MOSFET MODELING FOR VLSI SIMULATION - Theory and Practice Episode 2 ppt

16 2 Basic Semiconductor and pn Junction Theory 0 FREE ELECTRON (-) 0 HOLE (+I Fig. 2.1 Energy band diagram of a semiconductor (silicon) separate bands of allowed energies, called the valence band and the conduction band. The energy levels in the valence bands are mostly filled with electrons forming the covalent bonds. The energy levels in the conduc- tion bands are nearly empty. Electrons which occupy the energy levels in the conduction band are called free electrons, (or conduction electrons). The very closely spaced energy levels in the valence and conduction bands are often separated by a energy range where there are no allowed quantum states or energy levels, known as the energy gap E, (or band gap). This energy gap between the two allowed energy bands is often referred to as the forbidden band or forbidden gap. Although the energy is a complex function of momentum in three dimensions and there are so many energy levels, and so many electrons, the energy band picture will be tedious to draw out if all the energy levels are shown. Thus, only the edge levels of each of the allowed energy bands are shown in the energy band diagram (see Figure 2.1). The electron energy is considered a positive quantity and is plotted upward on the energy-band diagram. If E, and E, are the energy levels for the lower edge of the conduction band and upper edge of the valence band respectively, then the band gap E, is' E, = E, - E, (ev). (2.1) Note that the unit of E, is electron-Volt eV (or qV). By definition, an electron-Volt is the energy an electron of charge q acquires when it moves through a potential difference of 1 Volt (V). When E, in eV is divided by q (in units of electron charge, not Coulombs), the charge q of the electron cancels out, and the result is Volts (V). Thus E, and E,/q have the same numerical value but diferent units. In general any physical quantity expressed in eV can be converted into Volts by simply dividing the quantity by the charge q and vice versa. If E, is in Joules, then E,/q is in Volts where q = 1.602 x 10-'9C. 2.2 Intrinsic Semiconductor 17 When a valence electron is given sufficient additional energy (2 Eg), it can break out of the chemical bonding state and become a free electron that moves about freely in the lattice. A hole (the absence of an electron) is left where the electron was bonded. Since net positive charge is now associated with the atom from which the electron broke away, a hole is associated with a positive charge. Note that both the electron and the hole are generated simultaneously. In the energy band diagram holes are normally represented as circles and electrons as dots. Experimental results show that the band gap energy E, ofmost semicon- ductors, including silicon, decreases as the temperature increases, because the crystal lattice spacing increases due to the thermal expansion. The temperature dependence of the band gap E, for silicon can be modeled using the following polynomial equation [ 151 1.206-2.73 x lOP4T (2.2a) 1.1785-9.025 x 10-5T-3.05 x 10p7T2 (2.2b) 1.17+ 1.059 x 10p5T-6.05 x 10-7T2 (for T 2 250 K) (for 300 K > T > 170 K) (for T i 170 K) (2.2c) where Tis the temperature in Kelvin (K). Equation (2.24 fits the experimental data to within 1 meV in the temperature 250-415 K, while at temperatures below 250 K, E, is modeled to within 2 meV [l5]. Note that both Eqs. (2.2a) and (2.2b) are accurate for 250 < T < 300 K. Although Eq. (2.2) has been used for low-temperature device modeling work [ 16,171, the circuit simulator SPICE uses the following equation for E, [lS] (eV) (SPICE) 7.02 x 10-~~2 1108 + T E, = 1.160 - giving E, = 1.115eV at 300K, which is somewhat lower than the more accurate value of 1.124 eV predicted by Eq. (2.2). 2.2 Intrinsic Semiconductor A pure, single crystal semiconductor in which all the electrons in the conduction band are thermally excited from the valence band is called an intrinsic semiconductor. In other words, in an intrinsic semiconductor, at a given temperature, the number of holes in the valence band equals the number of electrons in the conduction band. Thus if n and p are the free electron and hole concentrations (per cm3) respectively, then 18 2 Basic Semiconductor and pn Junction Theory Table 2.1. Effective mass ratios .for silicon at 300 K Density-of-states effective mass Conduction effective mass m:lm mp Electrons 1.08 Holes 0.81 0.26 0.386 np = n; (2.4b) where ni by definition is the free electron (or hole) concentration in intrinsic silicon, often called the intrinsic carrier concentration. Effective Mass of Electron and Hole. The electrons in the conduction band