Generation-Recombination and Mobility

Chapter 20 Generation-Recombination and Mobility 20.1 Introduction This chapter illustrates the main contributions to the transitions of the inter-band type, that give rise to the generation-recombination terms in the continuity equations for electrons and holes, and to those of the intra-band type, that give rise to the electron and hole mobilities in the current-density equations. The inter-band transitions that are considered are the net thermal recombinations (of the direct and trap-assisted type), Auger recombinations, impact-ionization generations, and net-optical recombinations. The model for each type of event is first given as a closed-form function of the semiconductor-device model’s unknowns, like carrier concentrations, electric field, or current densities. Such functions contain a number of coefficients, whose derivation is successively worked out in the complements by means of a microscopic analysis. The case of semiconductors having a distribution of traps within the gap, like, e.g., polycrystalline silicon, is treated as well. Some discussion is devoted to the optical-generation and recombination events to show how the concepts of semiconductor laser, solar cell, and optical sensor may be derived as particular cases of nonequilibrium interaction between the material and an electromagnetic field. The intra-band transitions are treated in a similar manner: two examples, the collisions with acoustic phonons and ionized impurities, are worked out in some detail; the illustration then follows of how the contributions from different scattering mechanisms are combined together in the macroscopic mobility models. The material is supplemented with a brief discussion about advanced modeling methods. © Springer International Publishing AG 2018 M. Rudan, Physics of Semiconductor Devices, DOI 10.1007/978-3-319-63154-7_20 507 508 20 Generation-Recombination and Mobility 20.2 Net Thermal Recombinations As anticipated in Sect. 19.5.5, it is customary to separate the net generation rates Wn , Wp into two contributions, namely, those deriving from the phonon collisions and those of the other types (e.g., electron-electron collisions, electron-photon collisions, and so on). The separate contributions are defined in (19.132); this section deals with the net thermal recombination rates Un , Up . In the calculations carried out below, the nonequilibrium carrier concentrations are derived by integrating over the bands’ energy. This is consistent with the general definitions (19.31) and (19.109). In fact, considering the nonequilibrium electron concentration n as defined in (19.31), one introduces the variable transformation illustrated in Sect. B.5 and replaces the quantities appearing in it as follows: .u; v; w/ S .k1 ; k2 ; k3 / ; n; s  f D Q˚ ; .r; t/ ;  E; (20.1) ; sN P; (20.2) b where Q, .E/ are, respectively, the density of states in the phase space r; k and the combined density of states in energy and r space, while ˚.r; k; t/, P.r; E; t/ are the nonequilibrium occupation probabilities in the phase space and, respectively, in energy; the integration in energy is carried out over the range corresponding to the conduction band’s branch. The hole concentration is treated in the same manner. In conclusion, n.r; t/ D p.r; t/ D ZZZ ZZZ C1 3 1 Q˚ d k D C1 1 Q .1  ˚/ d3 k D Z Z ECU P dE ; (20.3) EC EV EVL .1  P/ dE : (20.4) 20.2.1 Direct Thermal Recombinations To begin, a graphic example of thermal transitions is shown in Fig. 20.1, where the edges of the conduction and valence bands are indicated with the same symbols used in Sect. 18.2; the transition marked with a is a recombination event, in which an electron belonging to an energy state of the conduction band transfers to an empty state of the valence band. The energy difference between the initial and final state is released to the lattice in the form of a phonon. The opposite transition, where the electron’s energy increases due to phonon absorption, is an electron-hole generation and is marked with b in the figure. The transitions of type a and b are 20.2 Net Thermal Recombinations Fig. 20.1 A graphic example of direct thermal recombination (a) and generation (b). The edges of the conduction and valence bands are indicated with the same symbols used in Sect. 18.2. The same drawing applies also to the description of the direct optical recombinations and generations (Sect. 20.4) 509 E CU EC a b EV E VL called direct thermal recombination and direct thermal generation, respectively. Let ra be the number of direct thermal recombination per unit volume and time, and rb the analogue for the generations; considering the conduction band as a reference, the difference ra  rb provides the contribution to the net thermal recombination rate Un due to the direct thermal transitions. When the valence band is considered instead, the rates of electrons transitions reverse; however, for the valence band the transitions of holes must be considered: as consequence, the contribution to Up is again ra  rb . In conclusion, UDT D UDTn D UDTp D ra  rb ; (20.5) where D stands for “direct” and T for “thermal.” The expressions of ra , rb are determined by a reasoning similar to that used in Sect. 19.3.1 to express the collision term of the BTE; here, however, the analysis is carried out directly in the energy space instead of the k space.1 Let P.r; E; t/ be the occupation probability of a state at energy E; then, let C be the probability per unit time and volume (in r) of an electron transition from a state of energy E to a state of energy E0 belonging to a different band, induced by the interaction with a phonon.2 Such a probability depends on the phonon energy „ ! (Sect. 12.5), and also on the position in r if the semiconductor is nonuniform. Typically, the equilibrium distribution is assumed for the phonons, which makes C independent of time; as the collisions are point-like 1 A more detailed example of calculations is given below, with reference to collisions with ionized impurities. 2 The units of C are ŒC D m 3 s 1 . Remembering that the phonon energy equals the change in energy of the electron due to the transition (Sect. 14.8.2), it is C D 0 for „ ! < EC EV D EG (refer also to Fig. 20.1). 510 20 Generation-Recombination and Mobility (Sect. 19.3.2), the spatial positions of the initial and final states coincide, whence C D C.r; „ !; E ! E0 /. Indicating with g.E/ the density of states of the band where the initial state belongs, the product g dE P is the number of electrons within the elementary interval dE around the initial state; such a product is multiplied by C to find the number of unconditional E ! E0 transitions per unit time and volume. On the other hand, the transitions take place only if the final states around E0 are empty; as the empty states in that interval are g0 dE0 .1  P0 /, the number of actual transitions per unit time and volume from dE to dE0 turns out to be g dE P C g0 dE0 .1  P0 /. Now, to calculate the ra or rb rate it is necessary to add up all transitions: for ra one lets E range over the conduction band and E0 over the valence band; the converse is done for rb . As the calculation of the latter is somewhat easier, it is shown first: rb D Z EV Z g dE P EVL ECU EC   C g0 dE0 1  P0 : (20.6) As in normal operating conditions the majority of the valence-band states are filled, while the majority of the conduction-band states are empty, one lets P ' 1 and 1  P0 ' 1, whence, using symbol GDT for rb , GDT .r; „ !/ D Z EV EVL g dE Z ECU C g0 dE0 : (20.7) EC Thus, the generation rate is independent of the carrier concentrations. To proceed, one uses the relation g D  , with the combined density of states in energy and volume, given by (15.65), and the definition (20.4) of the hole concentration. Thus, the recombination rate is found to be ra D Z ECU g dE P EC Z EV EVL C g0 dE0 .1  P0 / D p Z ECU K g P dE ; (20.8) EC where K.r; „ !; E/, whose units are ŒK D s 1 , is the average of  C over the valence band, weighed by g0 .1  P0 /: KD R EV  C g0 .1  P0 / dE0 : R EV 0 0 0 EVL g .1  P / dE EVL (20.9) Strictly speaking, K is a functional of P0 ; however, the presence of P0 in both numerator and denominator of (20.9) makes such a dependence smoother, so that one can approximate K using the equilibrium distribution instead of P0 . By the same token one uses the definition of the electron concentration (20.3) to find 20.2 Net Thermal Recombinations ra D ˛DT n p ; 511 ˛DT .r; „!/ D R ECU  K g P dE ; R ECU EC g P dE EC (20.10) where the integrals are approximated using the equilibrium probability. In conclusion, UDT D ˛DT n p  GDT ; (20.11) where ˛DT is the transition coefficient of the direct thermal transitions, with units Œ˛DT  D m3 s 1 , and GDT their generation rate (ŒGDT  D m 3 s 1 ). As in the equilibrium case it is ra D rb , namely, GD D ˛D neq peq , it follows UDT D ˛DT .np neq peq /. 