Appendices .... .... A Transformation of Wi (ke, q) To be general, we assume an arbitrary angle between electric field vector and rotary axis of the energy ellipsoid. The coordinate system is chosen such that the z-axis (II-axis) coincides with the field direction (see Fig. A1). The energy surface is given then by the expression liZ Ec(k) = + 2 [(k- ko), m- I (k- ko)] (A1) with the effective mass tensor (A2) F Fig. A.1 Field vector F, valley vector 1<0, and coordinate system {ell, en, eJ..z}
33
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Appendices - Springer978-3-7091-6494-5/1.pdf · 326 Appendices We notice that ( ILII) I ( ILII) I I I-mil mll= l-mv mv=mll+mv' (B. 13) and therefore, the arguments of the Airy functions
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Appendices
.... .... A Transformation of Wi (ke, q)
To be general, we assume an arbitrary angle between electric field vector and rotary axis of the energy ellipsoid. The coordinate system is chosen such that the z-axis (II-axis) coincides with the field direction (see Fig. A1). The energy surface is given then by the expression
liZ Ec(k) = E~ + 2 [(k- ko), m-I (k- ko)] (A1)
with the effective mass tensor
(A2)
F
Fig. A.1 Field vector F, valley vector 1<0, and coordinate system {ell, en, eJ..z}
A Transformation of Wi (kO, q) 321
Using k- ko = (kll-koll)ell + (kJ..l -koJ..l)eJ..l + (k.L2 -kO.l2)e.l2, we find explicitly
j li2 [(k ll - kOIl)2 (kJ..l - kOJ..l)2 (k.l2 - kO.l2)2 Ec(k) = E g +-2 + +
New components ko..l1, ko.lz, and new masses m.l1. m.lZ have been defined by (A8) and (A9). The fixed vectors e..l1 and en are the eigenvectors of the tensor ~ -1 m.l
(A 10)
From Fig. Al it follows that ko..l1 = 0 and kon == ko.l, and consequently
(All)
Let the eigenvalues of the effective mass tensor in the main diagonal system of the rotary ellipsoid be mIl = m2"l == mt l and m31 = mil. We have to express the vectors ell, e..l1 and e.lZ within the main diagonal system {el, ez, e3} of the energy surface. We define the angle if between the rotary axis (e3) and the field vector (ell) if = ~ (ell, e3). Then kOIl = ko cos if and ko.l = ko sin if (see Fig. A2). The vector ell is given by spherical coordinates in the main diagonal system {el , ez, e3}
ell = (coscp sin if, sincp sin if, cos if) .
If we turn the system {el , ez, e3} so that cp = 0, we get (see Fig. A2)
ell = (sin if, 0, cos if)
e..l1 = (0, 1, 0)
e.lZ = (-cosif, 0, sinif).
(A 12)
(A 13)
B Evaluation of a Double Integral 323
All masses can be expressed by the longitudinal and transverse effective masses of the ellipsoid now:
1 ( A-I) mil = ell, m ell
.."L = (e.l1 , m- l e.l1) = ...1.. , m-11 mt
m- l = (e1.2, m- l e1.2) = ml cos2 7J + ...!.. sin2 7J , -12 t ml
m~ll = (ell, m-le.l1) = 0,
m~12 = (ell, m-len) = (~t - ~J cos7J sin7J .
