《电子工程师手册》学习资料（英文版）chapter 22 Semiconductors

Gildenblat, G.S., Gelmont, B, Milkovic, M, Elshabini-Riad, A, Stephenson, FW Bhutta. LA. Look. D. C "Semiconductors The electrical Engineering Handbook Ed. Richard C. dorf Boca Raton CRC Press llc. 2000

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 Gennady Sh. Gildenblat Energy Bands. Electrons and Holes. Transport Prope Hall Boris elmont Effect. Electrical Breakdown.Optical Properties and Recombination Processes.Nanostructure Engineering Disordered Semiconductors Miram ilkovic 22.2 Diodes Analog Technology Consultants pn-Junction Diode.pn-Junction with Applied Voltage. Forward Biased Diode. Ip Vo Characteristic. DC and Large-Signal Aicha elshabini-Riad Model. High Forward Current Effects. Large-Signal Piecewise irginia Polytechnic Institute and Linear Model. Small-Signal Incremental Model. Large-Signal State University Switching Behavior of a pn-Diode. Diode Reverse Breakdown Zener and avalanche diodes. varactor Diodes Tunnel F.W. Stephenson Diodes. Photodiodes and Solar Cells. Schottky Barrier Diode ginia Polytechnic Institute and 2.3 Electrical Equivalent Circuit Models and Device Simulators oran Overview of Equivalent Circuit Models. Overview of RAPP Semiconductor Device Simulators David C. look 22.4 Electrical Characterization of Semiconductors Theory. Determination of Resistivity and Hall Coefficient. Data Wright State University Analysis. Sources of Error 22.1 Physical Properties Gennady Sh. Gildenblat and Boris Elmont 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 um. 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 Eis f=[1+ exp(e- F)/kBr Here kB=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+ 3kg T, then f(e)<0.05 and these states are mostly empty. Similarly, the states with E< F-3ka T are mostly occupied by electrons. In a typical metal Fig. 22. 2(a)], the c 2000 by CRC Press LLC

© 2000 by CRC Press LLC 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 • ID-VD 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 kB = 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 + 3kBT, then f(E) < 0.05 and these states are mostly empty. Similarly, the states with E < F – 3kBT are mostly occupied by electrons. In a typical metal [Fig. 22.2(a)], the f E F kT =+ − 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

FIGURE 22 1 Crystalline lattice of silicon, a=5.43 A at 300oC. energy level E= Fis allowed, and only one energy band is partially filled. (In metals like aluminum, the partially filled band in ig. 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 the charge transfer. It is a fundamental result of solid-state physics that energy bands that are completely led 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 he 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) yo=uK(r) exp(ikr) (22.2) where r is the radius vector of electron, the modulating function uu(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 ne dispersion law E= E,(k). Since(see Fig 22. 2b)in the conduction band only the states with energies close to the bottom, E, are occupied, it suffices to consider the e(k)dependence near E 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 AW,=45 mev below the conduction band edge, E [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 e 2000 by CRC Press LLC

© 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) Cbk = ubk(r) exp(ikr) (22.2) where r is the radius vector of electron, the modulating function ubk(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 = Eb(k). Since (see Fig. 22.2b) in the conduction band only the states with energies close to the bottom, Ec, are occupied, it suffices to consider the E(k) dependence near Ec. 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 DWd ª 45 meV below the conduction band edge, Ec [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 Å at 300°C. a

FIGURE 22.2 Band diagrams of metal (a)and semiconductor(b);, electron; o, missing electron(he 11 FIGURE 22.3 Simplified E(k) dependence for Si(a)and GaAs(b) At room temperature E Si)=1.12 eV,E(GaAs)=1. 43 ev, and A=0.31 eV; (1)and(2)indicate direct and indirect band-to-band transitions While considering the contribution j, 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 uk) be the velocity of electron described by the wave function(20.2). Then i=-9∑k)=-∑vk)-∑k) (223) 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 e 2000 by CRC Press LLC

© 2000 by CRC Press LLC While considering the contribution jp 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); ●, electron; C, missing electron (hole). FIGURE 22.3 Simplified E(k) dependence for Si (a) and GaAs (b). At room temperature Eg(Si) = 1.12 eV, Eg(GaAs) = 1.43 eV, and D = 0.31 eV; (1) and (2) indicate direct and indirect band-to-band transitions. jp = -q q v k = - v k - v k q v k È Î Í Í Í ˘ ˚ ˙ ˙ ˙ Â Â ( ) Â ( ) Â ( ) = ( ) empty states filled all states states empty states