20.2.2 Trap-Assisted Thermal Recombinations An important contribution to the thermal generation and recombination phenomena is due to the so-called trap-assisted transitions. As mentioned in Sect. 19.3, among the possible collisions undergone by electrons or holes are those with lattice defects. The latter may originate from lattice irregularities (e.g., dislocations of the material’s atoms occurring during the fabrication process, Sect. 24.1), or from impurities that were not eliminated during the semiconductor’s purification process, or were inadvertently added during a fabrication step. Some defects may introduce energy states localized in the gap; such states, called traps, may capture an electron from the conduction band and release it towards the valence band, or vice versa. The phenomena are illustrated in Fig. 20.2, where four traps located in the energy gap are shown in order to distinguish among the different transition events, that are: a) capture of a conduction-band electron by a trap, b) release of a trapped electron towards the conduction band, c) release of a trapped electron towards the valence band (more suitably described as the capture of a valence-band hole by the trap), Fig. 20.2 Different types of trap-assisted transitions E CU E a Et E Fi E EC b c d EV E VL 512 20 Generation-Recombination and Mobility and d) capture of a valence-band electron from the valence band (more suitably described as the release of a hole towards the valence band). Each transition is accompanied by the absorption or emission of a phonon. Thus, transitions of type a and b contribute to the net thermal recombination Un of the conduction band, while those of type c and d contribute to the net thermal recombination Up of the valence band. Also, a sequence of two transitions, one of type a involving a given trap, followed by one of type c involving the same trap, produces an electronhole recombination and is therefore called trap-assisted thermal recombination; similarly, a sequence of two transitions, one of type d involving a given trap, followed by one of type b involving the same trap, produces an electron-hole generation and is therefore called trap-assisted thermal generation. To calculate the contribution of the trap-assisted transitions to Un and Up it is necessary to distinguish between two kinds of traps: those of donor type, that are electrically neutral when the electron is present in the trap and become positively charged when the electron is released, and those of acceptor type, that are electrically neutral when the electron is absent from the trap and become negatively charged when the electron is captured. In this respect, the traps are similar to the dopants’ atoms. Instead, a strong difference is made by the position of the traps’ energy within the gap. Consider, for instance, traps localized near the gap’s midpoint (the latter is indicated by the intrinsic Fermi level EFi in Fig. 20.2); the phonon energy necessary for the transition is about EG =2 in all cases, to be compared with the value EG necessary for a direct transition. On the other hand, the equilibriumphonon distribution (Sect. 16.6) is the Bose-Einstein statistics (15.55); it follows that the number dNph of phonons in the interval d! is dNph D gph .!/ d! ; expŒ„ !=.kB T/  1 (20.12) with „ ! the energy and gph the density of states of the phonons. Due to (20.12), dNph =d! rapidly decreases as the phonon energy increases, thus making the probability of an electron-phonon interaction much larger at lower energies. For this reason, even in an electronic-grade semiconductor, where the concentration of defects is very small (Sect. 19.3.2), the traps are able to act as a sort of “preferred path” in energy for the inter-band transitions, to the extent that the contribution to Un , Up of the trap-assisted transitions is largely dominant over that of the direct transitions. Therefore, in the continuity equations (20.13) below, and in the subsequent derivation of the trap-assisted, thermal-transition rates, symbols Un , Up refer only to the latter transitions, not any more to the sum of the trap-assisted and direct ones. The net thermal-recombination terms Un , Up appear in (19.129) and (19.130) after replacing Wn , Wp with (19.132); this yields @n 1 C Un  divJn D Gn ; @t q @p 1 C Up C divJp D Gp : @t q (20.13) 20.2 Net Thermal Recombinations 513 To introduce the trap-assisted transitions one formally duplicates (20.13) as if the acceptor and donor traps formed two additional bands; as the acceptor traps are either neutral or negatively charged, the charge and current densities of the band associated with them are thought of as due to electrons; instead, the charge and current densities of the band associated with the donor traps are thought of as due to holes. In summary, the two additional equations read @nA 1 C UnA  divJnA D GnA ; @t q @pD 1 C UpD C divJpD D GpD ; @t q (20.14) with a and d standing for “acceptor” and “donor,” respectively. To ease the calculation it is assumed that the nonthermal phenomena are absent, whence Gn D Gp D GnA D GpD D 0. Combining (20.13) with (20.14) and observing that J D Jp C JpD C Jn C JnA is the total current density of the semiconductor yield @Œq .p C pD  n  nA / C divJ D q .Un C UnA /  q .Up C UpD / : @t (20.15) As the net dopant concentration N is independent of time, it is @Œq .p C pD  n  nA /=@t D @Œq .p C pD  n  nA C N/=@t D @%=@t; thus, the left-hand side of (20.15) vanishes due to (4.23), and3 Un C UnA D Up C UpD : (20.16) The two continuity equations (20.14) are now simplified by observing that in crystalline semiconductors the current densities JpD , JnA of the traps are negligible. In fact, the trap concentration is so low that inter-trap tunneling is precluded by the large distance from a trap to another; the reasoning is the same as that used in Sect. 18.7.2 with respect to the impurity levels.4 Letting JpD D JnA D 0 makes the two equations (20.14) local: @nA D UnA ; @t @pD D UpD : @t (20.17) In steady-state conditions the traps’ populations are constant, thus yielding UnA D UpD D 0 and, from (20.16), Un D Up . In equilibrium all continuity equations reduce to the identity 0 D 0, whence the net-recombination terms vanish eq eq eq eq independently, Un D UnA D Up D UpD D 0. 3 The result expressed by (20.16) is intuitive if one thinks that adding up all continuity equations amounts to counting all transitions twice, the first time in the forward direction (e.g., using the electrons), the second time in the backward direction (using the holes). The reasoning is similar to that leading to the vanishing of the intra-band contribution in (19.63). 4 In a polycrystalline semiconductor with a large spatial concentration of traps it may happen that the traps’ current densities are not negligible; in fact, the whole system of equations (20.13) and (20.14) must be used to correctly model the material [26–28]. The conduction phenomenon associated with these current densities is called gap conduction. 514 20 Generation-Recombination and Mobility 20.2.3 Shockley-Read-Hall Theory The Shockley-Read-Hall theory describes the trap-assisted, net thermal-recombination term in a crystalline semiconductor based upon the steady-state relation Un D Up . In fact, the outcome of the theory is used also in dynamic conditions; this approximation is acceptable because, due to the smallness of the traps’ concentration, the contribution of the charge density stored within the traps is negligible with respect to that of the band and dopant states; the contribution of the time variation of the traps’ charge density is similarly negligible. The theory also assumes that only one trap level is present, of energy Et ; with reference to Fig. 20.2, the trap levels must be thought of as being aligned with each other. If more than one trap level is present, the contributions of the individual levels are summed up at a later stage. In the theory it is not important to distinguish between acceptor-type or donor-type traps; however, one must account for the fact that a trap can accommodate one electron at most. Still with reference to Fig. 20.2, let ra be the number of type-a transitions per unit volume and time, and similarly for rb , rc , rd . The derivation of these rates is similar to that of the direct transitions and is shown in the complements; here the expressions of the net thermal-recombination terms are given, that read Un D ra  rb D ˛n n Nt .1  Pt /  en Nt Pt ; (20.18) Up D rc  rd D ˛p p Nt Pt  ep Nt .1  Pt / ; (20.19) where Nt is the concentration of traps of energy Et , Pt the trap-occupation probability, ˛n , ˛p the electron- and hole-transition coefficients, respectively, and en , ep the electron- and hole-emission coefficients, respectively.5 The ratios en =˛n , ep =˛p are assumed to vary little from the equilibrium to the nonequilibrium case. eq eq From Un D Up D 0 one derives en D neq ˛n   1  1 ; eq Pt ep D peq ˛p   1  1 eq Pt 1 : (20.20) The occupation probability at equilibrium is the modified Fermi-Dirac statistics (compare with (18.21) or (18.36)) 5     Et  EF 1 C1 exp dt kB T 1   1 Et  EF 1 ; exp  1 D eq dt kB T Pt (20.21) with dt the degeneracy coefficient of the trap. It follows, after introducing the shorthand notation nB D en =˛n , pB D ep =˛p , eq Pt D It is Œ˛n;p  D m3 s 1 , Œen;p  D s 1 . ; 20.2 Net Thermal Recombinations 515   Et  EF neq ; exp nB D dt kB T  EF  Et pB D p dt exp kB T eq  : (20.22) Note that nB pB D neq peq . Replacing (20.22) into (20.18), (20.19) and letting Un D Up yield ˛n n .1  Pt /  ˛n nB Pt D ˛p p Pt  ˛p pB .1  Pt /; (20.23) whence Pt D ˛n n C ˛p pB ; ˛n .n C nB / C ˛p .p C pB / 1  Pt D ˛n nB C ˛p p : ˛n .n C nB / C ˛p .p C pB / (20.24) In this way one expresses the trap-occupation probability as a function of two of the unknowns of the semiconductor-device model, namely, n and p, and of a few parameters. Among the latter, nB and pB are known (given the trap’s energy) because they are calculated in the equilibrium condition. In conclusion, replacing (20.24) into (20.18) or (20.19) yields, for the common value USRH D Un D Up , USRH D n p  neq peq ; .n C nB /=.Nt ˛p / C .p C pB /=.Nt ˛n / (20.25) where the indices stand for “Shockley-Read-Hall.” Eventually, the only unknown parameters turn out to be the products Nt ˛p and Nt ˛n which, as shown in Sect. 25.2, can be obtained from measurements. The expression obtained so far, (20.25), has been derived considering a single trap level Et . Before adding up over the levels it is convenient to consider how sensitive USRH is to variations of Et ; in fact, one notes that the numerator of (20.25) is independent of Et , whereas the denominator D has the form D D c C 2  cosh  ; D 1 Et  EF C log  ; kB T 2 (20.26) where 1 cD Nt  p n C ˛p ˛n  ; 1 D Nt s neq peq ; ˛p ˛n D 1 neq =˛p : dt2 peq =˛n (20.27) The denominator has a minimum where  D 0; thus, USRH has a maximum there. Moreover, the maximum is rather sharp due to the form of the hyperbolic cosine. It follows that the trap level EtM that most efficiently induces the trap-assisted transitions is found by letting  D 0. The other trap levels have a much smaller efficiency and can be neglected; in conclusion, it is not necessary to add up over the trap levels.6 With this provision, one finds 6 This simplification is not applicable in a polycrystalline or amorphous semiconductor. 