and the holes in the valence bands move freely throughout the crystal as if they were free particles, suffering only occasional scattering by impurities and defects present in the crystal. Thus electrons and holes are analogous to electrons moving in a vacuum. The difference is the presence of the potential or the Coulomb force experienced by the electrons due to the charged atomic cores of host atoms. These charged atomic cores are located on a regular lattice, giving rise to a periodic potential energy; in vacuum there is no such periodic potential. The effect of the periodic potential on the motion of electrons in the conduction band and holes in the valence band is represented by the effective masses of the electrons (m:) and holes (m;) respectively, and by the equivalent positive charge of a hole. It should be pointed out that effective mass is not a simple scalar quantity. For a given material and carrier type there are several effective masses encountered in practice [1]-[ lo]. Further, the effective mass required to calculate carrier (electron and hole) concentration, called the density of states’ effective mass, is different from the conductivity effective mass required to calculate carrier mobility. These effective masses are function of temperature as well. There is large variation in the values of m: and mp* reported in the literature [lS]. The commonly used values for the effective mass for electrons and holes at room temperature are summarized in the following Table 2.1 [6]. Intrinsic Currier Concentration. According to Barber [19], who has reviewed and correlated the theoretical and experimental data on ni for * Density of states is the total number of energy levels per unit volume which are available for possible occupation by electrons. 2.2 Intrinsic Semiconductor exp [ 2kT 2kT0 19 (2.7) silicon we have I I 1.206 -__ n, = 3.1 x 10'6T3'2 exp ( 2kT) (cm-3) 1 I I where 1.206 is the extrapolated zero-degree band gap energy E,(O) [cf. Eq. (2.2a)], k is the Boltzmann constant (= 8.62 x lo-' eV/K) and T is the temperature in K. The term kT is called Boltzmannfactor. Since it has the dimension of energy, it is often called the thermal energy and the corresponding factor kTlq (V), the thermal voltage which we will denote by V,. The value of thermal energy at 300 K is 0.02586 eV. At T = 300 K, Eq. (2.5) yields n, = 1.19 x 1010cm-3. Recently Green [lS] has reported that at 300K a more accurate value is ni = 1.08 x 10" ~m-~. Note that ni increases rapidly with temperature, doubling roughly every 8 OC3 Another expression which has been often used for ni calculation as a function of temperature is (~m-~). If ni(To) is the value of ni at the nominal or reference temperature To (say 300 K), then using (2.6), n, at any other temperature T could be written as where E,(T) is given by Eq. (2.2). The above equation is used in SPICE for calculating ni at any temperature T with ni = 1.45 x 1010cm-3 at T = 300K. 2.2.1 Fermi Level The number of carriers available for conduction determines the electrical properties of a semiconductor. This number is found from the density of allowed states and the probability that these states are occupied. The probability that an available state with energy E is occupied by an electron under thermal equilibrium conditions is given by the Fermi-Dirac At 77K ni for silicon is -10-20cm-3, while at 400K its value is -1O'*~m-~ 20 flE)A 1.0. 0.5 0.0 2 Basic Semiconductor and pn Junction Theory probability density function f(E), also called the Fermi function [l]-[lO] (2.8) where E, is the Fermi energy or Fermi level defined as the energy level at which the probability offinding an electron, for T > 0 K, is exactly one-haEf: Note that the Fermi level is a purely mathematical parameter and provides a reference with which other energies can be compared. When E = E, we havef(E) = 1/2 which means that the electron is equally likely to have an energy above the Fermi level as below it. At T = OK, f(E) = 1 indicating thereby that the probability of finding an electron below E, is unity and above E, is zero. In other words all energy levels below E, are filled and all energy levels above E, are empty. As T is increased above zero, the function f(E) changes as shown in Figure 2.2. Thus, the probability that energy levels above E, are filled increases with temperature. It is important to note that the Fermi function (or Fermi energy) applies only under equilibrium condition^.