Furthermore we find
Kl. = K (k1.2 - kOl.) ,
and
(mt -mz)cos7J sin7J K=
mz sin2 7J + mt cos2 7J
With these results the energy nwi (ku, q) of Eq. (3.47) takes the form
Q'f(k-L,q)= f dkll f dtexp{ -~f d~[2:11 (kll_*F~+qll)2 -00 -00 0
+ !v (k ll - *F~)2 +Ci J} . (B.4)
We carry out the ~-integration in the exponent, re-arrange the terms and introduce a new integration variable to obtain
00
x f drexp{-i[~8~(r+p)3+~8~r3+C!rJ} -00
00
x f dtexp{-i[~8~(t+p)3+~8~t3+C!tJ} -00
(B.5)
with 8~ = q2 F2 /(2mlln) and 13 = 1Uj1l/(qF). In order to end up with the integral representation of the Airy function we again have to transform the exponents
1 1 C'f -8\r +fJ)3 + _83r 3 +-1:.r 3 II 3 v n
1 ( 8 3) 3 ( C'f 8 3
) = _83 r+-II 13 + 8 3132 +-1:.-_" 8 3132 r 3 r,lI 8 3 II n 8 3 II
D Asymptotic Forms and Interpolation of Cylinder Functions 327
which is Eq. (4.8). Here we have used the Wronskian Uo V~ - VoU~ = ,J2/1f ([App.3], p. 687) and the definition of the normalized momenta K~ = ks,xA, KM = kM,xAmejf/mM' If the absolute square in (C.9) is evaluated, the mixed terms are reordered, and again the Wronskian of the parabolic cylinder functions is used, we end up with
D Asymptotic Forms and Interpolation of Cylinder Functions
The asymptotic formulas ofthe parabolic cylinder functions U~B = U( -K~, ~B)' V~B = V(-K~'~B)' U~B = U'(-K~'~B)' VIB = V'(-K~'~B) are given by ([App.3], p. 690)
(D.l)
(D.2)
(D.3)
(D.4)
where the upper factor holds in the case ~B > 2Ks and the lower in the case ~B < 2Ks, respectively. S is the action integral
(D.S)
328 Appendices
The prime denotes the derivative with respect to the second argument. The interpolating functions with the same asymptotic behavior read
· (2;rr)1/4 Ir(l +K2) 1/6 UAi = V 2 S (~I SI) Ai [(3S /2)2/3] (D.6) ~B 1 e2 11/4 2 ' ~-K2
· (2;rr)1/41~1_KiI1/4 3 -1/6 ViB Ai = (-lSI) Bi/[(3S/2)2/3]. (D.9)
Jr(~ +K~) 2
E Energy Limit for Gaussian Approximation
A Taylor expansion of the action S(~B) in the vicinity of 'f/ = CPB in the range 2KS > ~B yields
4 (~2 )3/2 S(~B) ~ 3~B : -K~ , (E.1)
hence, the function Y becomes
( 2 )2/3 (~2 ) Y(S) ~ ~B : -K~ (E.2)
there. The energy limit is given by the maximum of the Gaussian (4.32), i.e. by Y(S) = to = -Itol. This leads to
~~ (~B)2/3 Emax-Ec= 4+"2 Itol· (E.3)
Changing to the variable 'f/ (normalized energy measured from 'f/ F,M), one obtains
( ~B )2/3 'f/max = CPB +"2 'f/).. Itol ,
( )1/3 2/3 =CPB+ 'f/F,M-'f/c+CPB 'f/).. Itol·
(E.4)
(E.5)
F WKB Approximation for the Range 1/ > 1/max 329
F WKB Approximation for the Range 11 > 11max
The WKB form of the transmission probability, valid for energies much larger than the maximum of the barrier, is most easily obtained from Eq. (4.31) inserting the asymptotic representations of the Airy functions for large negative arguments ([App.3], p. 448):
Inserting into Eq. (4.31) immediately yields (4.37). The limes 'Tl -+ 00 ofT' turns out to be
(F.7)
which actually has to approach unity, since the effective masses tend to the free electron mass for 'Tl -+ 00. The latter effect was not taken into account in the model, consequently the limit (F.7) expresses quantum reflection at the boundary of two media with different effective masses. For the purpose of analytical integration the WKB form T'WKB has to be approximated in the vicinity of 'Tlmax. Therefore, we write T'WKB as
4 mM J1]-({JB rr'WKB( 0) mc (7I+7IF,M) .L 'Tl+'TlF,M, = 2 .
(1+ mC(7I:~F'M)J1] -({JB)
(F. 8)
An integrable approximation is obtained, if 1] is neglected compared to 'TlF,M (because 'Tlmax « 'TlF,M can be assumed) and 1] is replaced by 1]max in the denominator (because T'WKB is only important for the lowly doped contacts, where contributions to the current originate from a range of a few kB T above the top of the barrier only).