516 20 Generation-Recombination and Mobility EtM   eq kB T 2 p =˛n : D EF C log dt eq 2 n =˛p (20.28) An estimate of EtM is easily obtained by considering the nondegenerate condition, whence neq D NC expŒ.EF  EC /=.kB T/ and peq D NV expŒ.EV  EF /=.kB T/ (compare with (18.28)). It follows EtM '   NV ˛p kB T EC C EV C log dt2 : 2 2 NC ˛n (20.29) Observing that the second term at the right-hand side of (20.29) is small, this result shows that the most efficient trap level is near the gap’s midpoint which, in turn, is near the intrinsic Fermi level EFi . In fact, combining (20.29) with (18.16) yields EtM ' EFi C   ˛p kB T log dt2 ' EFi : 2 ˛n (20.30) Defining the lifetimes p0 D 1 ; Nt ˛p n0 D 1 ; Nt ˛n (20.31) gives (20.25) the standard form USRH D n p  neq peq ; p0 .n C nB / C n0 .p C pB / (20.32) which is also called Shockley-Read-Hall recombination function. In equilibrium it is eq USRH D 0; in a nonequilibrium condition, a positive value of USRH , corresponding to an excess of the n p product with respect to the equilibrium product neq peq , indicates that recombinations prevail over generations, and vice versa. In a nonequilibrium condition it may happen that USRH D 0; this occurs at the boundary between a region where recombinations prevail and another region where generations prevail. In a nondegenerate semiconductor (20.22) become, letting Et D EtM D EFi and using (18.12), nB D ni ; dt pB D dt ni ; (20.33) whence nB pB D n2i . This result is useful also in a degenerate semiconductor for discussing possible simplifications in the form of USRH . 20.2 Net Thermal Recombinations 517 Limiting Cases of the Shockley-Read-Hall Theory The operating conditions of semiconductor devices are often such that the SRH recombination function (20.32) can be reduced to simpler forms. The first case is the so-called full-depletion condition, where both electron and hole concentrations are negligibly small with respect to nB and pB . Remembering that neq peq D nB pB one finds p r r nB pB nB nB nB pB USRH '  D ; g D p0 C n0 : p0 nB C n0 pB g pB pB (20.34) In a nondegenerate condition n whence , p take the simplified form (20.33), B B p nB =pB D ni and g D p0 =dt C n0 dt . In a full-depletion condition USRH is always negative, namely, generations prevail over recombinations; for this reason, g is called generation lifetime. The second limiting case of interest is the so-called weak-injection condition. This condition occurs when both inequalities below are fulfilled: jn  neq j  ceq ; jp  peq j  ceq ; (20.35) where ceq is the equilibrium concentration of the majority carriers in the spatial position under consideration. From the above definition it follows that the concept of weak injection is applicable only after specifying which carriers are the majority ones. Expanding the product n p to first order in n and p around the equilibrium value yields n p ' neq peq C neq .p  peq / C peq .n  neq /. As a consequence, the numerator of (20.32) becomes n p  neq peq ' neq .p  peq / C peq .n  neq / : (20.36) To proceed, it is necessary to distinguish between the n-type and p-type regions. Weak-Injection Condition, n-Type Semiconductor The weak-injection condition (20.35) reads jn  neq j  neq , jp  peq j  neq . As a consequence, one lets n ' neq in the denominator of (20.32) and neglects nB with respect to neq ; in fact, in a nondegenerate condition it is nB ' ni  neq , and the same inequality is also applicable in a degenerate condition. As the lifetimes are similar to each other, the term n0 .p C pB / in the denominator is negligible with respect to p0 neq , because p is a concentration of minority carriers and pB is similar to nB . In conclusion, the denominator of (20.32) simplifies to p0 neq , whence USRH ' n  neq p  peq C eq eq : p0 .n =p / p0 (20.37) 518 20 Generation-Recombination and Mobility The second term at the right-hand side of (20.37) is negligible7 because neq =peq  1; letting p D p0 finally yields USRH ' p  peq ; p (20.38) with p the minority-carrier lifetime in an n-doped region. Weak-Injection Condition, p-Type Semiconductor The weak-injection condition (20.35) reads jn  neq j  peq , jp  peq j  peq . As a consequence, one lets p ' peq in the denominator of (20.32) and neglects pB with respect to peq ; the other term in the denominator of (20.35) is neglected as above, thus simplifying the denominator to n0 peq . In conclusion, USRH ' p  peq .peq =neq / n0 C n  neq : n0 (20.39) The first term at the right-hand side of (20.39) is negligible because peq =neq  1; letting n D n0 finally yields USRH ' n  neq ; n (20.40) with n the minority-carrier lifetime in a p-doped region. The simplified expressions of USRH found here are particularly useful; in fact, in contrast to (20.32), the weak-injection limits (20.38) and (20.40) are linear with respect to p or n. Moreover, as (20.38) and (20.40) depend on one unknown only, they decouple the continuity equation of the minority carriers (the first one in (19.129) or in (19.130)) from the other equations of the semiconductor’s model; thanks to this it is possible to separate the system of equations. The simplification introduced by the full-depletion condition is even stronger, because (20.34) is independent of the model’s unknowns. On the other hand, all simplifications illustrated here are applicable only in the regions where the approximations hold; once the simplified model’s equations have been solved locally, it is necessary to match the solutions at the boundaries between adjacent regions. Considering for instance the example in Sect. 18.4.1, one has neq ' 1015 cm whence neq =peq ' 1010 . 7 3 , peq ' 105 cm 3 , 20.2 Net Thermal Recombinations 519 20.2.4 Thermal Recombination with Tail and Deep States In recent years, thin-film transistors made of amorphous silicon (˛-Si:H TFTs) or polycrystalline silicon (poly-TFTs) have acquired great importance in microelectronics; in fact, they can be fabricated with a low thermal budget and on large-area substrates and, at the same time, they are able to achieve performances adequate for the realization of complex circuits. Typical applications are in the area of solidstate image sensors, active-matrix liquid-crystal displays, charge-coupled devices, and static random-access memories. These materials are characterized by a large amount of defects, giving rise to localized states with a complex energy distribution within the gap; typically, the concentration of the acceptor-like states is larger in the upper half of the gap, while that of the donor-like states is larger in the lower half of the gap. Traps in amorphous silicon are due to the irregular distribution of atoms and to defects in the material; the lack of long-range order in the atomic structure produces a distribution of localized states with energies near the conduction- and valence-band edges (tail states); in turn, the defects give rise to a distribution of states localized near midgap (deep states). As for the spatial localization, traps in amorphous silicon are uniformly distributed in the semiconductor’s volume, while defects in polycrystalline silicon are located at the grain boundaries; for the latter material a simplifying hypothesis is used, that consists in describing the traps as uniformly distributed over the volume [51]. Given these premises, for both materials the energy distribution of traps in the gap can be modeled as the superposition of two distributions of acceptor and donor states; in turn, the densities of states per unit volume, D and A , for each group of donor- and acceptor-like states is approximated as the sum of two exponential functions, describing the deep and tail states, respectively [25, 103, 137]: D .E/ D A .E/ D TD TA exp  EV  E TD   E  EC exp TA C  C DD DA exp  EV  E DD  E  EC exp DA   ; (20.41) (20.42) (compare with 18.20, 18.35). In (20.41), (20.42), TD ; : : : and TD ; : : : are constants, with suffixes TD, DD standing for “Tail-Donor,” “Deep-Donor,” respectively, and the like for TA, DA. When the number of energy states in the gap is large, distinguishing between bands and gap seems meaningless; the distinction, however, is kept as long as the gap states, although dense in energy, are still much less dense in space than those of the bands, so that the contribution of the gap states to current transport is negligible (in other terms, the mobility of the carriers belonging to the gap states is much smaller than that of the band carriers). This condition is assumed here; thus, proceeding in the same manner as in Sect. 20.2.3 yields (compare with (20.18) and (20.19)) 