. The probability that an electron will occupy an allowed quantum state with energy E is (22.1) Here k B = 1/11,606 eV/K denotes the Boltzmann constant, T is the absolute temperature, and F is a parameter known as the Fermi level. If the energy E > F + 3k B T, then f(E) < 0.05 and these states are mostly empty. Similarly, the states with E < F – 3k B T are mostly occupied by electrons. In a typical metal [Fig. 22.2(a)], the fEFkT B =+ ? ? [ exp( ) ]1 1 / Gennady Sh. Gildenblat The Pennsylvania State University Boris Gelmont University of Virginia Miram Milkovic Analog Technology Consultants Aicha Elshabini-Riad Virginia Polytechnic Institute and State University F.W. Stephenson Virginia Polytechnic Institute and State University Imran A. Bhutta RFPP David C. Look Wright State University ? 2000 by CRC Press LLC energy level E = F is allowed, and only one energy band is partially filled. (In metals like aluminum, the partially filled band in Fig. 22.2(a) may actually represent a combination of several overlapping bands.) The remaining energy bands are either completely filled or totally empty. Obviously, the empty energy bands do not contribute to the charge transfer. It is a fundamental result of solid-state physics that energy bands that are completely filled also do not contribute. What happens is that in the filled bands the average velocity of electrons is equal to zero. In semiconductors (and insulators) the Fermi level falls within a forbidden energy gap so that two of the energy bands are partially filled by electrons and may give rise to electron current. The upper partially filled band is called the conduction band while the lower is known as the valence band. The number of electrons in the conduction band of a semiconductor is relatively small and can be easily changed by adding impurities. In metals, the number of free carriers is large and is not sensitive to doping. A more detailed description of energy bands in a crystalline semiconductor is based on the Bloch theorem, which states that an electron wave function has the form (Bloch wave) C bk = u bk (r) exp(ikr) (22.2) where r is the radius vector of electron, the modulating function u bk (r) has the periodicity of the lattice, and the quantum state is characterized by wave vector k and the band number b. Physically, (22.2) means that an electron wave propagates through a periodic lattice without attenuation. For each energy band one can consider the dispersion law E = E b (k). Since (see Fig. 22.2b) in the conduction band only the states with energies close to the bottom, E c , are occupied, it suffices to consider the E(k) dependence near E c . The simplified band diagrams of Si and GaAs are shown in Fig. 22.3. Electrons and Holes The concentration of electrons in the valence band can be controlled by introducing impurity atoms. For example, the substitutional doping of Si with As results in a local energy level with an energy about DW d ? 45 meV below the conduction band edge, E c [Fig. 22.2(b)]. At room temperature this impurity center is readily ionized, and (in the absence of other impurities) the concentration of electrons is close to the concentration of As atoms. Impurities of this type are known as donors. FIGURE 22.1 Crystalline lattice of silicon, a = 5.43 ? at 300°C. a ? 2000 by CRC Press LLC While considering the contribution j p of the predominantly filled valence band to the current density, it is convenient to concentrate on the few missing electrons. This is achieved as follows: let v(k) be the velocity of electron described by the wave function (20.2). Then (22.3) Here we have noted again that a completely filled band does not contribute to the current density. The picture emerging from (22.3) is that of particles (known as holes) with the charge +q and velocities corresponding to those of missing electrons. The concentration of holes in the valence band is controlled by adding acceptor- type impurities (such as boron in silicon), which form local energy levels close to the top of the valence band. At room temperature these energy levels are occupied by electrons that come from the valence band and leave FIGURE 22.2 Band diagrams of metal (a) and semiconductor (b); c, electron; C, missing electron (hole). FIGURE 22.3 Simplified E(k) dependence for Si (a) and GaAs (b). At room temperature E g (Si) = 1.12 eV, E g (GaAs) = 1.43 eV, and D = 0.31 eV; (1) and (2) indicate direct and indirect band-to-band transitions. jvk vkvkvk p qq q=- =- - é ? ê ê ê ù ? ú ú ú = ???? () () () () empty states all statesfilled states empty states ? 2000 by CRC Press LLC the holes behind. Assuming that the Fermi level is removed from both E c and E v by at least 3k B T (a nondegenerate semiconductor), the concentrations of electrons and holes are given by n = N c exp[(F – E c )/k B T] (22.4) and p = N v exp[(E v – F)/k B T] (22.5) where N c = 2 (2m* n pk B T) 3/2 /h 3 and N v = 2(2m* p pk B T) 3/2 /h 3 are the effective densities of states in the conduction and valence bands, respectively, h is Plank constant, and the effective masses m* n and m* p depend on the details of the band structure [Pierret, 1987]. In a nondegenerate semiconductor, np = N c N v exp(–E g /k B T) D = n 2 i is independent of the doping level. The neutrality condition can be used to show that in an n-type (n > p) semiconductor at or below room temperature n(n + N a )(N d – N a – n) –1 = (N c /2) exp(–DW d /k B T) (22.6) where N d and N a denote the concentrations of donors and acceptors, respectively. Corresponding temperature dependence is shown for silicon in Fig. 22.4. Around room temperature n = N d – N a , while at low temperatures n is an exponential function of temperature with the activation energy DW d /2 for n > N a and DW d for n < N a . The reduction of n compared with the net impurity concentration N d – N a is known as a freeze-out effect. This effect does not take place in the heavily doped semiconductors. For temperatures T > T i = (E g /2k B )/ln[ /(N d – N a )] the electron concentration n ? n i >> N d – N a is no longer dependent on the doping level (Fig. 22.4). In this so-called intrinsic regime electrons come directly from the valence band. A loss of technological control over n and p makes this regime unattractive for electronic FIGURE 22.4The inverse temperature dependence of electron concentration in Si; 1: N d = 10 17 cm –3 , N a = 0; 2: N d = 10 16 cm –3 , N a = 10 14 cm –3 . NN cv ? 2000 by CRC Press LLC applications. Since T i } E g the transition to the intrinsic region can be delayed by using widegap semiconductors. Both silicon carbide (several types of SiC with different lattice structures are available with E g = 2.2–2.86 eV) and diamond (E g = 5.5 eV) have been used to fabricate diodes and transistors operating in the 300–700°C temperature range. Transport Properties In a semiconductor the motion of an electron is affected by frequent collisions with phonons (quanta of lattice vibrations), impurities, and crystal imperfections. In weak uniform electric fields, %, the carrier drift velocity, v d , is determined by the balance of the electric and collision forces: m n *v d /t = –q% (22.7) where t is the momentum relaxation time. Consequently v d = –m n %, where m n = qt/m* n is the electron mobility. For an n-type semiconductor with uniform electron density, n, the current density j n = –qnv d and we obtain Ohm’s law j n = s% with the conductivity s = qnm n . The momentum relaxation time can be approximately expressed as 1/t = 1/t ii + 1/t ni + 1/t ac + 1/t npo + 1/t po + 1/t pe + . . . (22.8) where t ii , t ni , t ac , t npo , t po , t pe are the relaxation times due to ionized impurity, neutral impurity, acoustic phonon, nonpolar optical, polar optical, and piezoelectric scattering, respectively. In the presence of concentration gradients, electron current density is given by the drift-diffusion equation j n = qnm n % + qD n ? (22.9) where the diffusion coefficient D n is related to mobility by the Einstein relation D n = (k B T/q)m n . A similar equation can be written for holes and the total current density is j = j n + j p . The right-hand side of (22.9) may contain additional terms corresponding to temperature gradient and compositional nonunifor- mity of the material [Wolfe et al., 1989]. In sufficiently strong electric fields the drift velocity is no longer proportional to the electric field. Typical velocity–field dependencies for several semiconductors are shown in Fig. 22.5. In GaAs v d (%) dependence is not monotonic, which results in negative differential conductivity. Physically, this effect is related to the transfer of electrons from the conduction band to a secondary valley (see Fig. 22.3). The limiting value v s of the drift velocity in a strong electric field is known as the saturation velocity and is usually within the 10 7 –3·10 7 cm/s range. As semiconductor device dimensions are scaled down to the submi- crometer range, v s becomes an important parameter that determines the upper limits of device performance. FIGURE 22.5Electron (a) and hole (b) drift velocity versus electric field dependence for several semiconductors at N d = 10 17 cm –3 . (Source: R.J. Trew, J.-B. Yan, and L.M. Mack, Proc. IEEE, vol. 79, no. 5, p. 602, May 1991. ? 1991 IEEE.) ? 2000 by CRC Press LLC The curves shown in Fig. 22.5 were obtained for uniform semiconductors under steady-state conditions. Strictly speaking, this is not the case with actual semiconductor devices, where velocity can “overshoot” the value shown in Fig. 22.5. This effect is important for Si devices shorter than 0.1mm (0.25 mm for GaAs devices) [Shur, 1990; Ferry, 1991]. In such extreme cases the drift-diffusion equation (22.9) is no longer adequate, and the analysis is based on the Boltzmann transport equation (22.10) Here f denotes the distribution function (number of electrons per unit volume of the phase space, i.e., f = dn/d 3 rd 3 p), v is electron velocity, p is momentum, and (?f/?t) coll is the “collision integral” describing the change of f caused by collision processes described earlier. For the purpose of semiconductor modeling, Eq. (22.10) can be solved directly using various numerical techniques, including the method of moments (hydrodynamic modeling) or Monte Carlo approach. The drift-diffusion equation (22.9) follows from (22.10) as a special case. For even shorter devices quantum effects become important and device modeling may involve quantum transport theory [Ferry, 1991]. Hall Effect In a uniform magnetic field electrons move along circular orbits in a plane normal to the magnetic field B with the angular (cyclotron) frequency w c = qB/m* n .For a uniform semiconductor the current density satisfies the equation j = s(% + R H [jB]) (22.11) In the usual weak-field limit w c t << 1 the Hall coefficient R H = –r/nq and the Hall factor r depend on the dominating scattering mode. It varies between 3p/8 ? 1.18 (acoustic phonon scattering) and 315p/518 ? 1.93 (ionized impurity scattering). The Hall coefficient can be measured as R H = V y d/I x B using the test structure shown in Fig. 22.6. In this expression V y is the Hall voltage corresponding to I y = 0 and d denotes the film thickness. FIGURE 22.6Experimental setup for Hall effect measurements in a long two-dimensional sample. The Hall angle is determined by a setting of the rheostat that renders j y = 0. Magnetic field B = B z . (Source: K.W. B?er, Surveys of Semiconductor Physics, New York: Chapman & Hall, 1990, p. 760. With permission.) ? ? ? ? f t fqf f t +?+ ?= ? è ? ? ? ÷ v % p coll ? 2000 by CRC Press LLC Combining the results of the Hall and conductivity measurements one can extract the carrier concentration type (the signs of V y are opposite for n-type and p-type semiconductors) and Hall mobility m H = rm: m H = –R H s, n = –r/qR H (22.12) Measurements of this type are routinely used to extract concentration and mobility in doped semiconductors. The weak-field Hall effect is also used for the purpose of magnetic field measurements. In strong magnetic fields w c t >> 1 and on the average an electron completes several circular orbits without a collision. Instead of the conventional E b (k) dependence, the allowed electron energy levels in the magnetic field are given by (\= h/2p; s = 0, 1, 2, . . .) E s = \w c (s + 1/2) + 2 k 2 z /2m* n (22.13) The first term in Eq. (22.13) describes the so-called Landau levels, while the second corresponds to the kinetic energy of motion along the magnetic field B = B z . In a pseudo-two-dimensional system like the channel of a field-effect transistor the second term in Eq. (22.13) does not appear, since the motion of electrons occurs in the plane perpendicular to the magnetic field. 1 In such a structure the electron density of states (number of allowed quantum states per unit energy interval) is peaked at the Landau level. Since w c } B, the positions of these peaks relative to the Fermi level are controlled by the magnetic field. The most striking consequence of this phenomenon is the quantum Hall effect, which manifests itself as a stepwise change of the Hall resistance r xy = V y /I x as a function of magnetic field (see Fig. 22.7). At low temperature (required to establish the condition t << w c –1 ) it can be shown [von Klitzing, 1986] that r xy = h/sq 2 (22.14) where s is the number of the highest occupied Landau level. Accordingly, when the increased magnetic field pushes the sth Landau level above the Fermi level, r xy changes from h/sq 2 to h/(s–1)q 2 . This stepwise change of r xy is seen in Fig. 22.7. Localized states produced by crystal defects determine the shape of the r xy (B) dependence between the plateaus given by Eq. (22.14). They are also responsible for the disappearance of r xx = V x /I x between the transition points (see Fig. 22.7). The quantized Hall resistance r xy is expressed in terms of fundamental constants and can be used as a resistance standard that permits one to measure an electrical resistance with better accuracy than any wire resistor standard. In an ultraquantum magnetic field, i.e., when only the lowest Landau level is occupied, plateaus of the Hall resistance are also observed at fractional s (the fractional quantum Hall effect). These plateaus are related to the Coulomb interaction of electrons. Electrical Breakdown In sufficiently strong electric fields a measurable fraction of electrons (or holes) acquires sufficient energy to break the valence bond. Such an event (called impact ionization) results in the creation of an electron–hole pair by the energetic electron. Both the primary and secondary electrons as well as the hole are accelerated by the electric field and may participate in further acts of impact ionization. Usually, the impact ionization is balanced by recombination processes. If the applied voltage is high enough, however, the process of electron multiplication leads to avalanche breakdown. The threshold energy E th (the minimum electron energy required to produce an electron–hole pair) is determined by energy and momentum conservation laws. The latter usually results in E th > E g , as shown in Table 22.1. The field dependence of the impact ionization is usually described by the impact ionization coefficient a i , defined as the average number of electron–hole pairs created by a charge carrier per unit distance traveled. A simple analytical expression for a i [Okuto and Crowell, 1972] can be written as (22.15) 1 To simplify the matter we do not discuss surface subbands, which is justified as long as only the lowest of them is occupied. al i xaax=-+ ? è ? ? ( )exp/ 22 ? 