^ The Fermi level can be considered to be the chemical potential for electrons and holes. Since the condition for any system in equilibrium is that the chemical potential must be constant through out the system, it follows that the Fermi level must be constant throughout a semiconductor in equilibrium. The Fermi level in intrinsic silicon, often referred to as intrinsic Fermi level Ei, is only 0.0073 eV below midgap at T = 300K. Thus for all practical purposes it can be assumed that Ei is in the middle of the energy gap. Here ‘equilibrium’ means no applied voltage, no applied external fields or thermal gradients. Fig. 2.2 A Fermi-Dirac distribution function 2.3 Extrinsic or Doped Semiconductor 21 For all energy levels higher than 3kTabove E, the functionf(E) can be approximated by which is identical to the Maxwell-Boltzmann density function for classical gas particles. For most device applications, the function f(E) given by Eq. (2.9) is a good approximation. 2.3 Extrinsic or Doped Semiconductor When elemental impurities called dopants are added to ~ilicon,~ free carrier concentration of intrinsic silicon changes and the resulting silicon is called doped or extrinsic silicon. The most commonly used dopants in integrated circuit technology are boron(B), phosphorous(P), and arsenic(As). If the dopants are phosphorous or arsenic they are called donor atoms, since they donate an electron to the crystal lattice, and the doped silicon is called n-type material that contains excess electrons. However, if the dopant is boron, it is called an acceptor atom, since it can be thought of as accepting an electron from the valence band, and the doped silicon is called p-type that contains excess holes. In terms of energy band diagrams, donors add allowed electron states in the band-gap close to the conduction band edge as pictured in Figure 2.3a; acceptors add allowed states just above the valence band edge as shown in Figure 2.3b. Also shown in this figure are positions of the Fermi level due to donors (Fig. 2.3~) and acceptors (Fig. 2.3d). It is possible to dope silicon so that p = n. Material of this type is called compensated silicon. In practice, however, one impurity dominates so that semiconductor is either n-type or p-type. A semiconductor6 is said to be nondegenerate, if the Fermi level lies in the band gap more than a few kT (- 3kT) from either band edge. Conversely, if the Fermi level is within a few kT (-3kT) of either band edge, the semiconductor is said to be degenerate. In the nondegenerate case, the carrier concentration obeys Maxwell-Boltzrnann statistics (2.9). However, for the degenerate case where the dopant concentration is in excess of approximately lo1* ~rn-~ (heavy The silicon crystal contains 5 x lo2* atoms/cm3 [ = Avogadro number (6.02 x loz3) x Density (2.33)/Gram Molecular Weight (28.09)]. The doping concentration used in the devices ranges from 1014-1020cm-3 or from less than one atom in hundred million to a fraction of a percent. Through out this book the word semiconductor and silicon are used interchangeably. 22 = ni exp [&(4 - 4,)] (cm-3) 2 Basic Semiconductor and pn Junction Theory (2.1 Oa) CONDUCTION BAND + €, Ed 0-041 eV(P) 7 Eg=1-12eV I EV L VALENCE BAND (a) t VALENCE BAND (b) CONDUCTION BAND EC f L VALENCE BAND ACCEPTER HOLES (C 1 (d) Fig. 2.3 Energy-band diagram representation of (a) donor level Ed (phosphorous, P) in silicon (b) acceptor level E, (boron, B) (c) Fermi level with phosphorous doping concen- tration of 1015 cm13 and (d) Fermi level with boron doping concentration of 1015 cm13 d~ping)~ one must use the Fermi-Dirac distribution function given by Eq. (2.8). In what follows, unless otherwise specified, we will assume the semiconductor to be nondegenerate. In an n-type nondegenerate semiconductor the Fermi level E, (or Fermi potential 4, = - E,/q) lies above the intrinsic level Ei (or potential 4 = -Ei/q)8 by an amount given by the following equation (see Figure 2.3c), ' The carrier concentration greater than 10" cml (heavy doping) is normally represented as n+(electrons) or p+(holes). Note the negative sign in the 4 = - E/q relation; when the electron energy plotted upwards is positive, the positive potential must be plotted downwards because ofthe negative electron charge. 