520 20 Generation-Recombination and Mobility Un D Up D Z Z EC EV f A Œn ˛nA .1  PA /  enA PA  C D Œn ˛nD .1  PD /  enD PD g dE ; D (20.43)   p ˛pD PD  epD .1  PD / dE ; EC EV ˚ A   p ˛pA PA  epA .1  PA / C (20.44) where n, p are the concentration of electrons in the conduction band and that of holes in the valence band, (20.3) and (20.4) respectively. The meaning of the other symbols in (20.43), (20.44) is as follows: n ˛nA D Z ECU EC .0/ SCA .E0 ; E; r/ C .E 0 / PC .E0 ; r; t/ dE0 ; (20.45) with C , PC the density of states per unit volume and the nonequilibrium occupation probability of the conduction band; in turn, letting SCA .E0 ; r0 ; E; r/ be the unconditional probability per unit time of a transition from a conduction-band state of energy E0 and position r0 to an acceptor trap of energy E and position r, it is .0/ SCA D SCA ı.r0  r/. The Dirac symbol in the definition indicates that the change in position during the transition is negligible (compare with 19.40). Symbol n ˛nD has a similar meaning. Similarly, p ˛pA D Z EV EVL .0/ SAV .E; E0 ; r/ V .E 0   / 1  PV .E0 ; r; t/ dE0 ; (20.46) with V , PV the density of states per unit volume and the nonequilibrium occupation .0/ probability of the valence band; in turn it is SAV D SAV ı.r0  r/, where SAV is the unconditional probability per unit time of a transition from an acceptor trap of energy E and position r to a valence-band state of energy E0 and position r0 . Finally, the emission probability per unit time of an electron, from an acceptor state of energy E to the conduction band, is given by enA D ECU Z EC .0/ SAC .E; E0 ; r0 / C .E 0 / dE0 ; (20.47) and the like for enD ; in turn, the emission probability per unit time of a hole, from an acceptor state of energy E to the valence band, is given by epA D Z EV EVL .0/ SVA .E0 ; E ; r/ V .E 0 / dE0 ; (20.48) and the like for epD . The concentrations of electrons belonging to the acceptor traps and of holes belonging to the donor traps are, respectively (compare with (20.3) and (20.4)), 20.3 Auger Recombination and Impact Ionization nA .r; t/ D pD .r; t/ D Z EC Z A .E 0 ; r/ PA .E0 ; r; t/ dE0 ; 521 (20.49) EV EC D .E EV 0   ; r/ 1  PD .E0 ; r; t/ dE0 : (20.50) Observing that (20.14), (20.15), (20.16), and (20.17) hold also for the distribution of traps considered here, and replacing (20.17) into (20.16), one finds Up  @nA @pD D Un  : @t @t (20.51) The expressions of Up and Un are given by (20.43), (20.44), while those of @pD =@t and @nA =@t are obtained from (20.49), (20.50); introducing such expressions into (20.51) and letting DA D ˛nA n C enA C ˛pA p C epA ; DD D ˛nD n C enD C ˛pD p C epD ; (20.52) yield Z EC EV f A Œ.˛nA n C epA /  DA PA  PP A  C D Œ.˛nD n C epD /  DD PD  PP D g dE D 0 ; (20.53) with PP A D @PA =@t, PP D D @PD =@t. As equality (20.53) holds for any distributions of states, it follows @PA C DA PA D ˛nA n C epA ; @t @PD C DD PD D ˛nD n C epD : @t (20.54) The expressions (20.52) of the denominators DA , DD generalize that of the denominator in (20.24), which refers to the case of a single trap level; in turn, the steady-state form of (20.54) generalizes the first expression in (20.24). In conclusion, the continuity equations for a semiconductor having a distribution of traps within the gap, neglecting gap conduction, are (20.13) and (20.54), supplemented with the definitions (20.43), (20.44) of the net recombination rates. 20.3 Auger Recombination and Impact Ionization An important, nonthermal recombination mechanism is Auger recombination. The phenomenon is due to the electron-electron or hole-hole collision and is illustrated in Fig. 20.3. With reference to case a, two electrons whose initial state is the conduction band collide and exchange energy. The outcome of the collision is that one of the electrons suffers an energy loss equal or larger than the energy gap and 522 20 Generation-Recombination and Mobility Fig. 20.3 Auger recombinations initiated by electrons (a) and holes (c) a c E CU EC E Fi EV E VL makes a transition to an empty state of the valence band; the other electron absorbs the same amount of energy and makes a transition to a higher-energy state of the conduction band. The phenomenon is also indicated as an Auger recombination initiated by electrons. The analogue for holes is shown in case c of Fig. 20.3: two holes whose initial state is in the valence band collide and exchange energy. Remembering that hole energy increases in the opposite direction with respect to that of electrons (Sect. 19.2.3), the hole that suffers an energy loss equal or larger than the energy gap makes a transition to a filled state of the conduction band; the other hole absorbs the same amount of energy and makes a transition to a higher-energy state of the valence band. The phenomenon is indicated as an Auger recombination initiated by holes. The phenomenon dual to Auger recombination is illustrated in Fig. 20.4 and is called impact ionization. With reference to case b, an electron whose initial state is in the conduction band at high energy collides and exchanges energy with an electron whose initial state is in the valence band. The initial energy E of the electron in the conduction band is such that E  EC is equal or larger than the energy gap, whereas the initial energy of the electron in the valence band is near EV . The outcome of the collision is that although the high-energy electron suffers an energy loss equal or larger than the energy gap, its final state is still in the conduction band; the other electron absorbs the same amount of energy and makes a transition to the conduction band. The phenomenon is in fact an electron-hole pair generation and is also indicated as an impact-ionization event initiated by electrons. The analogue for holes is shown in case d of Fig. 20.4: a hole whose initial state is in the valence band at high energy collides and exchanges energy with a hole whose initial state is in the conduction band. The initial energy E of the hole in the valence band is such that jE  EV j is equal or larger than the energy gap, whereas the initial energy of the hole in the conduction band is near EC . The outcome of the collision is that although the high-energy hole suffers an energy loss equal or larger than the energy gap, its final state is still in the valence band; the other hole absorbs the same amount of energy 20.3 Auger Recombination and Impact Ionization Fig. 20.4 Impact-ionization transitions initiated by electrons (b) and holes (d) 523 b d E CU EC E Fi EV E VL and makes a transition to the valence band. The phenomenon is in fact an electronhole pair generation and is also indicated as an impact-ionization event initiated by holes. The derivation of the Auger and impact-ionization rates is shown in the complements; here the expressions of the net recombinations due to the Auger and impact-ionization events are given, that read UnAI D ra  rb D cn n2 p  In n ; UpAI D rc  rd D cp p2 n  Ip p ; (20.55) where UnAI refers to the electron-initiated transitions and UpAI to the hole-initiated ones. As usual, ra indicates the number of transitions of type a per unit time and volume; the same holds for rb , rc , and rd . In (20.55), cn , In are the transition coefficients for the Auger recombination and impact ionization initiated by electrons, and cp , Ip the analogue for holes; cn , cp are also called Auger coefficients.8 In equilibrium it is UnAI D UpAI D 0, whence In D cn neq peq , Ip D cp neq peq . The above holds also in a nonequilibrium case as long as the operating conditions are not too far from equilibrium; with these premises it follows UnAI D cn n .n p  neq peq / ; UpAI D cp p .np  neq peq / ; (20.56) When the operating condition departs strongly from equilibrium, the simplification leading to (20.56) is no longer applicable and the general expressions (20.55) must be used. Referring to all recombinations as due to transitions of electrons, their rate is easily found to be ra C rc ; similarly, the total generation rate is rb C rd . In conclusion, the net recombination rate due to the Auger and impact-ionization phenomena is given by 8 The units are Œcn;p  D cm6 s 1 and ŒIn;p  D s 1 . 524 20 Generation-Recombination and Mobility UAI D UnAI C UpAI : (20.57) For Auger recombination to occur it is necessary that an electron collides with another electron, or a hole collides with another hole. The probability of such an event is relatively small because in normal operating conditions and at room temperature there is a high probability that a carrier collides with a phonon; as a consequence, for the collisionless motion of an electron to be interrupted by a collision with another electron it is necessary that the electron concentration be very high. This situation occurs only in a heavily doped, n-type region; similarly, an Auger recombination initiated by holes can be significant only in a heavily doped, p-type region.9 Considering now the case of impact-ionization, for this phenomenon to occur it is necessary that an electron, or a hole, acquires a kinetic energy larger than the energy gap. This is a rare event as well,10 because in general the carrier undergoes a phonon collision when its kinetic energy is still significantly lower than the energy gap. The impact-ionization event occurs only if the carrier acquires a substantial energy over a distance much shorter than the average collisionless path, which happens only in presence of a strong electric field.11 The qualitative reasoning outlined above explains why the conditions for a strong Auger recombination are incompatible with those that make impact-ionization dominant; in fact, a large charge density, like that imposed by a heavy dopant concentration, prevents the electric field from becoming strong. Vice versa, a strong electric field prevents a large carrier concentration from building up. It is therefore sensible to investigate situations where only one term dominates within UAI . 