1000T FIGURE 22. 4 The inverse temperature dependence of electron concentration in Si; 1 N,=10 7cm-3N=0; 2: N,=10 6 cm3,Nn=1014cm-3 the holes behind. Assuming that the Fermi level is removed from both E and E, by at least 3kBT(a nondegenerate semiconductor), the concentrations of electrons and holes are given by N expI(F-edIkBTI p=N expI(E-F)/kgT (22.5) where N =2(2mtrkgn)2// and N,= 2(2mttkgn)// are the effective densities of states in the conduction and valence bands, respectively, h is Plank constant, and the effective masses m* and m* depend on the detai of the band structure [ Pierret, 19871 ( E /kg)2 n is independent of the doping neutrality condition can be used to show that in an n-type(n>p)semiconductor at or below room temperature n(n+Na)(Na-Na-n)-1=(N/2)exp(-△W∥k) (22.6) where Na and Na denote the concentrations of donors and acceptors, respectively Corresponding temperature dependence is shown for silicon in Fig. 22.4. Around room temperature n AW/2 for n >N, and AW, for n T:=(E/2kB)/n[VNN,/(Na-Na)] the electron concentration n=n>>No-N,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 e 2000 by CRC Press LLC

© 2000 by CRC Press LLC the holes behind.Assuming that the Fermi level is removed from both Ec and Ev by at least 3kBT (a nondegenerate semiconductor), the concentrations of electrons and holes are given by n = Nc exp[(F – Ec)/kBT] (22.4) and p = Nv exp[(Ev – F)/kBT] (22.5) where Nc = 2 (2m* npkBT)3/2/h3 and Nv = 2(2m* ppkBT)3/2/h3 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 = NcNv exp(–Eg /kBT) D = n2 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 + Na)(Nd – Na – n)–1 = (Nc/2) exp(–DWd /kBT) (22.6) where Nd and Na denote the concentrations of donors and acceptors, respectively. Corresponding temperature dependence is shown for silicon in Fig. 22.4. Around room temperature n = Nd – Na, while at low temperatures n is an exponential function of temperature with the activation energy DWd /2 for n > Na and DWd for n Ti = (Eg /2kB)/ln[ /(Nd – Na)] the electron concentration n ª ni >> Nd – Na 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 22.4 The inverse temperature dependence of electron concentration in Si; 1: Nd = 1017 cm–3, Na = 0; 2: Nd = 1016 cm–3, Na = 1014 cm–3. N Nc v

Diamond E(kv/cm) 107cm.(Source: R.J. Trew, J. B Yan, and LM. Mack, Proc. IEEE, voL. 79, no 5,P 602, May 1991.@ 1991 IEEE)aS applications. Since T: o Eg 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 Eg=2.2-2.86 ev) and diamond (Eg=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, 2, the carrier drift velocity, Va is determined by the balance of the electric and collision forces m*valt=-ga (22.7) where t is the momentum relaxation time. Consequently va=--, 6, where um gt/m is the electron mobility. For an n-type semiconductor with uniform electron density, n, the current density j,=-gnva and we obtain Ohms law in =08 with the conductivity o= qnu The momentum relaxation time can be approximately expressed as l/t=1/ta+1/tn+l/t+1/t+1/t+l/t。+ (228) where ta tna tas tnpoTpo Tpe 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 jn=q叫ng+qD2V (22.9) where the diffusion coefficient D, is related to mobility by the Einstein relation D,=(kgTlq)u, A similar equation can be written for holes and the total current density is j=jn+je 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 relocity-field dependencies for several semiconductors are shown in Fig. 22.5. In GaAs va 8)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, of the drift velocity in a strong electric field is known as the saturation velocity and is usually within the 10-3.10 cm/s range. As semiconductor device dimensions are scaled down to the submi crometer range, v, becomes an important parameter that determines the upper limits of device performance e 2000 by CRC Press LLC

