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 • ForwardBiased 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 acceptortype 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< N The reduction of n compared with the net impurity concentration.2 Na-Na, while at low temperatures n is an exponential function of temperature with the activation energ Na is known as a freeze-out effect. This effect does not take place in the heavily doped semiconductor For temperatures T> 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 < Na. The reduction of n compared with the net impurity concentration Nd – Na is known as a freeze-out effect. This effect does not take place in the heavily doped semiconductors. For temperatures T > 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