330 Appendices
G Probability of Resonant Tunneling
The transmission probability T,.es is determined by the component M22 of the transfer matrix M (e.g. [AppA])
To E x _ me,r(E) kl(E) 1 res( , ) - me,I(E) kr(E) I M 22(E,x)1 2 '
(G.1)
where M is composed as
M(E,x) = Mr(E)· Mt(E,x)· MI(E) (G.2)
with the product matrix Mt (E, x) = Mt,r (E, x)· Mt,l (E, x) containing the matching conditions at the trap potential walls at x ± rt, and the matrices Ml and Mr describing the matching at the gate-oxide and oxide-substrate interfaces, respectively. The component M22 can easily be evaluated from Eq. (G.2)
M22 = mi2m;1 (mil + m~2 m~l) + m~2m;2 (m~2 + m~l mb) , (G.3) m 2l m 22
The index s is either I or r, and ro = nl0o/(qF). The arguments of the Airy functions have the following explicit form:
Ai(~l) = Ai [I~~oO)] ,
Ai(~r) =Ai[I~~od)] ,
G Probability of Resonant Tunneling 331
Ai(t: )=Ai['E(E,x-rt)J st,1 lieo ' (G.7)
Ai(t: ) =Ai['E(E,x+rt)J St,r+ lieo '
Ai(t: )=Ai['E(E+Vr,x-rt)J st,l+ lieo '
Ai(t: ) =Ai['E(E+ Vt,x+rt)J St,r lieo ·
The arguments of the functions Ai', Bi, and Bi' were labeled in the same way. The quantity 'E (E , x) is given by Eq. (5.25) and Vt denotes the depth of the trap potential measured from the oxide conduction band edge. For the trap levels and field strengths considered here we have <l>t » lieo and also I <l>t - Vt I » lieo. Therefore, it follows for the arguments of the Airy functions at the resonance level Et(x) = <l>1-qFx - <l>t:
'E(Et(x),O)>> lieo ,
'E(Et(x),x±rt)>> lieo , (G.8)
'E(Et(x) + Vt,x ±rt)« -lieo .
That allows to use the respective asymptotic forms [App.3] at the gate-oxide interface and at the trap potential walls. Only at the oxide-substrate interface the full Airy functions have to be applied. For the matrix elements of MI we get
± sin (St,r - St,I+) (1~/I~/r+ 11/4 _I~/I+/;"r 11/4)J ' (G. 11) /;"l+~"r ~"I~"r+
it = =~}. e±(S/,r++S/,I) [cos (St,r - St,I+)
x (I ~"I/;"r 11/4 _I ~"l+~/,r+ 11/4) /;/ ,I+/;/ ,r+ ~"I~' ,r
± sin (St,r - St,I+) (I ~"I~"r+ 11/4 + II;',I+/;"r 11/4)J . ~',I+~"r ~,,/I;,,r+
(G. 12)
332 Appendices
The actions Sin (G.ll) and (G.12) can be developed with respect to the small potential drop q Frt across the trap radius
(G. 13)
with ~t,in = (~t,l+ + ~t,r) /2. Accordingly
rt 'E(E,x) rt ~ St + - St I ~ - 2- = -2- t:t t ,r, i:;r.::\ 'j ,ou ,
ro f£obo ro (G. 14)
with ~t,out = (~t,l + ~t,r+)/2. Developing the algebraic factors in (G.11) and (G.12) as well and neglecting the quadratic Stark effect gives
St,lSt,r ~ 1 + __ rt I
t: t: 11/4 1
~t,l+~t,r+ 2~t,out ro ' (G. 15)
st,l+st,r+ ~ 1-__ ...!... I
t: t: 11/4 1 r
~t,l~t,r 2~t,out ro ' (G.16)
1 11/4
~t,l~t,r+ ~ ~t,l+~t,r
(G. 17)
The diagonal elements of M t become
mil = e1"2J~t.out~ cos2 a ( ~,ou.t =f tana) ( 22 ~t,m
-~t,in ± t ) -- ana ~t,out
(G.18)
with a = J -~t ,in rtf r o. The bound state of the square-well potential is reproduced by the resonance condition
~t,out t --= ana -~t,in
(G.19)
and the symmetry relation m~2 ( - F) = m~l (F) ensures that the same level occurs if the polarity of the field is changed. The off-diagonal elements that determine the damping of the transmission probability, turn into
t 2} ±2St 2 [ 1 rt (~t,out + ~t,in) ± (~t,out - ~t,in) ] "'i2 = ·e cos a --- , 21 1/2 ~t,out r 0 ~t,in ~t,in
(G.20)
G Probability of Resonant Tunneling 333
which holds in the vicinity of the resonance energy. Inserting (GA), (G.9), and (G.lD) into Eq. (G.3) we obtain
1M 12 n 2S1 [ t 2 ( - 2 .+ 2) t 2 ( - 2 .+ 2) 22 = 4"e mll r Ai + 1 Ai +m21 r Bi + lBi
- 2milm~1 (rAirSi + I;;'jjti) ] n - 2S1 [ t 2 ( + 2 ._ 2) t 2 ( + 2 ._ 2) + 16 e m22 r Bi + 1 Bi + m12 r Ai + 1 Ai
- 2m~2mi2 (r"t/t + jSijAi) ] n [( _ t - t) (+ t + t) + 4" r AimU - r Bim21 r Bim22 - r Aim 12
+ (jtim~l - Itmil) (jsimk - jAimb)] , (G.21)
with the abbreviations
(G.22)
(G.23)
and the corresponding definitions for r~i and j~i' For not too small field strengths, i.e. as long as S[ » Sr can still be assumed, various terms in (G.22) are negligible. The remaining are
(G. 24)
The last but one term accounts for the shift of the resonance level, if the trap is located very close to the gate-oxide interface. The last term describes the respective shift for a trap situated very close to the oxide-substrate interface. These shifts are due to the delocalization of the wave function as one potential barrier becomes very thin. At the same time, the damping term for those traps strongly increases (second line in (G.24», which reduces the total transmission probability. Therefore, we skip the last two terms of (G.24).