© 2000 by CRC Press LLC applications. Since Ti } Eg 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 Eg = 2.2–2.86 eV) and diamond (Eg = 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, vd, is determined by the balance of the electric and collision forces: mn *vd /t = –q% (22.7) where t is the momentum relaxation time. Consequently vd = –mn%, where mn= qt/m* n is the electron mobility. For an n-type semiconductor with uniform electron density, n, the current density jn= –qnvd and we obtain Ohm’s law jn = s% with the conductivity s = qnmn. The momentum relaxation time can be approximately expressed as 1/t = 1/tii + 1/tni + 1/tac + 1/tnpo + 1/tpo + 1/tpe + . . . (22.8) where tii, tni, tac, tnpo, tpo, tpe 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 jn = qnmn% + qDn—n (22.9) where the diffusion coefficient Dn is related to mobility by the Einstein relation Dn = (kBT/q)mn. A similar equation can be written for holes and the total current density is j = jn + jp. 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 vd(%) 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 vs of the drift velocity in a strong electric field is known as the saturation velocity and is usually within the 107 –3·107 cm/s range. As semiconductor device dimensions are scaled down to the submi￾crometer range, vs becomes an important parameter that determines the upper limits of device performance. FIGURE 22.5 Electron (a) and hole (b) drift velocity versus electric field dependence for several semiconductors at Nd = 1017 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.)

FIGURE 22.6 Experimental setup for Hall effect measurements in a long two-dimensional sample. The Hall angl determined by a setting of the rheostat that renders j, =0. Magnetic field B= B(Source: K W Boer, Surveys of Semicond Physics, New York: Chapman Hall, 1990, P 760. With permission.) 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. 1um(0. 25 um 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 +vVf +gvF= df (22.10) Here f denotes the distribution function (number of electrons per unit volume of the phase space, i. e, f dn/rd,p), v is electron velocity, p is momentum, and (af/dn)ool is the"collision integral "describing the change of f caused by collision processes described earlier. For the purpose of semiconductor modeling, Eq (22.10) 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 [Fery;1991] In a uniform magnetic field electrons move along circular orbits in a plane normal to the magnetic field B with ne angular(cyclotron) frequency @.= gB/m. For a uniform semiconductor the current density satisfies the j=σ(g+R[jB]) (22.11) In the usual weak-field limit o,t < I the Hall coefficient RH =-ring and the Hall factor r depend on the dominating scattering mode. It varies between 3 /8=1. 18(acoustic phonon scattering) and 3157/518=1.93 The Hall coefficient can be measured as rh= V d/I B using the test structure shown in Fig 22. 6. In this expression V, is the Hall voltage corresponding to 1=0 and d denotes the film thickness. e 2000 by CRC Press LLC

© 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/d3 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 wc = qB/m* n . For a uniform semiconductor the current density satisfies the equation j = s(% + RH[jB]) (22.11) In the usual weak-field limit wct << 1 the Hall coefficient RH = –r/nq and the Hall factor r depend on the dominating scattering mode. It varies between 3p/8 ª 1.18 (acoustic phonon scattering) and 315p/518 ª 1.93 (ionized impurity scattering). The Hall coefficient can be measured as RH = Vy d/IxB using the test structure shown in Fig. 22.6. In this expression Vy is the Hall voltage corresponding to Iy = 0 and d denotes the film thickness. FIGURE 22.6 Experimental setup for Hall effect measurements in a long two-dimensional sample. The Hall angle is determined by a setting of the rheostat that renders jy = 0. Magnetic field B = Bz. (Source: K.W. Böer, Surveys of Semiconductor Physics, New York: Chapman & Hall, 1990, p. 760. With permission.) ¶ ¶ ¶ ¶ f t f q f f t + — + — = Ê Ë Á ˆ ¯ v % p ˜ coll

Combining the results of the Hall and conductivity measurements one can extract the carrier concentration type(the signs of V, are opposite for n-type and p-type semiconductors) and Hall mobility HH=ru LH=-R,o, n=-rqru (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 o t > I and on the average an electron completes several circular orbits without collision. Instead of the conventional Eb(k)dependence, the allowed electron energy levels in the magnetic field are given by (h= h/2T;s=0, 1, 2,... E4=ho(s+1/2)+h2k2/2mx (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_ 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. In such a structure the electron density of states(number of these peaks relative to the Fermi level are controlled by the magnetic e evel. Since o. B, the positions of The most striking consequence of this phenomenon is the quantum Hall effect, which manifests itself as a stepwise change of the Hall resistance Pxy= V/ as a function of magnetic field(see Fig. 22.7). At low temperature(required to establish the condition teo as shown in Table 22.1 The field dependence of the impact ionization is usually described by the impact ionization coefficient ap defined as the average number of electron-hole pairs created by a charge carrier per unit distance traveled.A simple analytical expression for a, [Okuto and Crowell, 1972] can be written as a,=(/x)exp(a -?+x To simplify the matter we do not discuss surface subbands, which is justified as long as only the lowest of them is occupie e 2000 by CRC Press LLC