2000 by CRC Press LLC where x = q%l/E th , a = 0.217 (E th /E opt ) 1.14 , l is the carrier mean free path, and E opt is the optical phonon energy (E opt = 0.063 eV for Si at 300°C). An alternative breakdown mechanism is tunneling breakdown, which occurs in highly doped semiconductors when electrons may tunnel from occupied states in the valence band into the empty states of the conduction band. Optical Properties and Recombination Processes If the energy of an incident photon \w > E g , then the energy conservation law permits a direct band-to-band transition, as indicated in Fig. 22.2(b). Because the photon’s momentum is negligible compared to that of an electron or hole, the electron’s momentum \k does not change in a direct transition. Consequently, direct transitions are possible only in direct-gap semiconductors where the conduction band minimum and the valence band maximum occur at the same k. The same is true for the reverse transition, where the electron is transferred FIGURE 22.7Experimental curves for the Hall resistance r xy = % y /j x and the resistivity r xx = % x /j x of a heterostructure as a function of the magnetic field at a fixed carrier density. (Source: K. von Klitzing, Rev. Modern Phys., vol. 58, no. 3, p. 525, 1986. With permission.) TABLE 22.1Impact Ionization Threshold Energy (eV) Semiconductor Si Ge GaAs GaP InSb Energy gap, E g 1.1 0.7 1.4 2.3 0.2 E th , electron-initiated 1.18 0.76 1.7 2.6 0.2 E th , hole-initiated 1.71 0.88 1.4 2.3 0.2 ? 2000 by CRC Press LLC from the conduction to the valence band and a photon is emitted. Direct-gap semiconductors (e.g., GaAs) are widely used in optoelectronics. In indirect-band materials [e.g., Si, see Fig. 22.3(a)], a band-to-band transition requires a change of momen- tum that cannot be accomplished by absorption or emission of a photon. Indirect band-to-band transitions require the emission or absorption of a phonon and are much less probable than direct transitions. For \w < E g [i.e., for l > l c = 1.24 mm/E g (eV) – cutoff wavelength] band-to-band transitions do not occur, but light can be absorbed by a variety of the so-called subgap processes. These processes include the absorption by free carriers, formation of excitons (bound electron–hole pairs whose formation requires less energy than the creation of a free electron and a free hole), transitions involving localized states (e.g., from an acceptor state to the conduction band), and phonon absorption. Both band-to-band and subgap processes may be responsible for the increase of the free charge carriers concentration. The resulting reduction of the resistivity of illuminated semiconductors is called photoconductivity and is used in photodetectors. In a strong magnetic field (w c t >> 1) the absorption of microwave radiation is peaked at w = w c . At this frequency the photon energy is equal to the distance between two Landau levels, i.e., \w = E S+1 – E S with reference to Eq. (22.13). This effect, known as cyclotron resonance, is used to measure the effective masses of charge carriers in semiconductors [in a simplest case of isotropic E(k) dependence, m* n = qB/w c ]. In indirect-gap materials like silicon, the generation and annihilation (or recombination) of electron–hole pairs is often a two-step process. First, an electron (or a hole) is trapped in a localized state (called a recombi- nation center) with the energy near the center of the energy gap. In a second step, the electron (or hole) is transferred to the valence (conduction) band. The net rate of recombination per unit volume per unit time is given by the Shockley–Read–Hall theory as (22.16) where t n , t p , p 1 , and n 1 are parameters depending on the concentration and the physical nature of recombination centers and temperature. Note that the sign of R indicates the tendency of a semiconductor toward equilibrium (where np = n 2 i , and R = 0).For example, in the depleted region np < n 2 i and R < 0, so that charge carriers are generated. Shockley–Read–Hall recombination is the dominating recombination mechanism in moderately doped silicon. Other recombination mechanisms (e.g., Auger) become important in heavily doped semiconductors [Wolfe et al., 1989; Shur, 1990; Ferry, 1991]. The recombination processes are fundamental for semiconductor device theory, where they are usually modeled using the continuity equation (22.17) Nanostructure Engineering Epitaxial growth techniques, especially molecular beam epitaxy and metal-organic chemical vapor deposition, allow monolayer control in the chemical composition process. Both single thin layers and superlattices can be obtained by such methods. The electronic properties of these structures are of interest for potential device applications. In a single quantum well, electrons are bound in the confining well potential. For example, in a rectangular quantum well of width b and infinite walls, the allowed energy levels are E s (k) = p 2 s 2 2 /(2m* n b 2 ) + 2 k 2 /(2m* n ), s = 1, 2, 3, . . . (22.18) where k is the electron wave vector parallel to the plane of the semiconductor layer. The charge carriers in quantum wells exhibit confined particle behavior. Since E s } b –2 , well structures can be grown with distance R np n pp nn i np = - +++ 2 11 tt()() ? ? n t div q R n =- j ? 2000 by CRC Press LLC between energy levels equal to a desired photon energy. Furthermore, the photoluminescence intensity is enhanced because of carrier confinement. These properties are advantageous in fabrication of lasers and photodetectors. If a quantum well is placed between two thin barriers, the tunneling probability is greatly enhanced when the energy level in the quantum well coincides with the Fermi energy (resonant tunneling). The distance between this “resonant” energy level and the Fermi level is controlled by the applied voltage. Consequently, the current peaks at the voltage corresponding to the resonant tunneling condition. The resulting negative differential resistance effect has been used to fabricate microwave generators operating at both room and cryogenic temperatures. Two kinds of superlattices are possible: compositional and doping. Compositional superlattices are made of alternating layers of semiconductors with different energy gaps. Doping superlattices consist of alternating n- and p-type layers of the same semiconductor. The potential is modulated by electric fields arising from the charged dopants. Compositional superlattices can be grown as lattice matched or as strained layers. The latter are used for modification of the band structure, which depends on the lattice constant to produce desirable properties. In superlattices energy levels of individual quantum wells are split into minibands as a result of electron tunneling through the wide-bandgap layers. This occurs if the electron mean free path is larger than the superlattice period. In such structures the electron motion perpendicular to the layer is quantized. In a one- dimensional tight binding approximation the miniband can be described as (22.19) where a is the superlattice period and E o is the half-width of the energy band. The electron group velocity v = –1 ?E(k)/?k = (E o a/\) sin(ka) (22.20) is a decreasing function of k (and hence of energy) for k > p/2a. The higher energy states with k > p/2a may become occupied if the electrons are heated by the external field. As a result, a negative differential resistance can be achieved at high electric fields. The weak-field mobility in a superlattice may exceed that of the bulk material because of the separation of dopants if only barriers are doped. In such modulated structures, the increased spatial separation between electrons and holes is also responsible for a strong increase in recombi- nation lifetimes. Disordered Semiconductors Both amorphous and heavily doped semiconductors are finding increasing applications in semiconductor technol- ogy. The electronic processes in these materials have specific features arising from the lack of long-range order. Amorphous semiconductors do not have a crystalline lattice, and their properties are determined by the arrangement of the nearest neighboring atoms. Even so, experimental data show that the forbidden energy band concept can be applied to characterize their electrical properties. However, the disordered nature of these materials results in a large number of localized quantum states with energies within the energy gap. The localized states in the upper and lower half of the gap behave like acceptors and donors, respectively. As an example, consider the density of states in hydrogenated amorphous silicon (a-Si) shown in Fig. 22.8. The distribution of the localized states is not symmetrical with respect to the middle of the energy gap. In particular, the undoped hydrogenated amorphous silicon is an n-type semiconductor. Usually amorphous semiconductors are not sensitive to the presence of impurity atoms, which saturate all their chemical bonds in the flexible network of the host atoms. (Compare this with a situation in crystalline silicon where an arsenic impurity can form only four chemical bonds with the host lattice, leaving the fifth responsible for the formation of the donor state.) Consequently, the doping of amorphous semiconductors is difficult to accomplish. However, in hydrogenated a-Si (which can be prepared by the glow discharge decom- position of silane), the density of the localized states is considerably reduced and the conductivity of this material can be controlled by doping. As in crystalline semiconductors, the charge carrier concentration in hydrogenated Ek E ka o () [ cos()]=-1 ? 2000 by CRC Press LLC a-Si can also be affected by light and strong field effects. The a-Si is used in applications that require deposition of thin-film semiconductors over large areas [xerography, solar cells, thin-film transistors (TFT) for liquid- crystal displays]. The a-Si device performance degrades with time under electric stress (TFTs) or under illu- mination (Staebler–Wronski effect) because of the creation of new localized states. An impurity band in crystalline semiconductors is another example of a disordered system. Indeed, the impurity atoms are randomly distributed within the host lattice. For lightly doped semiconductors at room temperature, the random potential associated with charged impurities can usually be ignored. As the doping level increases, however, a single energy level of a donor or an acceptor is transformed into an energy band with a width determined by impurity concentrations. Unless the degree of compensation is unusually high, this reduces the activation energy compared to lightly doped semiconductors. The activation energy is further reduced by the overlap of the wave functions associated with the individual donor or acceptor states. For sufficiently heavy doping, i.e., for N d > N dc = (0.2/a B ) 3 , the ionization energy is reduced to zero, and the transition to metal-type conductivity (the Anderson–Mott transition) takes place. In this expression the effective electron Bohr radius a B = \/, wher E i is the ionization energy of the donor state. For silicon, N dc ? 3.8 · 10 18 cm –3 . This effect explains the absence of freeze-out in heavily doped semiconductors. Defining Terms Conduction/valence band: The upper/lower of the two partially filled bands in a semiconductor. Donors/acceptors: Impurities that can be used to increase the concentration of electrons/holes in a semicon- ductor. Energy band: Continuous interval of energy levels that are allowed in the periodic potential field of the crystalline lattice. Energy gap: The width of the energy interval between the top of the valence band and the bottom of the conduction band. FIGURE 22.8 Experimentally determined density of states for a-Si. A and B are acceptor-like and donor-like states, respectively. The arrow marks the position of the Fermi level e fo in undoped hydrogenated a-Si. The energy spectrum is divided into extended states E, band-tail states T, and gap states G. (Source: M.H. Brodsky, Ed., Amorphous Semiconductors, 2nd ed., Berlin: Springer-Verlag, 1985. With permission.) 2mE ni * ? 2000 by CRC Press LLC Hole:Fictitious positive charge representing the motion of electrons in the valence band of a semiconductor; the number of holes equals the number of unoccupied quantum states in the valence band. Phonon: Quantum of lattice vibration. Photon:Quantum of electromagnetic radiation. Related Topic 52.1 Introduction References D.K. Ferry, Semiconductors, New York: Macmillan, 1991. Y. Okuto and C.R. Crowell, Phys. Rev., vol. B6, p. 3076, 1972. R.F. Pierret, Advanced Semiconductor Fundamentals, Reading, Mass.: Addison-Wesley, 1987. M. Shur, Physics of Semiconductor Devices, Englewood Cliffs, N.J.: Prentice-Hall, 1990. K. von Klitzing, Rev. Modern Phys., vol. 58, p. 519, 1986. C.M. Wolfe, N. Holonyak, and G.E. Stilman, Physical Properties of Semiconductors, Englewood Cliffs, N.J.: Prentice-Hall, 1989. Further Information Engineering aspects of semiconductor physics are often discussed in the IEEE Transactions on Electron Devices, Journal of Applied Physics, and Solid-State Electronics. 22.2 Diodes Miran Milkovic Diodes are the most widely used devices in low- and high-speed electronic circuits and in rectifiers and power supplies. Other applications are in voltage regulators, detectors, and demodulators. Rectifier diodes are capable of conducting several hundred amperes in the forward direction and less than 1 mA in the reverse direction. Zener diodes are ordinary diodes operated in the Zener or avalanche region and are used as voltage regulators. Varactor diodes are ordinary diodes used in reverse biasing as voltage-dependent capacitors. Tunnel diodes and quantum well devices have a negative differential resistance and are capable of operating in the upper gigahertz region. Photodiodes are ordinary diodes operated in the reverse direction. They are sensitive to light and are used as light sensors. Solar cells are diodes which convert light energy into electrical energy. Schottky diodes, also known as metal-semiconductor diodes, are extremely fast because they are majority carrier devices. pn-Junction Diode A pn-diode is a semiconductor device having a p-region, a n-region, and a junction between the regions. Modern planar semiconductor pn-junction diodes are fabricated by diffusion or implantation of impurities into a semiconductor. An n-type semiconductor has a relatively large density of free electrons to conduct electric current, and the p-type semiconductor has a relatively large concentration of “free” holes to conduct electric current. The pn-junction is formed during the fabrication process. There is a large concentration of holes in the p-semiconductor and a large concentration of electrons in the n-semiconductor. Because of their large concentration gradients, holes and electrons start to diffuse across the junction. As holes move across the junction, negative immobile charges (acceptors) are uncovered on the p side, and positive immobile charges (donors) are uncovered on the n side due to the movement of electrons across the junction. When sufficient numbers of the immobile charges on both sides of the junction are uncovered, a potential energy barrier voltage V 0 is created by the uncovered acceptors and donors. This barrier voltage prevents further diffusion of holes and electrons across the junction. The charge distribution of acceptors and donors establishes an opposing ? 