2.3 Extrinsic or Doped Semiconductor 23 while in a p-type semiconductor the Fermi level E, (Fermi potential 4,) lies below the intrinsic level Ei (potential 4) by an amount (see Figure 2.3d) The Eqs. (2.10a) and (2.10b) for n and p, respectively, are often referred to as Boltzmann’s relations. At room temperature, the available thermal energy is sufficient to ionize nearly all acceptor and donor atoms due to their low ionization energies. Hence it is a safe approximation to say that in nondegenerate silicon at room temperature n z N, (n-type) (2.1 la) P Na ( P-~Y (2.11b) where N, is the concentration of donor atoms (~m-~) and N, is the concentration of acceptor atoms (~m-~). In an n-type material, where N, >> ni, electrons are majority carriers whose concentration is given by Eq. (2.11a), while the hole concentration p, is9 n,? p,z- (~m-~) (2.12) Nd remembering that pn = nf [cf. Eq. (2.4b)l. The hole concentration p, is much smaller than n,. Thus holes are minority carriers in an n-type semiconductor. Similarly, in a p-type semiconductor where Na >> ni, holes are majority carriers given by Eq. (2.1 lb), while electron concentration np is given by (2.13) Since np << p, electrons are minority carriers in p-type semiconductor. Consequently, we often use the terminology of majority carriers and minority carriers. Taking Ei as the zero reference level and making use of Eq. (2.11a) in (2.10a) we can write the electron concentration n in terms of 4f, for an n-type semiconductor, as 44, (2.14a) It is common practice to represent carrier concentrations with the subscript denoting the type of semiconductor. Thus pn denotes hole concentration p in an n-type semiconductor and likewise n, (or simply n) denotes electron concentration n in an n-type semiconductor. 24 2 Basic Semiconductor and pn Junction Theory Similarly, the hole concentration p in a p-type semiconductor becomes p = N, = niexp (=). q4r (2.14b) Rearranging Eq. (2.14a) or (2.14b) for 4f we get (2.15) where the (+) sign is for p-type semiconductors (N, = N,) and the (-) sign is for n-type semiconductors (N, = Nd). Note that 4f is not only ufunction of currier concentration, but is also dependent on temperature. The variation of Fermi potential I 4r I at T = 300 K as a function of substrate concentration Nb( = N, or Nd) is shown in Figure 2.4. As the temperature increases, ni increases [cf. Eq. (2.5)] and therefore 4f decreases (the increase in ni with temperature is much faster than the increase in thermal energy kT). Thus, with an increase of temperature, the Fermi level approaches the midgap position i.e. the intrinsic Fermi level; showing thereby that the semicon- ductor becomes intrinsic at high temperature. Thus doped or extrinsic silicon will become intrinsic ifthe temperature is high enough. The temperature at which this happens depends upon the dopant concentration. When the material becomes intrinsic, the device can no longer function and therefore the intrinsic region is avoided in device operation. 06 1 Q,<O n-TYPE 0.5 2 0.1 1014 1015 1016 1017 1018 1019 BULK CONCENTRATION N b (cm -3) Fig. 2.4 Fermi potential df in silicon as a function of substrate concentration N, 2.3 Extrinsic or Doped Semiconductor 25 The temperature coefficient of +f can be obtained by differentiating Eq. (2.15) giving (2.16) where we have made use of Eq. (2.5) for ni. This gives d@,/dT - - 1 mV/K. If we use Eq. (2.7) for ni, 4f at any temperature T can be written in terms of its value at a nominal temperature To as (2.17) This is the equation used in SPICE for the temperature dependence of 4f. 2.3.1 Generation-Recombination Under thermal equilibrium, the condition pn = nt is maintained. This condi- tion may be disturbed by the introduction of free carriers (only electrons, only holes, or electron-hole pairs) in the semiconductor. This process of introducing additional carriers (excess carriers) is called carrier injection and can occur in different ways (optical, electrical, etc). Note that the injection refers to any increments of carriers due to a nonthermal source, irrespective of the nature of this source. If the injected carrier density is small compared to the majority carrier density at equilibrium, so that the latter remains essentially unchanged while the minority carrier density is equal to the excess carrier density, then the process is called low-leuel injection. If the injected carrier density is comparable to or exceeds the majority carrier density before injection, then it is called high leuel injection. Although in semiconductor device operation it is generally low level injection which is important, one also encounters high level injection. Let us take an example. Suppose in n-type silicon, N, = 10l6 cm-3 then from Eq. (2.1 1) the majority carrier concentration n,, = 10l6 ~m-~, while from Eq. (2.12) minority carrier concentration" pno = 2 x 104cm-3. NOW The majority and minority carrier concentrations calculated using Eqs. (2.1 1)-(2.13) are thermal equilibrium values of the carrier concentration and are often denoted by adding the subscript 0 to any other subscript in order to distinguish them from the new carrier concentration after injection. Thus, thermal equilibrium values of the majority electrons and minority holes in a n-type silicon will be represented by the symbols n,, and pn, respectively. [...]