20.3.1 Strong Impact Ionization As indicated in Sect. 20.3, far from equilibrium the approximations In D cn neq peq , Ip D cp neq peq are not valid, and the general expressions (20.55) must be used. Here the situation where impact ionization dominates over the other generationrecombination mechanisms is considered, using the steady-state case. If impact ionization is dominant, it is Un  Gn D Up  Gp ' UAI ' In n  Ip p. The continuity equations (the first ones in (19.129) and (19.130)) then become 9 In fact, Auger recombination becomes significant in the source and drain regions of MOSFETs and in the emitter regions of BJTs, where the dopant concentration is the highest. 10 In principle, high-energy electrons or hole exists also in the equilibrium condition; however, their number is negligible because of the exponentially vanishing tail of the Fermi-Dirac statistics. 11 The high-field conditions able to produce a significant impact ionization typically occur in the reverse-biased p-n junctions like, e.g., the drain junction in MOSFETs and the collector junction in BJTs. 20.4 Optical Transitions 525 divJn D q In n  q Ip p ; divJp D q In n C q Ip p : (20.58) As outlined in Sect. 20.3, impact-ionization dominates if the electric field is high. For this reason, the transport equations in (19.129) and (19.130) are simplified by keeping the ohmic term only, to yield Jn ' q n n E and Jp ' q p p E. As a consequence, the electron and hole current densities are parallel to the electric field. Let e.r/ be the unit vector of the electric field, oriented in the direction of increasing field, E D jEj e; it follows Jn D Jn e and Jp D Jp e, with Jn and Jp strictly positive. Extracting n, p from the above and replacing them into (20.58) yield  divJn D kn Jn C kp Jp ; divJp D kn Jn C kp Jp ; (20.59) where the ratios kn D In ; n jEj kp D Ip ; p jEj (20.60) whose units are Œkn;p  D m 1 , are the impact-ionization coefficients for electrons and holes, respectively. Equations (20.59) form a system of differential equations of the first order, whose solution in the one-dimensional case is relatively simple if the dependence of the coefficients on position is given (Sect. 21.5). 20.4 Optical Transitions The description of the optical transitions is similar to that of the direct thermal transitions given in Sect. 20.2.1; still with reference to Fig. 20.1, the transition marked with a can be thought of as an optical-recombination event if the energy difference between the initial and final state is released to the environment in the form of a photon. The opposite transition (b), where the electron’s energy increases due to photon absorption from the environment, is an optical electron-hole generation. The expression of the net optical-recombination rate is similar to (20.11) and reads UO D ˛O n p  GO ; (20.61) whose coefficients are derived in the same manner as those of UDT (Sect. 20.2.1). In normal operating conditions the similarity between the direct-thermal and optical generation-recombination events extends also to the external agent that induces the transitions. In fact, the distribution of the phonon energies is typically the equilibrium one, given by the Bose-Einstein statistics (15.55) at the lattice temperature; as for the photons, the environment radiation in which the semiconductor is immersed can also be assimilated to the equilibrium one, again given by the BoseEinstein statistics at the same temperature. 526 20 Generation-Recombination and Mobility The conditions of the optical generation-recombination events drastically change if the device is kept far from equilibrium. Consider for instance the case where the electron concentration of the conduction band is artificially increased with respect to the equilibrium value at the expense of the electron population of the valence band, so that both n and p in (20.61) increase. This brings about an excess of recombinations; if the probability of radiative-type generation-recombination events is high,12 the emission of a large number of photons follows. The angular frequencies of the emitted photons is close to .EC  EV /=„, because the majority of the electrons in the conduction band concentrate near EC , and the final states of the radiative transitions concentrate near EV . In this way, the energy spent to keep the artificially high concentration of electron-hole pairs is transformed into that of a nearly monochromatic optical emission. In essence, this is the description of the operating principle of a laser.13 Another method for keeping the device far from equilibrium is that of artificially decreasing both the concentration of electrons of the conduction band and the concentration of holes of the valence band. The outcome is opposite with respect to that described earlier: the decrease of both n and p in (20.61) brings about an excess of generations, which in turn corresponds to the absorption of photons from the environment. The absorption may be exploited to accumulate energy (thus leading to the concept of solar cell), or to provide an electrical signal whose amplitude depends on the number of absorbed photons (thus leading to the concept of optical sensor). In a nonequilibrium condition the amount of energy exchanged between the semiconductor and the electromagnetic field is not necessarily uniform in space. Consider, by way of example, the case of an optical sensor on which an external radiation impinges; as the nonequilibrium conditions are such that the absorption events prevail, the radiation intensity within the material progressively decreases at increasing distances from the sensor’s surface. Therefore, it is important to determine the radiation intensity as a function of position. It is acceptable to assume that the absorption events are uncorrelated from each other. Thus, one can limit the analysis to a monochromatic radiation; the effect of the whole spectrum is recovered at a later stage by adding up over the frequencies. When absorption prevails, (20.61) simplifies to UO ' GO , where GO is a function of the radiation’s frequency  and possibly of position. If the radiation’s intensity varies with time, GO depends on time as well.14 When the radiation interacts with the external surface of the material, part of the energy is reflected; moreover, the radiation is refracted at the boundary, so that the propagation direction outside the material differs in general from that inside. Letting  be the propagation direction inside the material, consider an elementary volume with a side d aligned with  12 As indicated in Sect. 17.6.6, among semiconductors this is typical of the direct-gap ones. In fact, LASER is the acronym of Light Amplification by Stimulated Emission of Radiation. 14 In principle, a time dependence of the intensity is incompatible with the hypothesis that the radiation is monochromatic. However, the frequency with which the intensity may vary is extremely small with respect to the optical frequencies. 13 20.4 Optical Transitions 527 Fig. 20.5 Sketch of photon absorption in a material layer Φ ( ξ) dξ Φ ( ξ+ d ξ) dA and a cross-section dA normal to it (Fig. 20.5). The monochromatic radiation can be described as a flux of photons of equal energy h , with h the Planck constant, and a momentum’s direction parallel to . Let ˚./ be the flux density of photons entering the volume from the face corresponding to , and ˚. C d/ the flux density leaving it at  C d; the following holds, ˚ D K uf , where K./ is the concentration of the photons and uf their constant phase velocity. Then, @˚ @K @K D D : @ @.=uf / @t (20.62) The derivatives in (20.62) are negative because the photon concentration decreases in time due to absorption; as the loss of each photon corresponds to the loss of an energy quantum h , the loss of electromagnetic energy per unit volume and time is h  .@˚=@/. By a similar token one finds15 that the energy absorbed by the optical-generation events per unit time and volume is h  GO . The latter is not necessarily equal to h  .@˚=@/; in fact, some photons crossing the elementary volume may be lost due to collisions with nuclei (this, however, is a rare event), or with electrons that are already in the conduction band, so that no electron-hole pair generation occurs. To account for these events one lets GO D  @˚ > 0; @ (20.63) with 0 <  < 1 the quantum efficiency. In moderately doped semiconductors  is close to unity because the concentration of the conduction-band electrons is small; instead, the efficiency degrades in degenerate semiconductors. The spatial dependence of the generation term can be derived from (20.63) if that of the photon flux is known. To proceed, one defines the absorption coefficient as 15 It is implied that h   EC EV , and that two-particle collisions only are to be considered. 528 20 Generation-Recombination and Mobility kD 1 @˚ > 0; ˚ @ (20.64) with Œk D m 1 . In general it is k D k.