2000 by CRC Press LLC electric field, E, which at equilibrium prevents a further diffusion of carriers across the junction. This equilib- rium can be regarded as the flow of two equal and opposite currents across the junction, such that the net current across the junction is equal to zero. Thus, one component represents the diffusion of carriers across the junction and the other component represents the drift of carriers across the junction due to the electric field E in the junction. The barrier voltage V 0 is, according to the Boltzmann relation, [Grove, 1967; Foustad, 1994] (22.21) In this equation, p p is the concentration of holes in the p-material and p n is the concentration of holes in the n-material. V T is the thermal voltage. V T = 26 mV at room temperature (300 K). With p p ? N A and p n ? where n i is the intrinsic concentration, the barrier voltage V 0 becomes approximately [Sze, 1985; Fonstad, 1994] (22.22) Here N A denotes the concentration of immobile acceptors on the p side of the junction and N D is the concentration of immobile donors on the n side of the junction. A depletion layer of immobile acceptors and donors causes an electric field E across the junction. For silicon, V 0 is at room temperature T = 300°K, typically V 0 = 0.67 V for an abrupt junction with N A = 10 17 at/cm 3 and N D = 10 15 at/cm 3 . The depletion layer width is typically about 4 mm, and the electric field E is about 60 kV/cm. Note the magnitude of the electric field across the junction. pn-Junction with Applied Voltage If the externally applied voltage V D to the diode is opposite to the barrier voltage V 0 , then p p in the Boltzmann relation in Eq. (22.21) is altered to p p = p n exp(V 0 – V D )/V T (22.23) This implies that the effective barrier voltage is reduced and the diffusion of carriers across the junction, is increased. Accordingly the concentration of diffusing holes into the n material is at x = 0, p n (x = 0) = p n exp V D /V T (22.24) and accordingly the concentration of electrons n n (x = 0) = n n exp V D /V T (22.25) Most modern planar diodes are unsymmetrical. Figure 22.9 shows a pn-diode with the n region W n much shorter than the diffusion length L pn of holes in the n-semiconductor region. This results in a linear concen- tration gradient of injected diffusing holes in the n region given by dp/dx = –(p n expV D /V T – p n )/W n (22.26) VVpp Tpn0 =/ln[ ] n N i D 2 VVNNn TADi0 2 = ln[ ]/ ? 2000 by CRC Press LLC The diffusion gradient is negative since the concentration of positive holes decreases with distance due to the hole–electron recombinations. The equation for the hole diffusion current is I p = –qA j D p dp/dx (22.27) where A j is the junction area, D p is the diffusion constant for holes, and q is the elementary charge. By combining of above equations we obtain I p = (qA j D p p n /W n ) (exp V D /V T – 1) (22.28) In the p-semiconductor we assume that L np << W p ; then dn/dx = n p exp(V D /V T – 1) (22.29) By substituting this into the electron diffusion equation, I n = qA j D n dn/dx (22.30) we obtain I n = (qA j D n n p )/L np (exp V D /V T – 1) (22.31) Thus, the total junction diffusion current is I D = I p + I n = {qA j D p p n /W n + qA j D n n p /L np } (exp V D /V T – 1) (22.32) Since the recombination of the injected carriers establishes a diffusion gradient, this in turn yields a flow of current proportional to the slope. For *–V D * >> V T , i.e., V D = –0.1 V, I S = (qA i D p p n /W n + qA i D n n p /L np ) (22.33) FIGURE 22.9Planar diodes are fabricated in planar technology. Most modern diodes are unsymmetrical; thus W n << L pn . The p-type region is more highly doped than the n region. ? 2000 by CRC Press LLC Here I S denotes the reverse saturation current. In practical junctions, the p region is usually much more heavily doped than the n region; thus n p << p n . Also, since W n << L np in Eq. (22.33), we obtain I S = qA j D p p n /W n = qA j D p n i 2 /W n N D (22.34) The reverse saturation current in short diodes is mainly determined by the diffusion constant D p and the width W n of the n region, by intrinsic concentration n i , by the doping concentration N D in the n region, and by the diode area A j . (In reality, I S is also slightly dependent on the reverse voltage [Phillips, 1962].) If V D is made positive, the exponential term in Eq. (22.32) rapidly becomes larger than one; thus I D = I S expV D /V T (22.35) where I D is the diode forward current and I S is the reverse saturation current. Another mechanism predominates the reverse current I S in silicon. Due to the recombination centers in the depletion region, a generation-recombination hole–electron current I G is generated in the depletion region [Phillips, 1962; Sze, 1985]. I G = KqA j eX d (22.36a) Here e is the generation rate unit volume, A j is the junction area, q is the elementary charge, X d is the depletion layer thickness, and K is a dimensional constant. I G is proportional to the thickness X d of the depletion layer and to the junction area A j . Since X d increases with the square root of the reverse voltage, I G increases accordingly, yielding a slight slope in the reverse I-V characteristic. The forward I-V characteristic of the practical diode is only slightly affected (slope m = 2) at very small forward currents (I D = 1 nA to 1 mA). In practical diodes n ? 1 at small to medium currents (I D = 1 mA to 10 mA). At large currents (I D > 10 mA), m = 1 to 2 due to the high current effects [Phillips, 1962] and due to the series bulk resistance of the diode. The reverse current I R in silicon is voltage dependent. The predominant effect is the voltage dependence of the generation-recombination current I G and to a smaller extent the voltage dependence of I S . The total reverse current of the diode is thus equal to I R = I G +I S (22.36b) Forward-Biased Diode For most practical applications I D = I S expV D /mV T (22.37) where I S is the reverse saturation current (about 10 –14 A for a small-signal diode); V T = kT/q is the thermal voltage equal to 26 mV at room temperature; k = Boltzmann’s constant, 1.38 · 10 –23 J/K; T is the absolute temperature in kelvin; q is the elementary charge 1.602·10 –19 C; m is the ideality factor, m = 1 for medium currents, m = 2 for very small and very large currents; I S is part of the total reverse current I R of the diode I R = I S + I G ; and I S is the reverse saturation current and I G is the generation- recombination current, also called diode leakage current because I G is not a part of the carrier diffusion process in the diode. I D is exponentially related to V D in Fig. 22.10. FIGURE 22.10I D versus V D of a diode. ? 2000 by CRC Press LLC Temperature Dependence of V D Equation (22.37) solved for V D yields V D = mV T ln(I D /I S ) (22.38) at constant current I D , the diode voltage V D is temperature dependent because V T and I S are temperature dependent. Assume m = 1. The reverse saturation current I S from Eq. (22.34) is I S = qA j n i 2 D p /W n N D = B 1 n i 2 D p = B 2 n i 2 m p where D p = V T m p . With m p = B 3 T –n and for n i 2 n i 2 = B 4 T g exp(–V G0 /V T ) (22.39) where g = 4 – n, and V G0 is the extrapolated bandgap energy [Gray and Meyer, 1993]. With Eq. (22.39) into Eq. (22.38), the derivative dV D /dT for I D = const yields dV D /dT = (V D – V G0 )/T – gk/q (22.40) At room temperature (T = 300 K), and V D = 0.65 V, V G0 = 1.2 V, g = 3, V T = 26 mV, and k/q = 86 mV/degree, one gets dV D /dT ? –2.1 mV/degree.The temperature coefficient TC of V D is thus TC =dV D /V D dT = 1/T – V G0 /V D T – gk/qV D (22.41) For the above case TC ? –0.32%/degree. In practical applications it is more convenient to use the expression V D (d 2 ) = V D (d 1 ) – TC(d 2 – d 1 ) (22.42) where d 1 and d 2 are temperatures in degrees Celsius. For TC = –0.32%/degree and V D = 0.65 V at d 1 = 27°C, V D = 0.618 V at d 2 = 37°C. Both dV D /dT and TC are I D dependent. At higher I D , both dV D /dT and TC are smaller than at a lower I D , as shown in Fig. 22.11. I D -V D Characteristic From the I D -V D characteristic of the diode one can find for m = 1 I D1 = I S exp(V D1 /V T ) and I D2 = I S exp(V D2 /V T ) (22.43) FIGURE 22.11(a) I D versus V D of a diode at three different temperatures d 3 > d 2 > d 1 . (b) V D = f(Temp), I DC > I DB > I DA . ? 2000 by CRC Press LLC Thus, the ratio of currents is I D2 /I D1 = exp (V D2 – V D1 )/V T (22.44) or the difference voltage V D2 – V D1 = V T ln(I D2 /I D1 ) (22.45) in terms of base 10 logarithm V D2 – V D1 = V T 2.3 log(I D2 /I D1 ) (22.46) For (I D2 /I D1 ) = 10 (one decade), V D2 – V D1 = ~60 mV, or V D2 – V D1 = 17.4 mV for (I D2 /I D1 ) = 2. In a typical example, m = 1, V D = 0.67 V at I D = 100 mA. At I D = 200 mA, V D = 0.67 V + 17.4 mV = 0.687 V. DC and Large-Signal Model The diode equation in Eq. (22.37) is widely utilized in diode circuit design. I S and m can sometimes be found from the data book or they can be determined from measured I D and V D . From two measurements of I D and V D , for example, I D = 0.2 mA at V D = 0.670 V and I D = 10 mA at V D = 0.772 V, one can find m = 1.012 and I S = 1.78·10 –15 A for the particular diode. A practical application of the large-signal diode model is shown in Fig. 22.13. Here, the current I D through the series resistor R and a diode D is to be found, I D = (V CC – V D )/R (22.47) The equation is implicit and cannot be solved for I D since V D is a function of I D . Here, V D and I D are determined by using iteration. By assuming V D = V D0 = 0.6 V (cut-in voltage), the first iteration yields I D (1) = (5 V – 0.6 V)/1 kW = 4.4 mA Next, the first iteration voltage V D (1) is calculated (by using m and I S above and I D1 = 4.4 mA), thus V D (1) = mV T [ln I D (1)/I S ] = 1.012 2 26 mV ln(4.4 mA/1.78 · 10 –15 A) = 0.751 V FIGURE 22.12I D versus V D of a diode on a semi-logarithmic plot. FIGURE 22.13Diode-resistor biasing circuit. ? 2000 by CRC Press LLC From the second iteration I D (2) = [V CC – V D (1)]/R = 4.25 mA and thus V D (2) = 0.75 V. The third iteration yields I D (3) = 4.25 mA, and V D (3) = 0.75 V. These are the actual values of I D and V D for the above example, since the second and the third iterations are almost equal. Graphical analysis (in Fig. 22.14) is another way to analyze the circuit in Fig. 22.13. Here the load line R is drawn with the diode I-V characteristic, where V CC = V D + I D R. This type of analysis is illustrative but not well suited for a numerical analysis. High Forward Current Effects In the pn-junction diode analysis it was assumed that the density of injected carriers from the p region into the n region is small compared to the density of majority carriers in that region. Thus, all of the forward voltage V D appears across the junction. Therefore, the injected carriers move only because of the diffusion. At high forward currents this is not the case anymore. When the voltage drop across the bulk resistance becomes comparable with the voltage across the junction, the effective applied voltage is reduced [Phillips, 1962]. Due to the electric field created by the voltage drop in the bulk (neutral) regions, the current is not only a diffusion current anymore. The drift current due to the voltage drop across the bulk region opposes the diffusion current. The net effect is that, first, the current becomes proportional to twice the diffusion constant, second, the high- level current becomes independent of resistivity, and, third, the magnitude of the exponent is reduced by a factor of two in Eq. (22.37). The effect of high forward current on the I-V characteristic is shown in Fig. 22.15. In all practical designs, m ? 2 at I D 3 20 mA in small-signal silicon diodes. FIGURE 22.14Graphical analysis of a diode-resistor circuit. FIGURE 22.15I D versus V D of a diode at low and high forward currents. FIGURE 22.16(a) Simplified piecewise linear model of a diode; (b) improved piecewise linear model of a diode. The diode cut-in voltage V D0 is defined as the voltage V D at a very small current I D typically at about 1 nA. For silicon diodes this voltage is typically V D0 = 0.6 V. ? 2000 by CRC Press LLC Large-Signal Piecewise Linear Model Piecewise linear model of a diode is a very useful tool for quick circuit design containing diodes. Here, the diode is represented by asymptotes and not by the exponential I-V curve. The simplest piecewise linear model is shown in Fig. 22.16(a). Here D i is an ideal diode with V D = 0 at I D 3 0, in series with V D0 , where V D0 is the diode cut-in or threshold voltage. The current in the diode will start to flow at V D 3 V D0 . An improved model is shown in Fig. 22.16(b), where V D0 is again the diode voltage at a very small current I D0 , r D is the extrapolated diode resistance, and I D1 is the diode current in operating point 1. Thus, the diode voltage is V D1 = V D0 + I D1 r D (22.48) where V D1 is the diode voltage at I D1 . V D0 for silicon is about 0.60 V. r D is estimated from the fact that V D in a real diode is changing per decade of current by m 2.3 V T . Thus, V D changes about 60 mV for a decade change of current I D at m = 1. Thus in a 0.1 to 10 mA current change, V D changes about 120 mV, which corresponds to anr D ? 