© 2000 by CRC Press LLC Combining the results of the Hall and conductivity measurements one can extract the carrier concentration type (the signs of Vy are opposite for n-type and p-type semiconductors) and Hall mobility mH = r m: mH = –RHs, n = –r/qRH (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 wct >> 1 and on the average an electron completes several circular orbits without a collision. Instead of the conventional Eb(k) dependence, the allowed electron energy levels in the magnetic field are given by (\= h/2p; s = 0, 1, 2, . . .) Es = \wc (s + 1/2) + \2k 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 = Bz. 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 wc } 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 rxy = Vy /Ix as a function of magnetic field (see Fig. 22.7). At low temperature (required to establish the condition t Eg, as shown in Table 22.1. The field dependence of the impact ionization is usually described by the impact ionization coefficient ai , defined as the average number of electron–hole pairs created by a charge carrier per unit distance traveled. A simple analytical expression for ai [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. a l i = x a - a + x Ê Ë ˆ ¯ ( / ) exp 2 2

FIGURE 22.7 Experimental curves for the Hall resistance Px =8 jx and the resistivity Px =6 jx 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 22.1 Impact lonization Threshold Energy(ev) Energy gap, E electron-initiated 1.18 0.76 1.7 2.6 0.2 Eh, hole-initiated 1710.88142.30.2 where x=qa/Eh, a=0.217(E, /En )-14, A is the carrier mean free path, and Eo is the optical phonon energy (Eopt =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 Optical Properties and Recombination Processes If the energy of an incident photon ho Eg, then the energy conservation law permits a direct band-to-band transition,as indicated in Fig 22. 2(b). Because the photons momentum is negligible compared to that of an electron or hole, the electrons momentum hk 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

© 2000 by CRC Press LLC where x = q%l/Eth, a = 0.217 (Eth/Eopt)1.14, l is the carrier mean free path, and Eopt is the optical phonon energy (Eopt = 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 > Eg, 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 22.7 Experimental curves for the Hall resistance rxy = %y /jx and the resistivity rxx = %x /jx 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 22.1 Impact Ionization Threshold Energy (eV) Semiconductor Si Ge GaAs GaP InSb Energy gap, Eg 1.1 0.7 1.4 2.3 0.2 Eth, electron-initiated 1.18 0.76 1.7 2.6 0.2 Eth, hole-initiated 1.71 0.88 1.4 2.3 0.2

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 hoA2=1.24 um/E (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(ot > 1)the absorption of microwave radiation is peaked at o=0, At this frequency the photon energy is equal to the distance between two Landau levels, ie,, ha= Es+I- Es with reference to Eq.(22. 13). This effect, known as cyclotron reson used to measure the effective masses of charge carriers in semiconductors [ in a simplest case of isotropic E(k)dependence, m*=q B/o 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 given by the Shockley-Read-Hall theory as R (22.16) t(P+Pi)+t, (n+n) where t,, te, Pi, and n, 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 ni, and R=O). For example, in the depleted region np f and R< 0, so that charge carriers ar 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 an diy )n-r (22.17) q Nanostructure engineeri 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、k)=π2s2h2/(2mxb2)+h2k2/(2mx),s=1,2,3, (22.18) where k is the electron wave vector parallel to the plane of the semiconductor layer. The charge carriers quantum wells exhibit confined particle behavior. Since E, o b-, well structures can be grown with distance e 2000 by CRC Press LLC

© 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 lc = 1.24 mm/Eg (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 (wct >> 1) the absorption of microwave radiation is peaked at w = wc. At this frequency the photon energy is equal to the distance between two Landau levels, i.e., \w = ES+1 – ES 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/wc]. 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 tn, tp, p1, and n1 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 = n2 i , and R = 0). For example, in the depleted region np < n2 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 Es (k) = p2s2\2/(2m* n b 2) + \2k2/(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 Es } b –2, well structures can be grown with distance R np n p p n n i n p = - + + + 2 1 1 t ( ) t ( ) ¶ ¶ n t div q R n = - j