˚; ; /; however, as the absorption effects are uncorrelated, the flux density lost per unit path d is proportional to the flux density available at . Then, k is independent of ˚; neglecting momentarily the dependence on  as well, one finds ˚./ D ˚B expŒk./  ; (20.65) with ˚B D ˚. D 0C / on account of the fact that due to the reflection at the interface, the flux density on the inside edge of the boundary is different from that on the outside edge. When k is independent of position, its inverse 1=k is called average penetration length of the radiation. When k depends on position, (20.64) is still separable and yields ˚./ D ˚B exp.km / ; km D 1  Z  0 k. 0 I / d 0 : (20.66) Combining (20.66) with (20.63), the optical-generation term is found to be " Z GO D  ˚B k.; / exp   0 k. ; / d 0 0 # : (20.67) 20.5 Macroscopic Mobility Models It has been shown in Sect. 19.5.2 that the carrier mobilities are defined in terms of the momentum-relaxation times. Specifically, in the parabolic-band approximation it is, for the electrons of the conduction band, n D .l C 2 t /=3, with l D q p =ml , t D q p =mt , where p is the electron momentum-relaxation time (19.87); similarly, for the holes of the valence band the carrier mobility is given by inserting (19.118) into the second relation of (19.121), namely, a linear combination of the heavy-hole and light-hole momentum-relaxation times. As, in turn, the inverse momentumrelaxation time is a suitable average of the inverse intra-band relaxation time, the Matthiessen rule follows (Sect. 19.6.5); in conclusion, the electron and hole mobilities are calculated by combining the effects of the different types of collisions (e.g., phonons, impurities, and so on) suffered by the carrier.16 In the case of electrons, the application of the Matthiessen rule is straightforward, leading to 16 As mentioned in Sect. 19.6.5, it is assumed that the different types of collisions are uncorrelated. 20.5 Macroscopic Mobility Models 1 mn D n q 529 1 ph p C 1 imp p C ::: ! ; (20.68) where the index refers to the type of collision, and 1=mn D .1=ml C 2=mt /=3. For holes a little more algebra is necessary, which can be avoided if the approximation ph ' pl is applicable. In the typical operating conditions of semiconductor devices the most important types of collisions are those with phonons and ionized impurities. For devices like surface-channel MOSFETs, where the flow lines of the current density are near the interface between semiconductor and gate insulator, a third type is also very important, namely, the collisions with the interface. The macroscopic mobility models are closed-form expressions in which mobility is related to a set of macroscopic parameters (e.g., temperature) and to some of the unknowns of the semiconductor-device model; the concept is similar to that leading to the expressions of the generation-recombination terms shown in earlier sections. 20.5.1 Example of Phonon Collision By way of example, a simplified analysis of the contribution to mobility of the electron-phonon collision is outlined below, starting from the definition of the ith component of the momentum-relaxation tensor pi given by (19.87); the simplifications are such that the first-order expansion f  f eq ' .df =d/eq  is not used here. Starting from the perturbative form (19.47) one considers the steadystate, uniform case and lets B D 0,  D v , to find f  f eq q : E  gradk f D „ v (20.69) Replacing f with f eq at the left-hand side of (20.69) and using the definition (17.52) of the group velocity yield gradk f eq D .df eq =dH/ „ u, with H the Hamiltonian function defined in Sect. 19.2.2. Inserting into (19.87) yields pi ZZZ C1 1 ui E  u .df eq =dH/ d3 k D ZZZ C1 1 ui E  u .df eq =dH/ v d3 k : (20.70) As the derivative df eq =dH is even with respect to k, the integrals involving velocity components different from ui vanish because the corresponding integrand is odd; as a consequence, only the ith component of the electric field remains, and cancels out. A further simplification is obtained by replacing the Fermi-Dirac statistics, with the Maxwell-Boltzmann distribution law, f eq ' Q expŒ.Ee C q '  EC C EF /=.kB T/, to find 530 20 Generation-Recombination and Mobility pi ZZZ C1 1 u2i expŒEe =.kB T/ d3 k D ZZZ C1 1 u2i expŒEe =.kB T/ v d3 k : (20.71) To proceed it is necessary to make an assumption about v . Remembering the definition of the relaxation time given by the first relation in (19.43), it is reasonable to assume that the scattering probability S0 increases with the kinetic energy Ee of the electron, so that the relaxation time decreases; a somewhat stronger hypothesis is that the relaxation time depends on Ee only, namely, the collision is isotropic.17 In this case, (20.71) is readily manipulated by a Herring-Vogt transformation. Following the same procedure as in Sect. 19.6.4, one finds that all numerical factors cancel out; as a consequence, one may replace the auxiliary coordinate 2i with 2 =3 D Ee =3, thus showing that pi D p is isotropic as well. One eventually finds p D R C1 3=2 v .Ee / Ee 0 R C1 0 3=2 Ee expŒEe =.kB T/ dEe expŒEe =.kB T/ dEe : (20.72) A simple approximation for the relaxation time is v D v0 .Ee =E0 / ˛ , where v0 , E0 , and ˛ are positive parameters independent of Ee (compare with Sect. 19.6.7). From (C.95) it follows p D v0 .5=2  ˛/ .5=2/  E0 kB T ˛ : (20.73) ph When the electron-phonon interaction is considered, v0 D v0 is found to be inversely proportional to kB T and to the concentration Nsc of semiconductor’s atoms; moreover, for acoustic phonons18 it is ˛ D 1=2 [78, Sects. 61,62], whence 4 pap D v0 .Nsc ; T/ p 3   E0 kB T 1=2 ; 1 ap n / Nsc .kB T/ 3=2 ; (20.74) where “ap” stands for “acoustic phonon.” More elaborate derivations, including also the contribution of optical phonons, still show that carrier-phonon collisions make mobility to decrease when temperature increases. 17 The first-principle derivation of the scattering probabilities is carried out by applying Fermi’s Golden Rule (Sect. 14.8.3) to each type of perturbation, using the Bloch functions for the unperturbed states [73]. Examples are given in Sect. 14.8.6 for the case of the harmonic perturbation in a periodic structure, and in this chapter (Sect. 20.5.2) for the case of ionizedimpurity scattering. 18 Acoustic phonons are those whose momentum and energy belong to the acoustic branch of the lattice-dispersion relation (Sect. 17.9.5); a similar definition applies to optical phonons (Sect. 17.9.6). 20.5 Macroscopic Mobility Models 531 20.5.2 Example of Ionized-Impurity Collision As a second example one considers the collisions with ionized impurities. The interaction with a single ionized impurity is a perturbation of the Coulomb type; due to the presence of the crystal, the more suitable approach is the screened Coulomb perturbation, an example of which is shown in Sect. 14.7, leading to the perturbation-matrix element (14.34): .0/ hkg D A=.2 /3 ; q2c C q2 AD  Z e2 : "0 (20.75) In (20.75), e > 0 is the elementary electric charge, Z a positive integer, "0 the vacuum permittivity, qc > 0 the inverse screening length,19 q D jqj D jk  gj and, finally,  D 1 .1/ in the repulsive (attractive) case. The wave vectors k and g correspond to the initial and final state of the transition, respectively. In principle, (20.75) should not be used as is because it holds in vacuo; in fact, the eigenfunctions of the unperturbed Hamiltonian operator used to derive (20.75) are plane waves. Inside a crystal, instead, one should define the perturbation matrix hkg .t/ using the Bloch functions wk D uk exp.i k  r/ in an integral of the form (14.24). However, it can be shown that the contribution of the periodic part uk can suitably be averaged and extracted from the integral, in the form of a dimensionless coefficient, whose square modulus G is called overlap factor.20 For this reason, the collisions with ionized impurities is treated starting from the definition (20.75) to calculate the perturbation matrix, with the provision that the result is to be multiplied by G and the permittivity "sc of the semiconductor replaces "0 in the second relation of (20.75). Like in Sect. 14.6, a Gaussian wave packet (14.27) centered on some wave vector b ¤ g is used as initial condition. In this case the perturbation is independent of .0/ time, hbg D hbg D const ¤ 0; as a consequence, the infinitesimal probability dPb that such a perturbation induces a transition, from the initial condition (14.27), to a final state whose energy belongs to the range dEg , is given by (14.32). In turn, the .0/ integral (14.31) providing Hb .Eg / is calculated in Prob. 14.1. Assuming that the duration tP of the interaction is large enough to make Fermi’s Golden Rule (14.44) applicable, and inserting the overlap factor, one finally obtains dPb  G 19  2 m „2 3=2 p 8  tP ı.Eb  Eg / A2 Eg dEg : 3 5 2 2 2  „ .2 / qc .qc C 8 m Eg =„ / (20.76) An example of derivation of the screening length is given in Sect. 20.6.4. An example of this procedure is given in Sect. 14.8.6 with reference to the case where the spatial part of the perturbation has the form of a plane wave. 20 532 20 Generation-Recombination and Mobility where the relation Eg D „2 g2 =.2 m/ has been used. Integrating over Eg and dividing by tP provides the probability per unit time of a transition from the initial energy Eb to any final energy; letting Ec D „2 q2c =.2 m/, one finds p 4 Eb =Ec 1 P b/ D P.E : vc 1 C 4 Eb =Ec p 1 G A2 = 2  m D : vc 8  2 .2 Ec /3=2 (20.77) The above expression provides the contribution to the intra-band relaxation time of the scattering due to a single impurity. One notes that since A is squared, the effect onto (20.77) of a positive impurity is the same as that of a negative one. If the effect of each impurity is uncorrelated with that of the others,21 the probabilities add up; letting NI D NDC C NA be the total concentration of ionized impurities, the product NI d3 r is the total number of ionized impurities in the elementary volume d3 r; it follows that the probability per unit time and volume is given P b / NI . Considering that NI depends on position only, mobility inherits the by P.E inverse proportionality with NI ; letting “ii” indicate “ionized impurity,” one finds iin / 1=NI . The derivation of the dependence on NI shown above is in fact oversimplified, and the resulting model does not reproduce the experimental results with sufficient precision. One of the reasons for this discrepancy is that the inverse screening length qc depends on the dopant concentration as well, as is apparent, for instance, from the second relation of (20.105). In order to improve the model, while still keeping an analytical form, the expression is modified by letting 1=iin / NI˛ ; with ˛ a dimensionless parameter to be extracted from the comparison with experiments. One then lets  ˛ NI 1 1 D ; (20.78) iin .NI / iin .NR / NR with NR a reference concentration. 