120 mV/10 mA = 12 W. The foregoing method is an approximation; however, it is quite practical for first-hand calculations. To compare this with the above iterative approach let us assume m = 1, V D0 = 0.60 V, r D = 12 W, V CC = 5 V, R = 1 kW. The current I D1 = [V CC – V D0 ]/(R + r D ) = 4.34 mA compared with I D1 = 4.25 mA in the iterative approach. Small-Signal Incremental Model In the small-signal incremental model, the diode is represented by linear elements. In small-signal (incremental) analysis, the diode voltage signals are assumed to be about V T /2 or less, thus much smaller than the dc voltage V D across the diode. In the forward-biased diode, three elements are of practical interest: incremental resistance (or small-signal or differential resistance) r d , the diffusion capacitance C d , and the junction capacitance C j . Incremental Resistance, r d For small signals the diode represents a small-signal resistance (often called incremental or differential resis- tance) r d in the operating point (I D ,V D ) where r d = dV D /dI D = mV T /I S exp(V D /mV T ) = mV T /I D (22.49) In Fig. 22.17, r d is shown as the tangent in the dc operating point (V D , I D ). Note that r d is independent of the geometry of the device and inversely proportional to the diode dc current. Thus for I D = 1 mA, m = 1 and V T = 26 mV, the incremental resistance is r d = 26 W. Diffusion Capacitance, C d C d is associated with the injection of holes and electrons in the forward- biased diode. In steady state, holes and electrons are injected across the junction. Hole and electron currents flow due to the diffusion gradients on both sides of the junction in Fig. 22.18. In a short diode, holes are traveling a distance W n << L p n . For injected holes, and since w n <<L p n I p = dq p /dt = dq p v/dx (22.50) where v is the average carrier velocity, D p is the diffusion constant for holes and W n is the travel distance of holes. By integrating of Eq. (22.50) one gets FIGURE 22.17Small-signal incre- mental resistance r d of a diode. Idxvdq p W p Q np 00 òò = ? 2000 by CRC Press LLC and the charge Q p of holes becomes Q p = I p W n /v = I p t p . (22.51) t p = W n /v is the transit time holes travel the distance W n . Similarly, for electron charge Q n , since W p >> L np Q n = I n L np /v = I n t n . (22.52) Thus the total diffusion charge Q d is Q d = Q p + Q n , (22.53) and the total transit time is t F = t p + t n , (22.54) and with I p + I n = I D = I S exp V D /mV T and Eqs. (22.51), (22.52), and (22.54) one gets Q d = t F I S exp V D /mV T = t F I D . (22.55) The total diffusion capacitance is C d = C p + C n = dQ d /dV D = Q d /mV T (22.56) and from Eqs. (22.55) and (22.56) C d = I D t F /mV T . (22.57) C d is thus directly proportional to I D and to the carrier transit time t F . For an unsymmetrical diode with W n << L pn and N A >> N D [Gray and Meyer, 1984] t F ? W 2 n /2D p (22.58) t F is usually given in data books or it can be measured. For W n = 6 m and D p = 14 cm 2 /s, t F ? 13 ns, I D = 1 mA, V T = 26 mV, and m = 1, the diffusion capacitance is C d = 500 pF. FIGURE 22.18 Minority carrier charge injection in a diode. n-regionp + -region n,p Q p Q n L n p L p n XW p X = 0 X = W n ? 2000 by CRC Press LLC Depletion Capacitance, C j The depletion region is always present in a pn-diode. Because of the immobile ions in the depletion region, the junction acts as a voltage-dependent plate capacitor C j [Gray and Meyer, 1993; Horenstein, 1990] (22.59) V D is the diode voltage (positive value for forward biasing, negative value for reverse biasing), and C j0 is the zero bias depletion capacitance; A j is the junction diode area: C j0 = KA j (22.60) K is a proportionality constant dependent on diode doping, and A j is the diode area. C j is voltage dependent. As V D increases, C j increases in a forward-biased diode in Fig. 22.19. For V 0 = 0.7 V and V D = –10 V and C j0 = 3 pF, the diode depletion capacitance is C j = 0.75 pF. In Fig. 22.20 the small-signal model of the diode is shown. The total small-signal time constant t d is thus (by neglecting the bulk series diode resistance R BB ) t d = r d (C d + C j ) = r d C d + r d C j = t F + r d C j (22.61) t d is thus current dependent. At small I D the r d C j product is predominant. For high-speed operation r d C j must be kept much smaller than t F . This is achieved by a large operating current I D . The diode behaves to a first approximation as a frequency-dependent element. In the reverse operation, the diode behaves as a high ohmic resistor R p ? V R /I G in parallel with the capacitor C j . In forward small-signal operation, the diode behaves as a resistor r d in parallel with the capacitors C j and C d (R p is neglected). Thus, the diode is in a first approximation, a low-pass network. FIGURE 22.19Depletion capacitance C j of a diode versus diode voltage V R . FIGURE 22.20Simplified small-signal model of a diode. CCVV jj D =- 00 ? 2000 by CRC Press LLC Large-Signal Switching Behavior of a pn-Diode When a forward-biased diode is switched from the forward into the reverse direction, the stored charge Q d of minority carriers must first be removed. The charge of minority carriers in the forward-biased unsymmetrical diode is from Eqs. (22.55) and (22.58) Q d = I D t F = I D W 2 n /2D p (22.62) where W n << L pn is assumed. t F is minimized by making W n very small. Very low-lifetime t F is required for high-speed diodes. Carrier lifetime t F is reduced by adding a large concentration of recombination centers into the junction. This is common practice in the fabrication of high-speed computer diodes [Phillips, 1962]. The charge Q d is stored mainly in the n region in the form of a concentration gradient of holes in Fig. 22.21(a). The diode is turned off by moving the switch from position (a) into position (b) [Fig. 22.21(a)]. The removal of carriers is done in three time intervals. During the time interval t 1 , also called the recovery phase, a constant reverse current *I R * = V R /R flows in the diode. During the time interval t 2 – t 1 the charge in the diode is reduced by about 1/2 of the original charge. During the third interval t 3 – t 2 , the residual charge is removed. If during the interval t 1 , *I R * >> I D , then Q d is reduced only by flow of reverse diffusion current; no holes arrive at the metal contact [Gugenbuehl et al., 1962], and t 1 ? t F (I D /*I R *) 2 (22.63) During time interval t 2 – t 1 , when *I R * = I D , in Fig. 22.21(b), t 2 – t 1 ? t F I D /*I R * (22.64) The residual charge is removed during the time t 3 – t 2 ? 0.5 t F . FIGURE 22.21(a) Diode is switched from forward into reverse direction; (b) concentration of holes in the n region; (c) diode turns off in three time intervals. ? 2000 by CRC Press LLC Diode Reverse Breakdown Avalanche breakdown occurs in a reverse-biased plane junction when the critical electric field E crt at the junction within the depletion region reaches about 3·10 5 V/cm for junction doping densities of about 10 15 to 10 16 at/cm 3 [Gray and Meyer, 1984]. At this electric field E crt , the minority carriers traveling (as reverse current) in the depletion region acquire sufficient energy to create new hole–electron pairs in collision with atoms. These energetic pairs are able to create new pairs, etc. This process is called the avalanche process and leads to a sudden increase of the reverse current I R in a diode. The current is then limited only by the external circuitry. The avalanche current is not destructive as long as the local junction temperature does not create local hot spots, i.e., melting of material at the junction. Figure 22.22 shows a typical I-V characteristic for a junction diode in the avalanche breakdown. The effect of breakdown is seen by the large increase of the reverse current I R when V R reaches –BV. Here BV is the actual breakdown voltage. It was found that I RA = M I R , where I RA is the avalanche reverse current at BV, M is the multiplication factor, and I R is the reverse current not in the breakdown region. M is defined as M = 1/[1 – V R /BV] n (22.65) where n = 3 to 6. As V R = BV, M ? ￥ and I RA ? ￥. The above BV is valid for a strictly plane junction without any curvature. However, in a real planar diode as shown in Fig. 22.9, the p-diffusion has a curvature with a finite radius x j . If the diode is doped unsymmetrically, thus s p >> s n , then the depletion area penetrates mostly into the n region. Because of the finite radius, the breakdown occurs at the radius x j , rather than in a plane junction [Grove, 1967]. The breakdown voltage is significantly reduced due to the curvature. In very shallow planar diodes, the avalanche breakdown voltage BV can be much smaller than 10 V. Zener and Avalanche Diodes Zener diodes (ZD) and avalanche diodes are pn-diodes specially built to operate in reverse breakdown. They operate in the reverse direction; however, their operating mechanism is different. In a Zener diode the hole–elec- tron pairs are generated by the electric field by direct transition of carriers from valence band into the conductance band. In an avalanche diode, the hole–electron pairs are generated by impact ionization due to high-energy holes and electrons. Avalanche and Zener diodes are extensively used as voltage regulators and as overvoltage protection devices. T C of Zener diodes is negative at V Z ￡ 3.5 to 4.5 V and is equal to zero at about V Z ? 5 V. T C of a Zener diode operating above 5 V is in general positive. Above 10 V the pn-diodes operate as avalanche diodes with a strong positive temperature coefficient. The T C of a Zener diode is more predictable than that of the avalanche diode. Temperature-compensated Zener diodes utilize the positive T C of a 7-V Zener diode, which is compensated with a series-connected forward-biased diode with a negative T C . The disadvantage of Zener diodes is a relatively large electronic noise. Varactor Diodes The varactor diode is an ordinary pn-diode that uses the voltage-dependent variable capacitance of the diode. The varactor diode is widely used as a voltage-dependent capacitor in electronically tuned radio receivers and in TV. Tunnel Diodes The tunnel diode is an ordinary pn-junction diode with very heavy doped n and p regions. Because the junction is very thin, a tunnel effect takes place. An electron can tunnel through the thin depletion layer from the FIGURE 22.22Reverse break- down voltage of a diode at –V R = BV. ? 2000 by CRC Press LLC conduction band of the n region directly into the valence band of the p region. Tunnel diodes create a negative differential resistance in the forward direction, due to the tunnel effect. Tunnel diodes are used as mixers, oscillators, amplifiers, and detectors. They operate at very high frequencies in the gigahertz bands. Photodiodes and Solar Cells Photodiodes are ordinary pn-diodes that generate hole–electron pairs when exposed to light. A photocurrent flows across the junction, if the diode is reverse biased. Silicon pn-junctions are used to sense light at near- infrared and visible spectra around 0.9 mm. Other materials are used for different spectra. Solar cells utilize the pn-junction to convert light energy into electrical energy. Hole–electron pairs are generated in the semiconductor material by light photons. The carriers are separated by the high electric field in the depletion region across the pn-junction.The electric field forces the holes into the p region and the electrons into the n region. This displacement of mobile charges creates a voltage difference between the two semiconductor regions. Electric power is generated in an external load connected between the terminals to the p and n regions. The conversion efficiency is relatively low, around 10 to 12%. With the use of new materials, an efficiency of about 30% has been reported. Efficiency up to 45% was achieved by using monochromatic light. Schottky Barrier Diode The Schottky barrier diode is a metal-semiconductor diode. Majority carriers carry the electric current. No minority carrier injection takes place. When the diode is forward biased, carriers are injected into the metal, where they reside as majority carriers at an energy level that is higher than the Fermi level in metals. The I-V characteristic is similar to conventional diodes. The barrier voltage is small, about 0.2 V for silicon. Since no minority carrier charge exists, the Schottky barrier diodes are very fast. They are used in high-speed electronic circuitry. Defining Terms Acceptor: Ionized, negative-charged immobile dopant atom (ion) in a p-type semiconductor after the release of a hole. Avalanche breakdown: In the reverse-biased diode, hole–electron pairs are generated in the depletion region by ionization, thus by the lattice collision with energetic electrons and holes. Bandgap energy: Energy difference between the conduction band and the valence band in a semiconductor. Barrier voltage: A voltage which develops across the junction due to uncovered immobile ions on both sides of the junction. Ions are uncovered due to the diffusion of mobile carriers across the junction. Boltzmann relation: Relates the density of particles in one region to that in an adjacent region, with the potential energy between both regions. Carrier lifetime: Time an injected minority carrier travels before its recombination with a majority carrier. Concentration gradient: Difference in carrier concentration. Diffusion: Movement of free carriers in a semiconductor caused by the difference in carrier densities (con- centration gradient). Also movement of dopands during fabrication of diffused diodes. Diffusion capacitance: Change in charge of injected carriers corresponding to change in forward bias voltage in a diode. Diffusion constant: Product of the thermal voltage and the mobility in a semiconductor. Donor: Ionized, positive-charged immobile dopant atom (ion) in an n-type semiconductor after the release of an electron. Drift: Movement of free carriers in a semiconductor due to the electric field. Ideality factor: The factor determining the deviation from the ideal diode characteristic m = 1. At small and large currents m ? 2. Incremental model: Small-signal differential (incremental) semiconductor diode equivalent RC circuit of a diode, biased in a dc operating point. ? 2000 by CRC Press LLC Incremental resistance: Small-signal differential (incremental) resistance of a diode, biased in a dc operating point. Junction capacitance: Change in charge of immobile ions in the depletion region of a diode corresponding to a change in reverse bias voltage on a diode. Majority carriers: Holes are in majority in a p-type semiconductor; electrons are in majority in an n-type semiconductor. Minority carriers: Electrons in a p-type semiconductor are in minority; holes are in majority. Similarly, holes are in minority in an n-type semiconductor and electrons are in majority. Reverse breakdown: At the reverse breakdown voltage the diode can conduct a large current in the reverse direction. Reverse generation-recombination current: Part of the reverse current in a diode caused by the generation of hole–electron pairs in the depletion region. This current is voltage dependent because the depletion region width is voltage dependent. Reverse saturation current: Part of the reverse current in a diode which is caused by diffusion of minority carriers from the neutral regions to the depletion region. This current is almost independent of the reverse voltage. Temperature coefficient: Relative variation DX/X of a value X over a temperature range, divided by the difference in temperature DT. Zener breakdown: In the reverse-biased diode, hole–electron pairs are generated by a large electric field in the depletion region. Related Topic 5.1 Diodes and Rectifiers References C.G. Fonstad, Microelectronic Devices and Circuits, New York: McGraw-Hill, 1994. P.R. Gray and R.G. Meyer, Analysis and Design of Analog Integrated Circuits, New York: John Wiley & Sons, 1993. A.S. Grove, Physics and Technology of Semiconductor Devices, New York: John Wiley & Sons, 1967. W. Gugenbuehl, M.J.O. Strutt, and W. Wunderlin, Semiconductor Elements, Basel: Birkhauser Verlag, 1962. M.N. Horenstein, Microelectronic Circuits and Devices, Englewood Cliffs, N.J.: Prentice-Hall, 1990. A.B. Phillips, Transistor Engineering, New York: McGraw-Hill, 1962. S.M. Sze, Semiconductor Devices, Physics, and Technology, New York: John Wiley & Sons, 1985. Further Information A good classical introduction to diodes is found in P. E. Gray and C. L. Searle, Electronic Principles, New York: Wiley, 1969. Other sources include S. Soclof, Applications of Analog Integrated Circuits, Englewood Cliffs, N.J.: Prentice-Hall, 1985 and E. J. Angelo, Jr., Electronics: BJT’s, FET’s and Microcircuits, New York: McGraw-Hill, 1969. 