... negligible (see discussion in section 3 .2. 1, Eq 2. 64) 2 Basic Semiconductor and p n Junction Theory 44 b( - X,) = &?(X,) = 0 €(x) = (see Figure 2. 1 lc) to give 12 1-[ 12] qN -L ( X , + x) - x, II x 0 EOEsi (2. 45) €(x) = qNd - -( X, - x) 0 I x I x, EOEsi Since the field must be continuous at x = 0, we get from Eq (2. 45) the maximum field b,,, as (2. 46) or 1 qNuXp=qNdXn 1 (2. 47) which gives the distribution... aJ, q ax -= - R, + G, (2. 38a) 2 Basic Semiconductor and p n Junction Theory 38 Similarly for holes we have (2. 38b) where R, and G are recombination and generation rates for holes These , equations are called continuity equations for electrons and holes respectively and describe the time dependent relationship between current density, recombination and generation rates and distance They are used for solving... diffusion length for holes and electrons L, and L, respectively .22 The diode current I , is given by [2 ]-[ 12] (2. 55) where I , is called the reverse saturation current given by (2. 56) 22 The minority carrier dijjiision lenghts L, and L, are defined as L P = a and L,=& (cm) Physically, LJL,) is the mean distance traveled by the injected hole (electron) before it recombines with an electron (hole) and may range... the silicon purity and doping concentration 2. 6 Diode Current-Voltage Characteristics - Na p-TYPE D,lL"*~, -_ , ++I ++, I 41 n-TYPE Nd - Dp, Lpv cp Fig 2. 12 Excess minority carrier distribution in the bulk region when the diode is forward bias - Clearly, I , may be considered as arising from thermal generation of minority carriers in the bulk region At T = 300 K, V, 26 mV and for a forward bias V,(... current flows inside the semiconductor and therefore J, = Jp = 0 However, under nonequilibrium conditions J, and J , can be written as J, = - J, = - qnp ,- dv, dx (2. 36a) dqP 4PPp- (2. 36b) dx and are easily obtained by combining Eqs (2. 35) and (2. 19) for n and respectively (quasi-Fermi potentials) [I 2. 4.4 Continuity Equation When carriers diffuse through a certain volume of semiconductor, the current density... even when the thickness is 35 2. 4 Electrical Conduction - 1 0-4 1 0-3 1 0 -2 lo-' 100 10' lo* lo3 lo4 RESlSTlVrrY(nan) Fig 2. 8 Dopant density versus resistivity at 23 °C for silicon doped with phosphorous and boron (After Muller and Kamins [S].) not uniform, such as would occur for ion imp1anted"j or diffused layers Since resistivity p is a function of carrier concentration and mobility, both of which are... follows: (2. 39) where p is net space charge density (Coul/cm3), eO( 8.854 x 1 0- l4 F/cm2) = is permittivity of free space, and esi(= 11.7) is relative permittivity of silicon If n and p are free electron and hole concentrations and No- and N l are concentrations of ionized acceptors and donors respectively in the space charge region, we have d& dx -= 4 -[ p(x) E ~ E , ~ - n(x) + N,+(x )- N i ( x ) ] (2. 40)... Fermi level E , is related to equilibrium carrier concentration If qn and q p are the quasi-Fermi potentials for electrons and holes , corresponding to the quasi-Fermi level 5 and 2Fp respectively, then under 28 2 Basic Semiconductor and p n Junction Theory nonequilibrium condition we have (2. 19a) (2. 19b) so that in equilibrium Fn Fp E , and q n= qp= d, Note that under = = nonequilibrium conditions the... Remembering that & = - d4/dx and since at room temperature No- M N a and N: % N d , the Poisson equation in terms of potential 4 can be written as (2. 41) The current equation (2. 35),the continuity equation (2. 38) and the Poisson 39 2. 5 p n Junction at Equilibrium equation (2. 41) are one-dimensional equations; however, they can easily be extended to three-dimensional equations (see section 6.1) 2. 5 p n Junction... built-in voltage 4bj For an abrupt junction with uniform dopant concentration on the two sides we have n,, = N , and p p o = N, Making use of Eqs (2. 12) and (2. 13) for l9 2o Strictly speaking the depletion region is not devoid of free carriers but the number is so small compared to N , and N , that for all practical purposes it can be assumed to be depleted of free carriers The three terms, space-charge . for silicon can be modeled using the following polynomial equation [ 151 1 .20 6 -2 .73 x lOP4T (2. 2a) 1.178 5-9 . 025 x 1 0-5 T-3.05 x 10p7T2 (2. 2b) 1.17+ 1.059 x 10p5T-6.05 x 1 0-7 T2. temperature 25 0-4 15 K, while at temperatures below 25 0 K, E, is modeled to within 2 meV [l5]. Note that both Eqs. (2. 2a) and (2. 2b) are accurate for 25 0 < T < 300 K. Although Eq. (2. 2). by the Fermi-Dirac At 77K ni for silicon is -1 0 -2 0cm-3, while at 400K its value is -1 O'*~m-~ 20 flE)A 1.0. 0.5 0.0 2 Basic Semiconductor and pn Junction Theory probability