20.5.3 Bulk and Surface Mobilities Combining the phonon and ionized-impurity contributions using the Matthiessen ph rule yields 1=Bn .T; NI / D 1=n .T/ C 1=iin .NI /, namely, ph Bn .T; NI / D n .T/ ; 1 C c.T/ .NI =NR /˛ (20.79) 21 In silicon, this assumption is fulfilled for values of the concentration up to about 1019 cm [80, 106]. 3 20.5 Macroscopic Mobility Models 533 b = 1.00 b = 1.50 b = 3.00 0.8 ph µn ( T, NI ) / µn ( T ) 1 0.6 B 0.4 0.2 0 -6 -4 -2 0 2 4 6 r = log 10 (NI / NR ) Fig. 20.6 Graph of the theoretical mobility curve (20.80), normalized to its maximum, for different values of b, with b0 D 0. Each curve has a flex at r D rflex D b0 =b and takes the value 0:5 there. The slope at the flex is b=4 ph with c.T/ D n .T/=iin .NR /. In practical cases the doping concentration ranges over many orders of magnitude; for this reason, (20.79) is usually represented in a semilogarithmic scale: letting r D log10 .NI =NR /, b D ˛ loge 10, and b0 D loge c, (20.79) becomes ap Bn .T; NI / D n .T/ : 1 C exp.b r C b0 / (20.80) The curves corresponding to b D 1; 1:5; 3 and b0 D 0 are drawn in Fig. 20.6, using r as independent variable at a fixed T. Index “B” in the mobility defined in (20.79) or (20.80) stands for “bulk.” More generally, the term bulk mobility is ascribed to the combination of all contributions to mobility different from surface collisions. As mentioned at the beginning of this section, in surface-channel devices the degradation of mobility produced by the interaction of the carriers with the interface between channel and gate insulator is also very important. The macroscopic models of this effect are built up by considering that the carrier-surface interaction is more likely to occur if the flow lines of the current density are closer to the interface itself; such a closeness is in turn controlled by the intensity of the electric field’s component normal to the interface, E? . In conclusion, the model describes the contribution to mobility due to surface scattering as a decreasing function of E? , e.g., for electrons, 1 1 D s sn .E? / n .ER /  E? ER ˇ ; (20.81) 534 20 Generation-Recombination and Mobility with ER a reference field and ˇ a dimensionless parameter to be extracted from experiments. Combining the bulk and surface contributions using the Matthiessen rule yields 1=n .T; NI ; E? / D 1=Bn .T; NI / C 1=sn .E? /, namely, n .T; NI ; E? / D Bn .T; NI / ; 1 C d.T; NI / .E? =ER /ˇ (20.82) with d.T; NI / D Bn .T; NI /=sn .ER /. 20.5.4 Beyond Analytical Modeling of Mobility In general the analytical approaches outlined above do not attain the precision necessary for applications to realistic devices. For this reason, one must often resort to numerical-simulation methods; in this way, the main scattering mechanisms are incorporated into the analysis (e.g., for silicon: acoustic phonons, optical phonons, ionized impurities, and impact ionization), along with the full-band structure of the semiconductor, which is included in the simulation through the density of states and group velocity defined in the energy space. The latter, in turn, are obtained directly from the corresponding functions in the momentum space by integrating the fullband system over the angles. The energy range considered to date allows for the description of carrier dynamics up to 5 eV. As mentioned above, the ionized-impurity collisions can be treated as interactions between the carrier and a single impurity as long as the impurity concentration is below some limit. When the limit is exceeded, impurity clustering becomes relevant and must be accounted for [80]. In fact, at high doping densities the carrier scatters with a cluster of K ions, where K is a function of the impurity concentration. Finally, different outcomes are found for majority- or minoritymobility calculations: e.g., minority-hole mobility is found to be about a factor 2 higher than the majority-hole mobility for identical doping levels. Figures 20.7 and 20.8 show the outcome of electron- and hole-mobility calculations for bulk silicon, obtained from the spherical-harmonics method illustrated in [140]. The method incorporates the models for the scattering mechanisms listed above. The electron and hole mobility have been calculated as a function of the total ionized-dopant concentration NI , using the lattice temperature T as a parameter; in the figures, they are compared with measurements taken from the literature. To include the surface effects in the analysis it is necessary to account for the fact that in modern devices the thickness of the charge layer at the interface with the gate insulator is so small that quantum confinement and formation of subbands must be considered. The typical collisions mechanisms to be accounted for at the semiconductor–insulator interface are surface roughness, scattering with ionized impurities trapped at the interface, and surface phonons. Figures 20.9 and 20.10 show the outcome of electron and hole surface-mobility calculations in silicon, 20.5 Macroscopic Mobility Models 2500 T = 250 K HARM Lombardi Klaassen Arora 2000 µ n (cm2 / V sec) Fig. 20.7 Electron mobility in silicon calculated with the spherical-harmonics expansion method (HARM) as a function of the total ionized-dopant concentration NI , using the lattice temperature T as parameter. The calculations are compared with measurements by Lombardi [91], Klaassen [80], and Arora [2] (courtesy of S. Reggiani) 535 T = 300 K 1500 1000 T = 400 K 500 T = 500 K 0 1014 1016 1018 10 20 NI (cm-3 ) 1000 µ n (cm2 / V sec) Fig. 20.8 Hole mobility in silicon calculated with the spherical-harmonics expansion method (HARM) as a function of the total ionized-dopant concentration NI , using the lattice temperature T as parameter. The calculations are compared with measurements by Lombardi [91], Klaassen [80], and Arora [2] (courtesy of S. Reggiani) HARM Lombardi Klaassen Arora T = 200 K T = 250 K 500 T = 300 K T = 400 K 0 1014 1016 1018 10 20 NI (cm-3 ) also obtained from the spherical-harmonics method [106]. The electron and hole mobility have been calculated as functions of the dopant concentration (NA and ND , respectively), at room temperature; in the figures, they are compared with measurements taken from the literature. 536 1000 Effective electron mobility (cm 2 V -1 sec -1) Fig. 20.9 Electron surface mobility in silicon calculated with the spherical-harmonics expansion method (HARM) method at room temperature, using the acceptor concentration NA as parameter. The calculations are compared with measurements by Takagi [131] (courtesy of S. Reggiani) 20 Generation-Recombination and Mobility NA = 3.9 1015 cm-3 NA = 2.0 1016 cm-3 NA = 7.2 1016 cm-3 NA = 3.0 1017 cm-3 100 NA = 7.7 1017 cm-3 NA = 2.4 1018 cm-3 HARM Fig. 20.10 Hole surface mobility in silicon calculated with the spherical-harmonics expansion method (HARM) at room temperature, using the donor concentration ND as parameter. The calculations are compared with measurements by Takagi [131] (courtesy of S. Reggiani) Effective hole mobility (cm 2 V -1 sec -1) 1 0 Effective electric field (MV cm-1) ND = 7.8 10 100 15 ND = 1.6 10 ND = 5.1 10 ND = 2.7 10 ND = 6.6 10 HARM 16 16 17 17 cm -3 cm cm cm cm -3 -3 -3 -3 0.10 1.00 -1 Effective Electric Field (MV cm ) 20.6 Complements 20.6.1 Transition Rates in the SRH Recombination Function The expressions of the transition rates ra , rb , rc , rd to be used in the calculation of the Shockley-Read-Hall recombination function (20.32) are determined by the same reasoning as that used in Sect. 20.2.1 for the direct thermal transitions. Let P.r; E; t/ be the occupation probability of a state at energy E, and C.E ! E0 / the probability per unit time and volume (in r) of a transition from a filled state of energy E to an empty state of energy E0 . Such a probability is independent of time; it depends on 20.6 Complements 537 the energy of the phonon involved in the transition, and possibly on position. Then, define P0 D P.r; E D E0 ; t/, Pt D P.r; E D Et ; t/, where Et is the energy of the trap. Finally, let .E/ be the combined density of states in energy and volume of the bands, and t .r; E/ the same quantity for the traps (the latter depends on position if the traps’ distribution is nonuniform). The number of transitions per unit volume and time, from states in the interval dE belonging to a band, to states in the interval dE0 belonging to the trap distribution, is obtained as the product of the number  .E/ dE P of filled states in the interval dE, times the transition probability per unit volume and time C, times the number  t .r; E0 / dE0 .1  P0 / of empty states in the interval dE0 . Thus, letting Et be an energy interval belonging