22.3 Electrical Equivalent Circuit Models and Device Simulators for Semiconductor Devices Aicha Elshabini-Riad, F. W. Stephenson, and Imran A. Bhutta In the past 15 years, the electronics industry has seen a tremendous surge in the development of new semicon- ductor materials, novel devices, and circuits. For the designer to bring these circuits or devices to the market in a timely fashion, he or she must have design tools capable of predicting the device behavior in a variety of circuit configurations and environmental conditions. Equivalent circuit models and semiconductor device simulators represent such design tools. ? 2000 by CRC Press LLC Overview of Equivalent Circuit Models Circuit analysis is an important tool in circuit design. It saves considerable time, at the circuit design stage, by providing the designer with a tool for predicting the circuit behavior without actually processing the circuit. An electronic circuit usually contains active devices, in addition to passive components. While the current and voltage behavior of passive devices is defined by simple relationships, the equivalent relationships in active devices are quite complicated in nature. Therefore, in order to analyze an active circuit, the devices are replaced by equivalent circuit models that give the same output characteristics as the active device itself. These models are made up of passive elements, voltage sources, and current sources. Equivalent circuit models provide the designer with reasonably accurate values for frequencies below 1 GHz for bipolar junction transistors (BJTs), and their use is quite popular in circuit analysis software. Some field-effect transistor (FET) models are accurate up to 10 GHz. As the analysis frequency increases, however, so does the model complexity. Since the equivalent circuit models are based on some fundamental equations describing the device behavior, they can also be used to predict the characteristics of the device itself. When performing circuit analysis, two important factors that must be taken into account are the speed and accuracy of computation. Sometimes, the computation speed can be considerably improved by simplifying the equivalent circuit model, without significant loss in computation accuracy. For this reason, there are a number of equivalent circuit models, depending on the device application and related conditions. Equivalent circuit models have been developed for diodes, BJTs, and FETs. In this overview, the equivalent circuit models for BJT and FET devices are presented. Most of the equivalent circuits for BJTs are based on the Ebers–Moll model [1954] or the Gummel–Poon model [1970]. The original Ebers–Moll model was a large signal, nonlinear dc model for BJTs. Since then, a number of improvements have been incorporated to make the model more accurate for various applications. In addition, an accurate model has been introduced by Gummel and Poon. There are three main types of equivalent circuit models, depending on the device signal strength. On this basis, the models can be classified as follows: 1.Large-signal equivalent circuit model 2.Small-signal equivalent circuit model 3.DC equivalent circuit model Use of the large-signal or small-signal model depends on the magnitude of the driving source. In applications where the driving currents or the driving voltages have large amplitudes, large-signal models are used. In circuits where the signal does not deviate much from the dc biasing point, small-signal models are more suitable. For dc conditions and very-low-frequency applications, dc equivalent circuit models are used. For dc and very- low-frequency analysis, the circuit element values can be assumed to be lumped, whereas in high-frequency analysis, incremental element values give much more precise results. Large-Signal Equivalent Circuit Model Depending on the frequency of operation, large-signal equivalent circuit models can be further classified as (1) high-frequency large-signal equivalent circuit model and (2) low-frequency large-signal equivalent circuit model. High-Frequency Large-Signal Equivalent Circuit Model of a BJT. In this context, high-frequency denotes frequencies above 10 kHz. In the equivalent circuit model, the transistor is assumed to be composed of two back-to-back diodes. Two current-dependent current sources are added to model the current flowing through the reverse-biased base-collector junction and the forward-biased base-emitter junction. Two junction capac- itances, C jE and C jC , model the fixed charges in the emitter-base space charge region and base-collector space charge region, respectively. Two diffusion capacitances, C DE and C DC , model the corresponding charge associated with mobile carriers, while the base resistance, r b , represents the voltage drop in the base region. All the above circuit elements are very strong functions of operating frequency, signal strength, and bias voltage. The high-frequency large-signal equivalent circuit model of an npn BJT is shown in Fig. 22.23, where the capacitances C jE , C jC , C DE , C DC are defined as follows: ? 2000 by CRC Press LLC (22.66) (22.67) (22.68) and (22.69) In these equations, V B￠E￠ is the internal base-emitter voltage, C jEO is the base-emitter junction capacitance at V B￠E￠ = 0, f E is the base-emitter barrier potential, and m E is the base-emitter capacitance gradient factor. Similarly, V B￠C￠ is the internal base-collector voltage, C jCO is the base-collector junction capacitance at V B￠C￠ = 0, f C is the base-collector barrier potential, and m C is the base-collector capacitance gradient factor. I CC and I EC denote the collector and emitter reference currents, respectively, while t F is the total forward transit time, and t R is the total reverse transit time. a R and a F are the large-signal reverse and forward current gains of a common base transistor, respectively. This circuit can be made linear by replacing the forward-biased base-emitter diode with a low-value resistor, r p , while the reverse-biased base-collector diode is replaced with a high-value resistor, r m . The junction and diffusion capacitors are lumped together to form C p and C m , while the two current sources are lumped into one source (g mF V F – g mR V R ), where g mF and g mR are the transistor forward and reverse transconductances, respectively. V F and V R are the voltages across the forward- and reverse-biased diodes, represented by r p and r m , respectively. r p is typically about 3 kW, while r m is more than a few megohms, and C p is about 120 pF. The linear circuit representation is illustrated in Fig. 22.24. The Gummel–Poon representation is very similar to the high-frequency large-signal linear circuit model of Fig. 22.24. However, the terms describing the elements are different and a little more involved. FIGURE 22.23High-frequency large-signal equivalent circuit model of an npn BJT. CV C v jE BE jEO BE E m E () ￠￠ ￠￠ = - ? è ? ? ? ÷ 1 f CV C v jC BC jCO BC C m C () ￠￠ ￠￠ = - ? è ? ? ? ÷ 1 f C I V DE FCC BE = ￠￠ t C I V DC REC BE = ￠￠ t ? 2000 by CRC Press LLC High-Frequency Large-Signal Equivalent Circuit Model of a FET. In the high-frequency large-signal equiv- alent circuit model of a FET, the fixed charge stored between the gate and the source and between the gate and the drain is modeled by the gate-to-source and the gate-to-drain capacitances, C GS and C GD , respectively. The mobile charges between the drain and the source are modeled by the drain-to-source capacitance, C DS . The voltage drop through the active channel is modeled by the drain-to-source resistance, R DS . The current through the channel is modeled by a voltage-controlled current source. For large signals, the gate is sometimes driven into the forward region, and thus the conductance through the gate is modeled by the gate conductance, G g . The conductance from the gate to the drain and from the gate to the source is modeled by the gate-to-drain and gate-to-source resistances, R GD and R GS , respectively. A variable resistor, R i , is added to model the gate charging time such that the time constant given by R i C GS holds the following relationship R i C GS = constant (22.70) For MOSFETs, typical element values are: C GS and C GD are in the range of 1–10 pF, C DS is in the range of 0.1–1 pF, R DS is in the range of 1–50 kW, R GD is more than 10 14 W, R GS is more than 10 10 W, and g m is in the range of 0.1–20 mA/V. Figure 22.25 illustrates the high-frequency large-signal equivalent model of a FET. Low-Frequency Large-Signal Equivalent Circuit Model of a BJT. In this case, low frequency denotes fre- quencies below 10 kHz. The low-frequency large-signal equivalent circuit model of a BJT is based on its dc characteristics. Whereas at high frequencies one has to take incremental values to obtain accurate analysis, at low frequencies, the average of these incremental values yields the same level of accuracy in the analysis. Therefore, in low-frequency analysis, the circuit elements of the high-frequency model are replaced by their average values. The low-frequency large-signal equivalent circuit model is shown in Fig. 22.26. FIGURE 22.24High-frequency large-signal equivalent circuit model (linear) of an npn BJT. FIGURE 22.25High-frequency large-signal equivalent circuit model of a FET. ? 2000 by CRC Press LLC Low-Frequency Large-Signal Equivalent Circuit Model of a FET. Because of their high reactance values, the gate-to-source, gate-to-drain, and drain-to-source capacitances can be assumed to be open circuits at low frequencies. Therefore, the low-frequency large-signal model is similar to the high-frequency large-signal model, except that it has no capacitances. The resulting circuit describing low-frequency operation is shown in Fig. 22.27. Small-Signal Equivalent Circuit Model In a small-signal equivalent circuit model, the signal variations around the dc-bias operating point are very small. Just as for the large-signal model, there are two types of small-signal models, depending upon the operating frequency: (1) the high-frequency small-signal equivalent circuit model and (2) the low-frequency small-signal equivalent circuit model. High-Frequency Small-Signal Equivalent Circuit Model of a BJT. The high-frequency small-signal equivalent circuit model of a BJT is quite similar to its high-frequency large-signal equivalent circuit model. In the small- signal model, however, in addition to the base resistance r b , the emitter and collector resistances, r e and r c , respectively, are added to the circuit. The emitter resistance is usually very small because of high emitter doping used to obtain better emitter injection efficiency. Therefore, whereas at large signal strengths the effect of r e is overshadowed by the base resistance, at small signal strengths this emitter resistance cannot be neglected. The collector resistance becomes important in the linear region, where the collector-emitter voltage is low. The high-frequency small-signal equivalent circuit model is shown in Fig. 22.28. High-Frequency Small-Signal Equivalent Circuit Model of a FET. For small-signal operations, the signal strength is not large enough to forward bias the gate-to-semiconductor diode; hence, no current will flow from the gate to either the drain or the source. Therefore, the gate-to-source and gate-to-drain series resistances, R GS and R GD , can be neglected. Also, since there will be no current flow from the gate to the channel, the gate conductance, G g , can also be neglected. Figure 22.29 illustrates the high-frequency small-signal equivalent circuit model of a FET. FIGURE 22.26Low-frequency large-signal equivalent circuit model of an npn BJT. FIGURE 22.27Low-frequency large-signal equivalent circuit model of a FET. ? 2000 by CRC Press LLC Low-Frequency Small-Signal Equivalent Circuit Model of a BJT.As in the low-frequency large-signal model, the junction capacitances, C jC and C jE , and the diffusion capacitances, C DE and C DC , can be neglected. Further- more, the base resistance, r b , can also be neglected, because the voltage drop across the base is not significant and the variations in the base width caused by changes in the collector-base voltage are also very small. The low-frequency small-signal equivalent circuit model is shown in Fig. 22.30. Low-Frequency Small-Signal Equivalent Circuit Model of a FET. Because the reactances associated with all the capacitances are very high, one can neglect the capacitances for low-frequency analysis. The gate conductance as well as the gate-to-source and gate-to-drain resistances can also be neglected in small-signal operation. The resulting low-frequency equivalent circuit model of a FET is shown in Fig. 22.31. FIGURE 22.28High-frequency small-signal equivalent circuit model of an npn BJT. FIGURE 22.29High-frequency small-signal equivalent circuit model of a FET. FIGURE 22.30Low-frequency small-signal equivalent circuit model of an npn BJT. ? 2000 by CRC Press LLC DC Equivalent Circuit Model DC Equivalent Circuit Model of a BJT. The dc equivalent circuit model of a BJT is based on the original Ebers–Moll model. Such models are used when the transistor is operated at dc or in applications where the operating frequency is below 1 kHz. There are two versions of the dc equivalent circuit model—the injection version and the transport version. The difference between the two versions lies in the choice of the reference current. In the injection version, the reference currents are I F and I R , the forward- and reverse-biased diode currents, respectively. In the transport version, the reference currents are the collector transport current, I CC , and the emitter transport current, I CE . These currents are of the form: (22.71) (22.72) (22.73) and (22.74) In these equations, I ES and I CS are the base-emitter saturation current and the base-collector saturation current, respectively. I S denotes the saturation current. In most computer simulations, the transport version is usually preferred because of the following conditions: 1.I CC and I EC are ideal over many decades. 