tween 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 nis"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 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 if the electro free perlattice 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 E(k)=Ell-cos(ka) (22.19) where a is the superlattice period and Eo is the half-width of the energy band. The electron group velocity h-lde(k)/ok=(e a/h) sin (ka) (2220) is a decreasing function of k(and hence of energy)for k> T/a. The higher energy states with k> T/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 recomb 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 crystallin 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 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 e 2000 by CRC Press LLC
© 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 nand 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 onedimensional tight binding approximation the miniband can be described as (22.19) where a is the superlattice period and Eo is the half-width of the energy band. The electron group velocity v = \–1¶E(k)/¶k = (Eoa/\) 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 recombination lifetimes. Disordered Semiconductors Both amorphous and heavily doped semiconductors are finding increasing applications in semiconductor technology. 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 decomposition 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 E k E ka o ( ) = [1 - cos( )]
E+T 75 6 FIGURE 22.8 Experimentally determined density of states for a-Si. A and B are acceptor- like and donor- like states, spectively. The arrow marks the position of the Fermi level Efs 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 per 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,> N=(0.2/ag)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 lectron Bohr radius ag=/2m E, where E, is the ionization energy of the donor state. For silicon, Na =3.8 108cm-. 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- Energy band: Continuous interval of energy levels that are allowed in the periodic potential field of the crystalline lattic Energy gap: The width of the energy interval between the top of the valence band and the bottom of the conduction band e 2000 by CRC Press LLC
© 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 liquidcrystal displays]. The a-Si device performance degrades with time under electric stress (TFTs) or under illumination (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 Nd > Ndc = (0.2/aB)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 aB = \/ , where Ei is the ionization energy of the donor state. For silicon, Ndc ª 3.8 · 1018 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 semiconductor. 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 efo 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.) 2m En i *
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. K von Klitzing, Rev. Modern Phys, vol 58, P 519, 19 s5s, N./. Prentice-Hall, 19 g R F. Pierret, Advanced Semiconductor Fundamentals, Reading, Mass.: Addison-Wesley, 1987 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 Ilkovic Diodes are the most widely used devices in low- and high-speed electronic circuits and in rectifiers and power upplies. 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 HA 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- unction 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 lo 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 e 2000 by CRC Press LLC
© 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 V0 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
electric field, E, which at equilibrium prevents a further diffusion of carriers across the junction. This equilib- ium 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 Vo is, according to the Boltzmann relation, [ Grove, 1967; Foustad, Vo= Vr Inlp, /p, (22.21) In this equation, Pe is the concentration of holes in the p-material and Pn is the concentration of holes in the n-material. Vr is the thermal voltage. Vr=26 mV at room temperature(300 K). with Pp= Pn N where n, is the intrinsic concentration, the barrier voltage Vo becomes approximately Sze, 1985; Fonstad, 1994] o= Vr In(NN, /nI (22.22) 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, Vo is at room temperature T Vo=0.67 V for an abrupt junction with NA=10 7 at/cm and Np= 10 5 at/cm. The depletion layer width is pically about 4 um, 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 Vp to the diode is opposite to the barrier voltage Vo then P, in the Boltzmann relation in Eq (22. 21)is altered to Pp=Pn exp(vo-VD)/ 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, Pn(x=0)=pn exp VD/Vr 22.24 and accordingly the concentration of electrons Most modern planar diodes are unsymmetrical. Figure 22.9 shows a pn-diode with the n region Wn much shorter than the diffusion length Lpn of holes in the n-semiconductor region. This results in a linear con tration gradient of injected diffusing holes in the n region given by dp/dx =-(P exp Vp/Vr-pn)/W (22.26) e 2000 by CRC Press LLC
© 2000 by CRC Press LLC electric field, E, which at equilibrium prevents a further diffusion of carriers across the junction. This equilibrium 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 V0 is, according to the Boltzmann relation, [Grove, 1967; Foustad, 1994] (22.21) In this equation, pp is the concentration of holes in the p-material and pn is the concentration of holes in the n-material. VT is the thermal voltage. VT = 26 mV at room temperature (300 K). With pp ª NA and pn ª where ni is the intrinsic concentration, the barrier voltage V0 becomes approximately [Sze, 1985; Fonstad, 1994] (22.22) Here NA denotes the concentration of immobile acceptors on the p side of the junction and ND 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, V0 is at room temperature T = 300°K, typically V0 = 0.67 V for an abrupt junction with NA = 1017 at/cm3 and ND = 1015 at/cm3 . 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 VD to the diode is opposite to the barrier voltage V0, then pp in the Boltzmann relation in Eq. (22.21) is altered to pp = pn exp(V0 – VD)/VT (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, pn(x = 0) = pn exp VD /VT (22.24) and accordingly the concentration of electrons nn(x = 0) = nn exp VD /VT (22.25) Most modern planar diodes are unsymmetrical. Figure 22.9 shows a pn-diode with the n region Wn much shorter than the diffusion length Lpn of holes in the n-semiconductor region. This results in a linear concentration gradient of injected diffusing holes in the n region given by dp/dx = –(pn expVD /VT – pn)/Wn (22.26) V V p p 0 = T p n ln[ / ] n N i D 2 V V N N n 0 T A D i 2 = ln[ / ]
A FIGURE 2 are fabricated in planar technology. Most modern diodes are unsymmetrical; thus W < Lpr The ghly doped than the n region. 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,=-qA, D, dp/. where A, is the junction area, D, is the diffusion constant for holes, and q is the elementary charge Dove equations we obtain IP=(qA DpP,/Wn)(exp Vp/Vr-1) (22.28) In the p-semiconductor we assume that Lmp < Wei ther /dx= n, exp(Vp/Vr-1) 2229) By substituting this into the electron diffusion equation In= gA, D dn/dx (22.30) h=(qA, Dn)/Ln(exp Vp/Vr-1) (22.31) Thus, the total junction diffusion current In=I,+L=gA D Pn/W+ qA Dn. /Lmpl(exp Vp/V-1) (2232) Since the recombination of the injected carriers establishes a diffusion gradient, this in turn yields a flow of current proportional to the slope For Fvp>> vr, i.e. VD=0.1 V I s=(qA D Pn/W,+gA D n, /LuD) (2233) e 2000 by CRC Press LLC
© 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 Ip = –qAj Dp dp/dx (22.27) where Aj is the junction area, Dp is the diffusion constant for holes, and q is the elementary charge. By combining of above equations we obtain Ip = (qAj Dp pn /Wn ) (exp VD /VT – 1) (22.28) In the p-semiconductor we assume that Lnp << Wp; then dn/dx = np exp(VD /VT – 1) (22.29) By substituting this into the electron diffusion equation, In = qAjDn dn/dx (22.30) we obtain In = (qAj Dnnp)/Lnp(exp VD /VT – 1) (22.31) Thus, the total junction diffusion current is ID = Ip + In = {qAj Dp pn /Wn + qAjDnnp /Lnp} (exp VD /VT – 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 *–VD * >> VT , i.e., VD = –0.1 V, IS = (qAiDp pn /Wn + qAi Dn np /Lnp) (22.33) FIGURE 22.9 Planar diodes are fabricated in planar technology. Most modern diodes are unsymmetrical; thus Wn << Lpn. The p-type region is more highly doped than the n region