to the gap and containing the traps, the transition rate from the conduction band to the traps is given by Z ra D ECU Z EC Et  .E/ dE P C.E ! E0 /  t .r; E0 / dE0 .1  P0 / : (20.83) By the same token, the transition rate from the valence band to the traps is Z rd D EV Z EVL Et  .E/ dE P C.E ! E0 /  t .r; E0 / dE0 .1  P0 / : (20.84) In turn, the number of transitions per unit volume and time, from states in the interval dE0 belonging the trap distribution, to states in the interval dE belonging to a band, is obtained as the product of the number  t .r; E0 / dE0 P0 of filled states in the interval dE0 , times C.r; E0 ! E/, times the number  .E/ dE .1  P/ of empty states in the interval dE. Thus, the transition rates from the traps to conduction or valence band are respectively given by ECU rb D Z Z EV rc D EC EVL Z  t .r; E0 / dE0 P0 C.r; E0 ! E/  .E/ dE .1  P/ ; (20.85) Z  t .r; E0 / dE0 P0 C.r; E0 ! E/  .E/ dE .1  P/ : (20.86) Et Et The combined density of states of the traps is treated in the same manner as that of the dopant atoms (compare with (18.20) and (18.35)) by letting t .r; E 0 / D Nt .r/ ı.E0  Et / ; (20.87) where Nt .r/ is the trap concentration. Thanks to this, the integrals over Et are easily evaluated to yield ra D Nt .1  Pt / 2 Z ECU EC P C.r; E ! Et / dE D Nt .1  Pt / ˛n n ; (20.88) 538 20 Generation-Recombination and Mobility EV Z rc D Nt Pt 2 .1  P/ C.r; Et ! E/ dE D Nt Pt ˛p p ; EVL (20.89) where the definitions (20.3), (20.4) of the electron and hole concentrations are used, and the transition coefficients for electrons and holes are defined as the weighed averages 2 ˛n D  R ECU EC R ECU EC P C dE 2 ; ˛p D  P dE R EV EVL R EV .1  P/ C dE .1  P/ dE EVL : (20.90) Like in the case of (20.10), the integrals in (20.90) are approximated using the equilibrium probability. The remaining transition rates rb , rd are determined in a similar manner, using also the approximation 1  P ' 1 in (20.85) and P ' 1 in (20.84). Like in Sect. 20.2.1, the approximation is justified by the fact that in normal operating conditions the majority of the valence-band states are filled, while the majority of the conduction-band states are empty. In conclusion, 2 rb D Nt Pt  Z ECU .1  P/ C.r; Et ! E/ dE ' Nt Pt en ; EC rd D Nt .1  Pt / 2 Z (20.91) EV EVL P C.r; E ! Et / dE ' Nt .1  Pt / ep ; (20.92) with the emission coefficients defined by en D 2 Z ECU C dE ; EC ep D 2 Z EV C dE : (20.93) EVL 20.6.2 Coefficients of the Auger and Impact-Ionization Events The expression of the coefficients cn , cp and In , Ip , to be used in the calculation of the net recombination rates (20.55) due to the Auger and impact-ionization phenomena, is found in the same way as the transition rates of the SRH recombination function (Sect. 20.2.3) or the direct thermal recombinations (Sect. 20.2.1). Let P.r; E; t/ be the occupation probability of a state of energy E, and Cn .E1 ; E2 ! E10 ; E20 / the combined probability per unit time and volume (in r) of an electron transition from a filled state of energy E1 in the conduction band to an empty state of energy E10 in the conduction band, and of another electron from a filled state of energy E2 to an empty state of energy E20 , where E2 and E20 belong to different bands. 20.6 Complements 539 Auger Coefficients In an Auger recombination it is E10 > E1 ; also, E2 belongs to the conduction band while E20 belongs to the valence band. Due to energy conservation it is22   Cn D Cn0 ı .E1  E10 / C .E2  E20 / ; (20.94) g1 dE1 P1 g2 dE2 P2 Cn g01 dE10 .1  P01 / g02 dE20 .1  P02 / ; (20.95) where E2  E20 ' EG ; it follows E10 ' E1 C EG . Then, define Pi D P.r; E D Ei ; t/, P0i D P.r; E D Ei0 ; t/, with i D 1; 2, and let .E/ be the combined density of states in energy and volume for the bands; in particular, let gi D  .Ei / and g0i D  .Ei0 /. From the above definitions one finds, for the rate ra of the Auger recombinations initiated by electrons, ra D Z R where indicates a fourfold integral that extends thrice over the conduction band and once over the valence band. Observing that P01  1 and integrating over E10 with Cn D Cn0 ı.E1 C EG  E10 / yield ra D Z ECU g1 dE1 P1 Cn0 gG EC Z ECU g2 dE2 P2 EC Z EV EVL g02 dE20 .1  P02 / ; (20.96) where gG D g.E1 C EG / and ŒCn0 gG  D s 1 m 3 . Thanks to (20.3) and (20.4), the second integral in (20.96) equals  n and the third one equals  p. Letting 3 cn D  R ECU EC Cn0 gG g1 P1 dE1 ; R ECU g P dE 1 1 1 EC (20.97) finally yields ra D cn n2 p. The derivation of rc D cp p2 n is similar. Impact Ionization’s Transition Coefficients Using the same symbols introduced at the beginning of Sect. 20.6.2, for an impactionization event induced by an electron it is E1 > E10 ; in turn, E2 belongs to the valence band and E20 belongs to the conduction band. It follows rb D 22 Z g1 dE1 P1 g2 dE2 P2 Cn g01 dE10 .1  P01 / g02 dE20 .1  P02 / ; The units of Cn0 are ŒCn0 D J s 1 m 3 . (20.98) 540 20 Generation-Recombination and Mobility where the fourfold integral extends thrice over the conduction band and once over the valence band. From the energy-conservation relation E1 C E2 D E10 C E20 and from E20  E2 ' EG it follows E10 ' E1  EG . Observing that P2 ' 1, P01  1, P02  1, and integrating over E10 with Cn D Cn0 ı.E1  EG  E10 / yield rb D Z ECU Cn0 gG g1 P1 dE1 EC Z EV g2 dE2 EVL Z ECU EC g02 dE20 ; (20.99) where gG D g.E1  EG /, and the product of the second and third integral is a dimensionless quantity that depends only on the semiconductor’s structure. Indicating such a quantity with n , and letting In D n R ECU EC Cn0 gG g1 P1 dE1 ; R ECU EC g1 P1 dE1 (20.100) finally yields rb D In n. The derivation of rd D Ip p is similar. 20.6.3 Total Recombination-Generation Rate The expressions for the most important generation-recombination terms have been worked out in this chapter. Only one of them, the SRH recombination function USRH , involves energy states different from those of the conduction and valence bands; in principle, such states would require additional continuity equations to be added to the semiconductor-device model. However, as discussed in Sect. 20.2.3, this is not necessary in crystalline semiconductors. The other mechanisms (direct thermal recombination-generation UDT , Auger recombination and impact ionization UAI , and optical recombination-generation UO ) do not involve intermediate states. As a consequence, with reference to (20.13) the generation-recombination terms of the electron-continuity equation are equal to those of the hole continuity equation. Finally, assuming that the different generation-recombination phenomena are uncorrelated and neglecting UDT with respect to USRH (Sect. 20.2.2) yield Un  Gn D Up  Gp ' USRH C UAI C UDO : (20.101) 20.6.4 Screened Coulomb Potential In the context of physics, the general meaning of screening is the attenuation in the electric field intensity due to the presence of mobile charges; the effect is treated here using the Debye-Hückel theory [39], which is applicable to a nondegenerate semiconductor where the dopants are completely ionized. For a 20.6 Complements 541 medium of permittivity ", with charge density %, the electric potential in the equilibrium condition is found by solving Poisson’s equation  " r 2' D % : (20.102) One starts by considering a locally neutral material, to which a perturbation is added due, for instance, to the introduction of a fixed charge Zc ec placed in the origin; this, in turn, induces a variation in %. The corresponding perturbation of ' is calculated to first order by replacing ' with ' C ı' and % with % C .@%=@'/ ı', where the derivative is calculated at ı' D 0; the perturbed form of Poisson’s equation reads:  " r 2 '  " r 2 ı' D % C @% ı' : @' (20.103) As the unperturbed terms cancel out due to (20.102), a Poisson equation in the perturbation is obtained, q2c D  r 2 ı' D q2c ı' ; @%=@' ; " (20.104) where 1=qc is the screening length or Debye length. The definition implies that @%=@' < 0; this is in fact true, as shown below with reference to a nondegenerate semiconductor with completely ionized dopants.23 Letting NDC D ND , NA D NA in (19.125), and using the nondegenerate expressions (18.60), (18.61), of the equilibrium concentrations, one finds that N D ND NA is left unaffected by the perturbation, while the electron concentration24 n transforms into n expŒe ı'=.kB T/ and the hole concentration p transforms into p expŒe ı'=.kB T/. From % D e .p  n C N/ one obtains, to first order, e2 @% D .n C p/ ; @' kB T q2c D e2 .n C p/ > 0: " kB T (20.105) The left-hand side of the Poisson equation in (20.104) is conveniently recast using a set of spherical coordinates r; ;  whose origin coincides with the center of symmetry of the perturbation; using (B.25) one finds r 2 @ 1 @2 r ı' D .r ı'/ C r @r2 sin  @ 2  @ı' sin  @  C r 2 @2 ı' : sin2  @ 2 (20.106) Considering a perturbation with a spherical symmetry, only the first term at the righthand side of (20.106) is left, whence (20.104) becomes an equation in the unknown r ı': 23 24 As shown by (A.118), the property @%=@' < 0 holds true also in the degenerate case. The electron charge is indicated here with e to avoid confusion with qc . 542 20 Generation-Recombination and Mobility d2 .r ı'/ D q2c .r ı'/ : dr2 (20.107) The general solution of (20.107) is r ı' D A1 exp.qc r/ C A2 exp.qc r/, where it must be set A2 D 0 to prevent the solution from diverging as r becomes large. In conclusion, ı' D A1 exp.qc r/ : r (20.108) The remaining constant is found by observing that for very small r the pure Coulomb case ı' ' A1 =r is recovered, whence A1 D Zc ec e=.4  "/. This makes (20.108) to coincide with (14.33).