2.I S can specify both reference currents at any given voltage. The dc equivalent circuit model of a BJT is shown in Fig. 22.32. DC Equivalent Circuit Model of a FET. In the dc equivalent circuit model of a FET, the gate is considered isolated because the gate-semiconductor interface is formed as a reverse-biased diode and therefore is open circuited. All capacitances are also assumed to represent open circuits. R GS , R GD , and R DS are neglected because FIGURE 22.31Low-frequency small-signal equivalent circuit model of a FET. II qV kT FES BE = ? è ? ? ? ÷ - é ? ê ê ù ? ú ú exp 1 II qV kT RCS BC = ? è ? ? ? ÷ - é ? ê ê ù ? ú ú exp 1 II qV kT CC S BE = ? è ? ? ? ÷ - é ? ê ê ù ? ú ú exp 1 II qV kT EC S BC = ? è ? ? ? ÷ - é ? ê ê ù ? ú ú exp 1 ? 2000 by CRC Press LLC there is no conductance through the gate and, because this is a dc analysis, there are no charging effects associated with the gate. The dc equivalent circuit of a FET is illustrated in Fig. 22.33. Commercially Available Packages A number of circuit analysis software packages are commercially available, one of the most widely used being SPICE. In this package, the BJT models are a combination of the Gummel–Poon and the modified Ebers–Moll models. Figure 22.34 shows a common emitter transistor circuit and a SPICE input file containing the transistor model. Some other available packages are SLIC, SINC, SITCAP, and Saber. Equivalent circuit models are basically used to replace the semiconductor device in an electronic circuit. These models are developed from an understanding of the device’s current and voltage behavior for novel devices where internal device operation is not well understood. For such situations, the designer has another tool available, the semiconductor device simulator. Overview of Semiconductor Device Simulators Device simulators are based on the physics of semiconductor devices. The input to the simulator takes the form of information about the device under consideration such as material type, device, dimensions, doping con- centrations, and operating conditions. Based on this information, the device simulator computes the electric field inside the device and thus predicts carrier concentrations in the different regions of the device. Device simulators can also predict transient behavior, including quantities such as current–voltage characteristics and frequency bandwidth. The three basic approaches to device simulation are (1) the classical approach, (2) the semiclassical approach, and (3) the quantum mechanical approach. Device Simulators Based on the Classical Approach The classical approach is based on the solution of Poisson’s equation and the current continuity equations. The current consists of the drift and the diffusion current components. FIGURE 22.32DC equivalent circuit model (injection version) of an npn BJT. FIGURE 22.33DC equivalent circuit model of a FET. ? 2000 by CRC Press LLC Assumptions. The equations for the classical approach can be obtained by making the following approxima- tions to the Boltzmann transport equation: 1.Carrier temperature is the same throughout the device and is assumed to be equal to the lattice tem- perature. 2.Quasi steady-state conditions exist. 3.Carrier mean free path must be smaller than the distance over which the quasi-Fermi level is changing by kT/q. 4.The impurity concentration is constant or varies very slowly along the mean free path of the carrier. 5.The energy band is parabolic. 6.The influence of the boundary conditions is negligible. For general purposes, even with these assumptions and limitations, the models based on the classical approach give fairly accurate results. The model assumes that the driving force for the carriers is the quasi-Fermi potential gradient, which is also dependent upon the electric field value. Therefore, in some simulators, the quasi-Fermi level distributions are computed and the carrier distribution is estimated from this information. Equations to Be Solved. With the assumption of a quasi-steady-state condition, the operating wavelength is much larger than the device dimensions. Hence, Maxwell’s equations can be reduced to the more familiar Poisson’s equation: (22.75) and, for a nonhomogeneous medium, ?·e (?y) = –r (22.76) FIGURE 22.34Common emitter transistor circuit and SPICE circuit file. ?=- 2 y r e ? 2000 by CRC Press LLC where y denotes the potential of the region under simulation, e denotes the permittivity, and r denotes the charge enclosed by this region. Also from Maxwell’s equations, one can determine the current continuity equations for a homogeneous medium as: (22.77) where (22.78) and (22.79) where (22.80) For nonhomogeneous media, the electric field term in the current expressions is modified to account for the nonuniform density of states and the bandgap variation [Lundstrom and Schuelke, 1983]. In the classical approach, the objective is to calculate the potential and the carrier distribution inside the device. Poisson’s equation is solved to yield the potential distribution inside the device from which the electric field can be approximated. The electric field distribution is then used in the current continuity equations to obtain the carrier distribution and the current densities. The diffusion coefficients and carrier mobilities are usually field as well as spatially dependent. The generation-recombination term U is usually specified by the Shockley–Read–Hall relationship [Yoshi et al., 1982]: (22.81) where p and n are the hole and electron concentrations, respectively, n ie is the effective intrinsic carrier density, t p and t n are the hole and electron lifetimes, and p t and n t are the hole and electron trap densities, respectively. The electron and hole mobilities are usually specified by the Scharfetter–Gummel empirical formula, as (22.82) where N is the total ionized impurity concentration, E is the electric field, and a, b, c, d, and e are defined constants [Scharfetter and Gummel, 1969] that have different values for electrons and holes. ?× - ? è ? ? ? ÷ =+Jq n t qU n ? ? JqEqDn nnn =+?×m ?× + ? è ? ? ? ÷ =-Jq p t qU p ? ? J q pE qD p pp p =-?×m Rn pn n nn pp ie ptnt = - +++ 2 tt()() mm=+ + + + + é ? ê ê ù ? ú ú - 0 2 2 12 1 N Na b Ec Ec d Ee () () () () / / / / / ? 2000 by CRC Press LLC Boundary Conditions. Boundary conditions have a large effect on the final solution, and their specific choice is a very important issue. For ohmic contacts, infinite recombination velocities and space charge neutrality conditions are assumed. Therefore, for a p-type material, the ohmic boundary conditions take the form (22.83) (22.84) and (22.85) where V appl is the applied voltage, k is Boltzmann’s constant, and N + D and N – A are the donor and acceptor ionized impurity concentrations, respectively. For Schottky contacts, the boundary conditions take the form (22.86) and (22.87) where E G is the semiconductor bandgap and f B is the barrier potential. For other boundaries with no current flow across them, the boundary conditions are of the form (22.88) where j n and j p are the electron and hole quasi-Fermi levels, respectively. For field-effect devices, the potential under the gate may be obtained either by setting the gradient of the potential near the semiconductor-oxide interface equal to the gradient of potential inside the oxide [Kasai et al., 1982], or by solving Laplace’s equation in the oxide layer, or by assuming a Dirichlet boundary condition at the oxide-gate interface and determining the potential at the semiconductor-oxide interface as: (22.89) y= + ? è ? ? ? ÷ V kT q n p ie appl ln p NN n NN DA ie DA = - ? è ? ? ? ÷ + é ? ê ê ê ù ? ú ú ú - - ? è ? ? ? ÷ +- +- 22 2 2 12/ n n p ie = 2 yf=+-V E G Bappl 2 nn E kTq ie GB = - ? è ? ? ? ÷ exp ()/ / 2 f ?y ? ?j ? ?j ?nnp n p ===0 ee Si Si Ox ?y ? yy y xz Tz GS = - * (,) () ? 2000 by CRC Press LLC where e Si and e Ox are the permittivities of silicon and the oxide, respectively, y G is the potential at the top of the gate, y* S (x,z) is the potential of the gate near the interface, and T(z) is the thickness of the gate metal. Solution Methods. Two of the most popular methods of solving the above equations are finite difference method (FDM) and finite element method (FEM). In FDM, the region under simulation is divided into rectangular or triangular areas for two-dimensional cases or into cubic or tetrahedron volumes in three-dimensional cases. Each corner or vertex is considered as a node. The differential equations are modified using finite difference approximations, and a set of equations is constructed in matrix form. The finite difference equations are solved iteratively at only these nodes. The most commonly used solvers are Gauss–Seidel/Jacobi (G-S/J) techniques or Newton’s technique (NT) [Banks et al., 1983]. FDM has the disadvantage of requiring more nodes than the FEM for the same structure. A new variation of FDM, namely the finite boxes scheme [Franz et al., 1983], however, overcomes this problem by enabling local area refinement. The advantage of FDM is that its computational memory requirement is less than that required for FEM because of the band structure of the matrix. In FEM, the region under simulation is divided into triangular and quadrilateral regions in two dimensions or into tetrahedra in three dimensions. The regions are placed to have the maximum number of vertices in areas where there is expected to be a large variation of composition or a large variation in the solution. The equations in FEM are modified by multiplying them with some shape function and integrating over the simulated region. In triangular meshes, the shape function is dependent on the area of the triangle and the spatial location of the node. The value of the spatial function is between 0 and 1. The solution at one node is the sum of all the solutions, resulting from the nearby nodes, multiplied by their respective shape functions. The number of nodes required to simulate a region is less than that in FDM; however, the memory requirement is greater. Device Simulators Based on the Semiclassical Approach The semiclassical approach is based upon the Boltzmann transport equation (BTE) [Engl, 1986] which can be written as: (22.90) where f represents the carrier distribution in the volume under consideration at any time t, v is the group velocity, E is the electric field, and q and h are the electronic charge and Planck’s constant, respectively. BTE is a simplified form of the Liouville–Von Neumann equation for the density matrix. In this approach, the free flight between two consecutive collisions of the carrier is considered to be under the influence of the electric field, whereas different scattering mechanisms determine how and when the carrier will undergo a collision. Assumptions. The assumptions for the semiclassical model can be summarized as follows: 1. Carrier-to-carrier interactions are considered to be very weak. 2. Particles cannot gain energy from the electric field during collision. 3. Scattering probability is independent of the electric field. 4. Magnetic field effects are neglected. 5. No electron-to-electron interaction occurs in the collision term. 6. Electric field varies very slowly, i.e., electric field is considered constant for a wave packet describing the particle’s motion. 7. The electron and hole gas is not degenerate. 8. Band theory and effective-mass theorems apply to the semiconductor. Equations to Be Solved. As a starting point, Poisson’s equation is solved to obtain the electric field inside the device. Using the Monte Carlo technique (MCT), the BTE is solved to obtain the carrier distribution function, f. In the MCT, the path of one or more carriers, under the influence of external forces, is followed, and from df dt f t v q h Ef f t rk =+×?± ×?= ? è ? ? ? ÷ ? ?p ? ?()/ coll 2 ? 2000 by CRC Press LLC this information the carrier distribution function is determined. BTE can also be solved by the momentum and energy balance equations. The carrier distribution function gives the carrier concentrations in the different regions of the device and can also be used to obtain the electron and hole currents, using the following expressions: (22.91) and (22.92) Device Simulators Based on the Quantum Mechanical Approach The quantum mechanical approach is based on the solution of the Schrodinger wave equation (SWE), which, in its time-independent form, can be represented as (22.93) where j n is the wave function corresponding to the subband n whose minimum energy is E n , V is the potential of the region, m is the particle mass, and h and q are Planck’s constant and the electronic charge, respectively. Equations to Be Solved. In this approach, the potential distribution inside the device is calculated using Poisson’s equation. This potential distribution is then used in the SWE to yield the electron wave vector, which in turn is used to calculate the carrier distribution, using the following expression: (22.94) where n is the electron concentration and N n is the concentration of the subband n. This carrier concentration is again used in Poisson’s equation, and new values of j n , E n , and n are calculated. This process is repeated until a self-consistent solution is obtained. The final wave vector is invoked to determine the scattering matrix, after which MCT is used to yield the carrier distribution and current densities. Commercially Available Device Simulation Packages The classical approach is the most commonly used procedure since it is the easiest to implement and, in most cases, the fastest technique. Simulators based on the classical approach are available in two-dimensional forms like FEDAS, HESPER, PISCES-II, PISCES-2B, MINIMOS, and BAMBI or three-dimensional forms like TRA- NAL, SIERRA, FIELDAY, DAVINCI, and CADDETH. Large-dimension devices, where the carriers travel far from the boundaries, can be simulated based on a one-dimensional approach. Most currently used devices, however, do not fit into this category, and therefore one has to resort to either two- or three-dimensional simulators. FEDAS (Field Effect Device Analysis System) is a two-dimensional device simulator that simulates MOSFETs, JFETs, and MESFETs by considering only those carriers that form the channel. The Poisson equation is solved everywhere except in the oxide region. Instead of carrying the potential calculation within the oxide region, the potential at the semiconductor-oxide interface is calculated by assuming a mixed boundary condition. FEDAS uses FDM to solve the set of linear equations. A three-dimensional variation of FEDAS is available for the simulation of small geometry MOSFETs. Jqvfrktdk n k =- ò (, ,) 3 Jqvfrktdk p k =+ ò (, ,) 3 (/ ) () h m EqV nn n 2 2 0 2 2 p jj?+ + = nN nn n = ? **j 2 ? 2000 by CRC Press LLC HESPER (HEterostructure device Simulation Program to Estimate the performance Rigorously) is a two- dimensional device simulator that can be used to simulate heterostructure photodiodes, HBTs, and HEMTs. The simulation starts with the solution of Poisson’s equation in which the electron and hole concentrations are described as functions of the composition (composition dependent). The recombination rate is given by the Shockley–Read–Hall relationship. Lifetimes of both types of carriers are assumed to be equal in this model. PISCES-2B is a two-dimensional device simulator for simulation of diodes, BJTs, MOSFETs, JFETs, and MESFETs. Besides steady-state analysis, transient and ac small-signal analysis can also be performed. Conclusion The decision to use an equivalent circuit model or a device simulator depends upon the designer and the required accuracy of prediction. To save computational time, one should use as simple a model as accuracy will allow. At this time, however, the trend is toward developing quantum mechanical models that are more accurate, and with faster computers available, the computational time for these simulators has been considerably reduced. Defining Terms Density of states: The total number of charged carrier states per unit volume. Fermi levels: The energy level at which there is a 50% probability of finding a charged carrier. Mean free path: The distance traveled by the charged carrier between two collisions. Mobile charge: The charge due to the free electrons and holes. Quasi-Fermi levels: Energy levels that specify the carrier concentration inside a semiconductor under non- equilibrium conditions. Schottky contact: A metal-to-semiconductor contact where, in order to align the Fermi levels on both sides of the junction, the energy band forms a barrier in the majority carrier path. Related Topics 2.3 Controlled Sources ? 35.1 Maxwell Equations References R. E. Banks, D. J. Rose, and W. Fitchner, “Numerical methods for semiconductor device simulation,” IEEE Trans. Electron Devices, vol. ED-30, no. 9, pp. 1031–1041, 1983. J. J. Ebers and J. L. Moll, “Large signal behavior of junction transistors,” Proc. IRE, vol. 42, pp. 1761–1772, Dec. 1954. W. L. Engl, Process and Device Modeling, Amsterdam: North-Holland, 1986. A. F. Franz, G. A. Franz, S. Selberherr, C. Ringhofer, and P. Markowich, “Finite boxes—A generalization of the finite-difference method suitable for semiconductor device simulation,” IEEE Trans. Electron Devices, vol. ED-30, no. 9, pp. 1070–1082, 1983. H. K. Gummel and H. C. Poon, “An integral charge control model of bipolar transistors,” Bell Syst. Tech. J., vol. 49, pp. 827–852, May-June 1970. R. Kasai, K. Yokoyama, A. Yoshii, and T. Sudo, “Threshold-voltage analysis of short- and narrow-channel MOSFETs by three-dimensional computer simulation,” IEEE Trans. Electron Devices, vol. ED-21, no. 5, pp. 870–876, 1982. M. S. Lundstrom and R. J. Schuelke, “Numerical analysis of heterostructure semiconductor devices,” IEEE Trans. Electron Devices, vol. ED-30, no. 9, pp. 1151–1159, 1983. D. L. Scharfetter and H. K. Gummel, “Large-signal analysis of a silicon read diode oscillator,” IEEE Trans. Electron Devices, vol. ED-16, no. 1, pp. 64–77, 1969. A. Yoshii, H. Kitazawa, M. Tomzawa, S. Horiguchi, and T. Sudo, “A three dimensional analysis of semiconductor devices,” IEEE Trans. Electron Devices, vol. ED-29, no. 2, pp. 184–189, 1982. ? 2000 by CRC Press LLC Further Information Further information about semiconductor device simulation and equivalent circuit modeling, as well as about the different software packages available, can be found in the following articles and books: C. M. Snowden, Semiconductor Device Modeling, London: Peter Peregrinus Ltd., 1988. C. M. Snowden, Introduction to Semiconductor Device Modeling, Teaneck, N.J.: World Scientific, 1986. W. L. Engl, Process and Device Modeling, Amsterdam: North-Holland, 1986. J.-H. Chern, J. T. Maeda, L. A. Arledge, Jr., and P. Yang, “SIERRA: A 3-D device simulator for reliability modeling,” IEEE Trans. Computer-Aided Design, vol. CAD-8, no. 5, pp. 516–527, 1989. T. Toyabe, H. Masuda, Y. Aoki, H. Shukuri, and T. Hagiwara, “Three-dimensional device simulator CADDETH with highly convergent matrix solution algorithms,” IEEE Trans. Electron Devices, vol. ED-32, no. 10, pp. 2038–2044, 1985. PISCES-2B and DAVINCI are softwares developed by TMA Inc., Palo Alto, California 94301. Hewlett-Packard’s first product, the model 200A audio oscillator (preproduction version). William Hewlett and David Packard built an audio oscillator in 1938, from which the famous firm grew. Courtesy of Hewlett- Packard Company.) 22.4 Electrical Characterization of Semiconductors David C. Look The huge electronics and computer industries exist primarily because of the unique electrical properties of semiconductor materials, such as Si and GaAs. These materials usually contain impurities and defects in their crystal lattices; such entities can act as donors and acceptors, and can strongly influence the electrical and optical properties of the charge carriers. Thus, it is extremely important to be able to measure the concentration and mobility of these carriers, and the concentrations and energies of the donors and acceptors. All of these quantities can, in principle, be determined by measurement and analysis of the temperature-dependent resis- tivity and Hall effect. On the simplest level, Hall-effect measurements require only a current source, a voltmeter, and a modest-sized magnet. However, the addition of temperature-control equipment and computer analysis produce a much more powerful instrument that can accurately measure concentrations over a range 10 4 to 10 20 cm –3 . Many commercial instruments are available for such measurements; this chapter section reveals how to make full use of the versatility of the technique. Theory A phenomenological equation of motion for electrons of charge –e moving with velocity v in the presence of electric field E and magnetic field B is (22.95) where m? is the effective mass, v eq is the velocity at equilibrium (steady state), and τ is the velocity (or momentum) relaxation time (i.e., the time in which oscillatory phase information is lost through collisions). Consider a rectangular sample, as shown in Fig. 22.35(a), with an external electric field E ex = E x x and magnetic field B = B z z. (Dimensions x and y are parallel to “H5129” and “w,” respectively, and z is perpendicular to both.) Then, if no current is allowed to flow in the y direction (i.e., v y = 0), the steady-state condition · ν = 0 requires that E y = –v x B z , and E y is known as the Hall field. For electron concentration n, the current density is j x = nev x ; thus, E y = –j x B z /en ≡ –j x B z R H , where R H = –1/en, the Hall coefficient. Thus, simple measurements of the quantities E y , j x , and B z yield a very important quantity: n. m eq ?=? +× ( ) ?? ? ˙ vEvB vv em τ ? 2000 by CRC Press LLC The above analysis assumes that all electrons are moving with the same velocity v (constant τ), which is not true in a semiconductor. A more detailed analysis, allowing for the energy E dependence of the electrons, gives (22.96) (22.97) where (22.98) This formulation is called the relaxation-time approximation (RTA) to the Boltzmann transport equation (BTE). Here, f 0 is the Fermi-Dirac distribution function and the second equality in Eq. (22.98) holds for non-degenerate electrons (i.e., those describable by Boltzmann statistics). The quantity μ c = e?τ?/m? is known as the conductivity mobility, since the quantity neμ c is just the conductivity σ. The Hall mobility is defined as μ H = R H σ = rμ c , and the Hall concentration as n H = n/r = –1/eR H . Thus, a combined Hall effect and conductivity measurement gives n H and μ H , although one would prefer to know n, not n H ; fortunately, however, r is usually within 20% of unity, and is almost never as large as 2. In any case, r can often be calculated or measured so that an accurate value of n can usually be determined. It should also be mentioned that one way to evaluate the expressions in Eq. (22.98) is to define a new variable, u = E/kT, and set u = 10 as the upper limit in the integrals. The relaxation time, τ(E), depends on how the electrons interact with the lattice vibrations, as well as with extrinsic elements such as charged impurities and defects. For example, acoustical-mode lattice vibrations scatter electrons through the deformation potential (τ ac ) and piezoelectric potential (τ pe ); optical-mode vibra- tions through the polar potential (τ po ); ionized impurities and defects through the screened coulomb potential FIGURE 22.35 Various patterns commonly used for resistivity and Hall-effect measurements. j ne m EneE xxcx = ? ≡? μ 2 τ R E jB ne r en H y x ==? =? 1 2 2 τ τ τ τ τ n n n f d f d d d E EE E E E E E EE E EE E E ( ) = ( ) ? ? ? ? → ( ) ∞ ∞ ∞ ? ? ∞ ∫ ∫ ∫ ∫ 0 32 0 32 0 0 0 32 32 0 e e kT kT ? 2000 by CRC Press LLC (τ ii ); and charged dislocations, also through the coulomb potential (τ dis ). The strengths of these various scattering mechanisms depend on certain lattice parameters, such as dielectric constants and deformation potentials, and extrinsic factors, such as donor, acceptor, and dislocation concentrations, N D , N A , and N dis , respectively [Rode, 1975; Wiley, 1975; Nag, 1980; Look, 1989; Look, 1998]. The total momentum scattering rate, or inverse relaxation time, is (22.99) and this expression is then used to determine ?τ n (E)? via Eq. (22.98), and hence, μ H = e?τ 2 ?/m??τ?. Formulae for τ ac , τ pe , τ po , τ ii , and τ dis , can be found in the literature, but are given below for completeness. For ionized impurity (or defect) scattering, in a non-degenerate, n-type material: (22.100) where y = 8ε 0 m?kTE/h 2 e 2 n. Here, ε 0 is the low-frequency (static) dielectric constant, k is Boltzmann’s constant, and h is Planck’s constant divided by 2π. [If the sample is p-type, let (2N A +n) → (2N D +p)]. For acoustic-mode deformation-potential scattering: (22.101) where ρ d is the density, s is the speed of sound, and E 1 is the deformation potential. For acoustic-mode piezoelectric-potential scattering: (22.102) where P is the piezoelectric coupling coefficient [P = (h pz 2 /ρs 2 ε 0 ) 1/2 ]. Finally, for polar optic-mode scattering, only a rough approximation can be given because the scattering is inelastic: (22.103) where T po is the Debye temperature and ε ∞ is the high-frequency dielectric constant. This formula for τ po (E) has the following property: if only p-o scattering existed, then an accurate BTE calculation of μ H vs. T [Rode, 1975] would give results almost identical to those obtained by the RTA analysis described above, i.e., by setting μ H = e ?τ 2 ?/m??τ?. However, when other scattering mechanisms are also important, then the RTA solution may not be as reliable. Fortunately, at low temperatures (e.g., below about 150K in GaN), p-o scattering weakens, and the RTA approach is quite accurate. This fact is important because we usually are interested in obtaining a good value of the acceptor concentration N A from the μ H vs. T fit, and N A appears directly only in the ii- scattering formula Eq. (22.100), which is usually dominant at low temperatures. ττττττ ?????? ( ) = ( ) + ( ) + ( ) + ( ) + ( ) 111111 EEEEEE ac pe po ii dis τ ε ii m Nn yy y E E ( ) = π? ( ) + ( ) + ( ) ?+ ( ) [] 2 211 92 0 2 12 32 4 e A ln τ ρ ac d s Em kT E E ( ) = π ( ) ? h 4212 12 1 2 32 2 * τ ε p e Pm kT E E ( ) = π ? ( ) 2 32 2 0 12 22 12 h e τ εε po TT po po po ekTkT ekT m po E EE ( ) = π? ? ? ? ? + ( ) ? ( ) ? ? ? ? ? ? ( ) ? ( ) ? ∞ ?? 2 1 0 762 0 824 0 235 32 2 12 12 12 2 12 1 0 1 h .. . * ? 2000 by CRC Press LLC Dislocation scattering in semiconductor materials is often ignored because it becomes significant only for dislocation densities N dis > 10 8 cm –2 (note that this is an arreal, not volume, density). Such high densities are rare in most semiconductor devices, such as those fabricated from Si or GaAs, but are indeed quite common in devices based on GaN or other materials that involve mismatched substrates. In GaN grown on Al 2 O 3 (sapphire), vertical threading dislocations, typically of concentration 10 10 cm –2 or higher, emanate from the interface up to the surface, and horizontally moving electrons or holes experience a scattering characterized by (22.104) where λ = (ε 0 kT/e 2 n) 1/2 . For high-quality GaN/Al 2 O 3 , N dis ≈ 10 8 cm –2 ; in the case of a sample discussed later in this chapter section, this value of N dis drops the 300-K Hall mobility only a minor amount, from 915 to 885 cm 2 /V s. However, if this same sample contained the usual concentration of dislocations found in GaN (about 10 10 cm –2 ), the mobility would drop to less than 100 cm 2 /V s, a typical value found in many other samples. Before going on, it should be mentioned that a very rough approximation of μ H , which avoids the integrations of Eq. (22.98), can be obtained by setting E ≈ kT and μ ≈ eτ/m? in Eq. (22.99). The latter step (i.e., μ –1 = μ 1 –1 + μ 2 –1 + μ 3 –1 + …) is known as Matthiessen’s Rule. However, with present-day computing power, even that available on PCs, it is not much more difficult to use the RTA analysis. The fitting of μ H vs. T data, described above, should be carried out in conjunction with the fitting of n vs. T, which is derived from the charge-balance equation (CBE): (22.105) where φ D = (g 0 /g 1 )N C ′exp(α D /k)T 3/2 exp(–E D0 /kT). Here, g 0 /g 1 is a degeneracy factor, N C ′ = 2(2πm n *k) 3/2 /h 3 , where h is Planck’s constant, E D is the donor energy, and E D0 and α D are defined by E D = E D0 – α D T. The above equation describes the simplest type of charge balance, in which the donor (called a single donor) has only one charge-state transition within a few kT of the Fermi energy. An example of such a donor is Si on a Ga site in GaN, for which g 0 = 1, and g 1 = 2. If there are multiple single donors, then equivalent terms are added on the right-hand side of Eq. (22.105); if there are double or triple donors, or more than one acceptor, proper variations of Eq. (22.105) can be found in the literature [Look, 1989]. For a p-type sample, the nearly equivalent equation is used: (22.106) where φ A = (g 1 /g 0 )N V ′exp(α A /k)T 3/2 exp(–E A0 /kT), N V ′ = 2(2πm p *k) 3/2 /h 3 , and E A = E A0 – α A T. Hall samples do not have to be rectangular, and other common shapes are given in Fig. 22.35(c)–(f); in fact, arbitrarily shaped specimens are discussed in the next section. However, the above analysis does assume that n and μ are homogeneous throughout the sample. If n and μ vary with depth (z) only, then the measured quantities are (22.107) τ ε λ λ dis dis c Nme m E E ( ) = +? ( ) h 3 0 22 4 2 32 4 18 * nN N n A D D += +1 φ pN N p D A A += +1 φ σσ sq dd zdz e nz zdz= ( ) = ( ) μ ( ) ∫∫ 00 ? 2000 by CRC Press LLC (22.108) where d is the sample thickness and where the subscript “sq” denotes a sheet (arreal) quantity (cm –2 ) rather than a volume quantity (cm –3 ). If some of the carriers are holes, rather than electrons, then the sign of e for those carriers must be reversed. The general convention is that R H is negative for electrons and positive for holes. In some cases, the hole and electron contributions to R Hsq σ sq 2 exactly balance at a given temperature, and this quantity vanishes. Determination of Resistivity and Hall Coefficient Consider the Hall-bar structure of Fig. 22.35(a) and suppose that current I is flowing along the long direction. Then, if V c and V H are the voltages measured along dimensions H5129 and w, respectively, and d is the thickness, one obtains E x = V c /H5129, E y = V H /w, j x = I/wd, and (22.109) (22.110) (22.111) (22.112) In MKS units, I is in amps (A), V in volts (V), B in Tesla (T), and H5129, w, and d in meters (m). By realizing that 1 T = 1 V–s m –2 , 1 A = 1 coulomb (C)/s, and 1 ohm (?) = 1 VA –1 then σ is in units of ? –1 m –1 , R H in m 3 C –1 , μ H in m 2 V –1 s –1 , and n H in m –3 . However, it is more common to denote σ in ? –1 cm –1 , R H in cm 3 C –1 , μ H in cm 2 V –1 s –1 , and n H in cm –3 , with obvious conversion factors (1 m = 10 2 cm). Because B is often quoted in Gauss (G), it is useful to note that 1 T = 10 4 G. Clearly, the simple relationships given above will not hold for the nonrectangular shapes shown in Fig. 22.35(c)–(f), several of which are very popular. Fortunately, van der Pauw [1958] has solved the potential problem for a thin layer of arbitrary shape. One of the convenient features of the van der Pauw formulation is that no dimension need be measured for the calculation of sheet resistance or sheet carrier concentration, although a thickness must of course be known for volume resistivity and concentration. Basically, the validity of the van der Pauw method requires that the sample be flat, homogeneous, and isotropic, a singly connected domain (no holes), and have line electrodes on the periphery, projecting to point contacts on the surface, or else have true point contacts on the surface. The last requirement is the most difficult to satisfy, so that much work has gone into determining the effects of finite contact size. Consider the arbitrarily shaped sample shown in Fig. 22.36(a). Here, a current I flows between contacts 1 and 2, and a voltage V c is measured between contacts 3 and 4. Let R ij,kl ≡ V kl /I ij , where the current enters contact i and leaves contact j, and V kl = V k – V l . (These definitions, as well as the contact numbering, correspond to ASTM Standard F76.) The resistivity, ρ, with B = 0, is then calculated as follows: Rnzzd Hsq sq d σ 22 0 = ( ) μ ( ) ∫ σρ=== ?1 j E I Vwd x xc l R E jB Vd IB H y x H == μ= = H H c R V VwB σ l neR H = ( ) ?1 ? 2000 by CRC Press LLC (22.113) where f is determined from a transcendental equation: (22.114) Here, Q = R 21,34 /R 32,41 if this ratio is greater than unity; otherwise, Q = R 32,41 /R 21,34 . A curve of f vs. Q, accurate to about 2%, is presented in Fig. 22.37 [van der Pauw, 1958]. Also useful is a somewhat simpler analytical procedure for determining f, due to Wasscher and reprinted in Weider [1979]. First, calculate α from (22.115) FIGURE 22.36 An arbitrary shape for van der Pauw measurements: (a) resistivity; (b) Hall effect. FIGURE 22.37 The resistivity-ratio function used to correct the van der Pauw results for asymmetric sample shape. ρ= π ( ) + ? ? ? ? ? ? ? ? d RR f ln ,, 2 2 21 34 32 41 Q Q f h f ? + = ( ) ( ) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 1 2 1 2 2 ln arccos exp ln Q = ? ( ) + ( ) ln ln 12 12 α α ? 2000 by CRC Press LLC and then calculate f from (22.116) It is of course required that –1/2 < α < 1/2, but this range of α covers Q = 0 to ∞. For example, a ratio Q = 4.8 gives a value α ≈ 0.25, and then f ≈ 0.83. Thus, the ratio must be fairly large before ρ is appreciably reduced. It is useful to further average ρ by including the remaining two contact permutations, and also reversing current for all four permutations. Then ρ becomes (22.117) where f A and f B are determined from Q A and Q B , respectively, by applying either Eq. (22.114) or Eq. (22.115). Here, (22.118) (22.119) The Hall mobility is determined using the configuration of Fig. 22.36(b), in which the current and voltage contacts are crossed. The Hall coefficient becomes (22.120) In general, to minimize magnetoresistive and other effects, it is useful to average over current and magnetic field polarities. Then, (22.121) Data Analysis The primary quantities determined from Hall effect and conductivity measurements are the Hall carrier concentration (n H or p H ) and mobility (μ H ). As already discussed, n H = –1/eR H , where R H is given by Eq. (22.120) (for a van der Pauw configuration), and μ H = R H σ = R H /ρ, where ρ is given by Eq. (22.117). Although simple 300-K values of ρ, n H , and μ H are quite important and widely used, it is in temperature-dependent Hall (TDH) measurements that the real power of the Hall technique is demonstrated, because then the donor and acceptor concentrations and energies can be determined. The methodology is illustrated with a GaN example. The GaN sample discussed here was a square (6 mm × 6 mm) layer grown on sapphire to a thickness of d = 20 μm. Small indium dots were soldered on the corners to provide ohmic contacts, and the Hall measurements were carried out in an apparatus similar to that illustrated in Fig. 22.38. Temperature control was achieved using a He exchange-gas dewar. The temperature dependencies of n H and μ H are shown in Figs. 22.39 and 22.40, f = ( ) + ( ) +? ( ) ln ln ln 14 12 12αα ρ= π ( ) [] ?+? ( ) +?+? ( ) dRRRRfRRRRf AB ln ,,,, ,,,, 28 21 34 12 34 32 41 23 41 43 12 34 12 14 23 41 23 Q RR RR A = ? ? 21 34 12 34 32 41 23 41 ,, ,, Q RR RR B = ? ? 43 12 34 12 14 23 41 23 ,, ,, R d B RR H = + ? ? ? ? ? ? ? ? 31 42 42 13 2 ,, R dBR BR BR BR BR B RBRBRB H = ( ) + ( ) ?+ ( ) ++ ( ) ?+ ( ) +? ( ) [ ?? ( ) +? ( ) ?? ( ) ] 31 42 13 42 42 13 24 13 13 42 31 42 24 13 42 13 8 ,, ,,, ,,, ? 2000 by CRC Press LLC FIGURE 22.38 A schematic diagram of an automated, high-impedance Hall effect apparatus. All components are com- mercially available. FIGURE 22.39 Hall concentration data (squares) and fit (solid line) vs. inverse temperature. ? 2000 by CRC Press LLC respectively. The data in these figures have been corrected for a very thin, strongly n-type layer between the sapphire substrate and GaN layer, as discussed by Look and Molnar [1997]. The solid lines are fits of n H and μ H , carried out using MATHCAD software on a personal computer. In many cases, it is sufficient to simply assume n = n H (i.e., r = 1) in Eq. (22.105), but a more accurate answer can be obtained by using the following steps: (1) let n = n H = 1/eR H at each T; (2) use Eq. (22.99), Eq. (22.98), and the expression μ H = e?τ 2 ?/m * ?τ? to fit μ H vs. T and get a value for N A ; (3) calculate r = ?τ 2 ?/?τ? 2 at each T; (4) calculate a new n = rn H at each T; and (5) fit n vs. T with Eq. (22.105) to get values of N D and E D . Further iterations can be carried out if desired, but usually add little accuracy. The following parameters were taken from the literature: P = 0.104, ε 0 = 10.4(8.8542 × 10 –12 ) F m –1 ; ε ∞ = 5.47(8.8542 × 10 –12 ) F m –1 ; T po = 1044 K; m* = 0.22(9.1095 × 10 –31 ) kG; ρ d = 6.10 × 10 3 kg m –3 ; s = 6.59 × 10 3 m s –1 ; g 0 = 1; g 1 = 2; α D = 0; and N C ′ = 4.98 × 10 20 m –3 . The best value for E 1 was found to be 14 eV = 2.24 × 10 –18 joules, although 9.2 eV is given by one literature source. The fitted parameters are: N D = 1.8 × 10 17 cm –3 , N A = 2 × 10 16 cm –3 , and E D = 18 meV. Sources of Error Contact Size and Placement Effects Much has been written about this subject over the past few decades [Look, 1989]. Indeed, it is possible to calculate errors due to contact size and placement for any of the structures shown in Fig. 22.35. For (a), (c), and (e), great care is necessary, while for (b), (d), and (f), large or misplaced contacts are not nearly as much of a problem. In general, a good rule of thumb is to keep contact size, and distance from the periphery, each below 10% of the smallest sample-edge dimension. For Hall-bar structures (a) and (b), in which the contacts cover the ends, the ratio H5129/w > 3 should be maintained. Thermomagnetic Errors Temperature gradients can set up spurious emfs that can modify the measured Hall voltage. Most of these effects, as well as misalignment of the Hall contacts in structure (b), can be averaged out by taking measurements at positive and negative values of both current and magnetic field, and then applying Eq. (22.117) and Eq. (22.121). Conductive Substrates If a thin film is grown on a conductive substrate, the substrate conductance may overwhelm the film conduc- tance. If so, and if μ sub and n sub are known, then Eq. (22.107) and Eq. (22.108) can be reduced to a two-layer problem and used to extract μ bulk and n bulk . If the substrate and film are of different types (e.g., a p-type film FIGURE 22.40 Hall mobility data (squares) and fit (solid line) vs. temperature. ? 2000 by CRC Press LLC on an n-type substrate), then a current barrier (p/n junction) will be set up, and the measurement can possibly be made with no correction. However, in this case, the contacts must not overlap both layers. Depletion Effects in Thin Films Surface states as well as film/substrate interface states can deplete a thin film of a significant fraction of its charge carriers. Suppose these states lead to surface and interface potentials of φ s and φ i , respectively. Then, regions of width w s and w i will be depleted of their free carriers, where (22.122) It is assumed that φ s(i) >> kT/e, and that eφ s(i) >> E C – E F . The electrical thickness of the film will then be given by d elec = d – w s – w i . Typical values of φ s and φ i are 1 V, so that if N D – N A = 10 17 cm –3 , then w s + w i ≈ 2000 ? = 0.2 μm in GaN. Thus, if d ≈ 0.5 μm, 40% of the electrons will be lost to surface and interface states, and d elec ≈ 0.3 μm. Inhomogeneity A sample that is inhomogeneous in depth can be analyzed according to Eq. (22.107) and Eq. (22.108), as mentioned above. However, if a sample is laterally inhomogeneous, it is nearly always impossible to carry out an accurate analysis. One indication of such inhomogeneity is a resistivity ratio Q >> 1 (Fig. 22.37) in a symmetric sample, which would be expected to have Q = 1. The reader should be warned to never attempt an f-correction (Fig. 22.37) in such a case, because the f-correction is valid only for sample-shape asymmetry, not inhomogeneity. Non-ohmic Contacts In general, high contact resistances are not a severe problem as long as enough current can be passed to get measurable values of V c and V H . The reason is that the voltage measurement contacts carry very little current. However, in some cases, the contacts may set up a p/n junction and significantly distort the current flow. This situation falls under the “inhomogeneity” category, discussed above. Usually, contacts this bad show variations with current magnitude and polarity; thus, for the most reliable Hall measurements, it is a good idea to make sure the values are invariant with respect to the magnitudes and polarities of both current and magnetic field. Defining Terms Acceptor: An impurity or lattice defect that can “accept” one or more electrons from donors or the valence band; in the latter case, free holes are left to conduct current in the valence band. Charge-balance equation (CBE): A mathematical relationship expressing the equality between positive and negative charges in a sample as a function of temperature. Dislocation: A one-dimensional line defect in a solid, which often extends through the entire lattice. An edge dislocation is essentially an extra half-lattice plane inserted into the lattice. Distribution function: A mathematical relationship describing the distribution of the electrons, as a function of temperature, among all the possible energy states in the lattice, including those arising from the conduction band, valence band, donors, and acceptors. Donor: An impurity or lattice defect that can “donate” one or more electrons to acceptors or to the conduction band; in the latter case, free electrons are available to conduct current. Effective mass: The apparent mass of an electron or hole with respect to acceleration in an electric field. Electrical thickness: The “thickness” of a layer in which the current actually flows. In a thin sample, this dimension may be much less than the physical thickness of the sample because some of the charge carriers may be immobilized at surface and interface states. w NN si si DA () () = ? ( ) ? ? ? ? ? ? ? ? 2 2 0 12 εφ ? 2000 by CRC Press LLC Hall coefficient: The ratio between the Hall electric field E y (a field that develops perpendicular to the plane formed by the current and magnetic field directions), and the current density j x multiplied by the magnetic field strength B z ; i.e., R H = E y /j x B z . The Hall coefficient is closely related to the carrier concentration. Hall mobility: The Hall coefficient multiplied by the conductivity. This mobility is often nearly equal to the conductivity mobility. Lattice vibrations: The collective motions of atoms (often called phonons) in a crystal lattice. The phonons can interact with the charge carriers and reduce mobility. Matthiessen’s Rule: The approximation that the inverse of the total mobility is equal to the inverses of the individual components of the mobility; that is, μ –1 = μ 1 –1 + μ 2 –1 + μ 3 –1 + …, where μ i –1 denotes the mobility that would result if only scattering mechanism i were present. Mobility: The ease with which charge carriers move in a crystal lattice. n-type: The designation of a sample that has a conductivity primarily controlled by electrons. p-type: The designation of a sample that has a conductivity primarily controlled by holes. Relaxation time: The time required to nullify a disturbance in the equilibrium energy or momentum distri- bution of the electrons and holes. Relaxation time approximation (RTA): A relatively simple analytical solution of the Boltzmann transport equation that is valid for elastic (energy-conserving) scattering processes. References Look, D. C., Electrical Characterization of GaAs Materials and Devices. Wiley, New York, 1989, Chap. 1. Look, D. C., Dislocation scattering in GaN, Phys. Rev. Lett., 82, 1237, 1999. Look, D. C. and Molnar, R. J., Degenerate layer at GaN/sapphire interface: influence on Hall-effect measure- ments, Appl. Phys. Lett., 70, 3377, 1997. Nag, B. R., Electron Transport in Compound Semiconductors, Springer-Verlag, Berlin, 1980. Rode, D. L., Low-field electron transport, in Semiconductors and Semimetals, Willardson, R. K. and Beer, A. C., Eds., Academic, New York, 1975, Chap. 1. van der Pauw, L. J., A method of measuring specific resistivity and Hall effect of discs of arbitrary shape, Philips Res. Repts., 13, 1, 1958. Wieder, H. H., Laboratory Notes on Electrical and Galvanomagnetic Measurements, Elsevier, Amsterdam, 1979. Wiley, J. D., Mobility of holes in III-V compounds, in Semiconductors and Semimetals, Willardson, R. K. and Beer, A. C., Eds., Academic, New York, 1975, Chap. 2. Further Information Good general references on semiconductor characterization, including techniques other than electrical, are the following: Runyan, W. R., Semiconductor Measurements and Instrumentation, McGraw-Hill, New York, 1975; Schroder, D. K., Semiconductor Material and Device Characterization, Wiley, New York, 1990; and Orton, J. W. and Blood, P., The Electrical Characterization of Semiconductors: Measurement of Minority Carrier Properties, Academic, New York, 1990. ? 2000 by CRC Press LLC