334 Appendices
In order to obtain a Lorentzian for Tres, we linearize mil in the energy and evaluate mi2 and m~l at the resonance level. That gives
(G.26)
Inserting (G.25) and (G.26) into Eq. (G.24), 1/IM2212 takes Lorentzian form and can be transformed into a delta function, since the resonance is extremely sharp (see Fig. 5.13)
1 (G.27)
The prefactor follows from comparison with Eq. (G.24) and inserting (G.22) and (G.23):
We now tum the square-well potential into a delta potential V (x) = -J2fi2/m e,o(Vt - <l>t) 8(x - Xt). In this limit the transition rate does not depend on the potential parameters and can be directly compared to the capture/emission process, where a 3D delta potential was used. With Vt ~ 00 and rt ~ 0 in Eq. (G.28) the transmission probability for resonant tunneling takes the form
where Air =Ai(~r) etc., Ez == Et(x) - Ee,z and Er == Et(x) - Ee,r' Furthermore, we have introduced the WKB probabilities 'II and 'I'r (Eqs. (5.23) and (5.24» for tunneling into and out of the trap well, respectively. If the field strength is such that ~r » 1 holds, we can apply the WKB approximation also at the oxide-substrate interface
Ai; + lieo Ai~ --+ ~_1_ (1 + me,o [-qF(d -x) + <l>tl) e-2Sr , Er 4Jr ,Jf; me,r Er
(G.30)
B.2 + lieoB./2 1 1 (1+me,o [-qF(d-x) + <l>t]) 2Sr Ir -- Ir --+ - -- -- e . Er Jr ,Jf; me,r Er
(G.31)
Inserting into (G.29) we obtain the final form of 'I'res:
References [App.1] D. E. Aspnes. Electric-Field Effects on Optical Absorption. Phys. Rev., 147:554-561,
1966.
[App.2] D. E. Aspnes. Electric Field Effects on the Dielectric Constant of Solids. Phys. Rev., 153:972-982, 1967.
[App.3] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York, 1972.
[AppA] Y. Ando and T. Itoh. Calculation of Transmission Tunneling Current Across Arbitrary Potential Barriers. J. Appl. Phys., 61 (4):1497-1502,1987.
Subject Index
Absorption coefficient, 28 Absorption edge
ofSi, 15,28 ofSi02,288
Absorption measurements, 15, 21, 28 Action, 190,258,288,289,327,331 Activation energy
Auger, 78 electron capture, 209 electron emission, 308 field-reduced, 209, 240 impact ionization, 84, 87 ofleakage current, 283, 307, 309, 310 of trapped carriers in oxides, 308 shallow impurity, 18 single-charged center, 73
Activation law impact ionization, 84, 87 multiphonon transition, 212
Wigner-Boltzmann equation, 3 Wigner function, 3 Work function, 292
Yukawa potential, 22
List of Figures
1.1 Summary of semiclassical approaches to modeling of carrier and energy transport in semiconductors beyond drift-diffusion. . . . . . . . . . . . . . . . . . . . . . . . .. 12
1.2 Temperature dependence of the electron transverse effective mass and the hole DOS effective mass.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15
1.3 Silicon band gap vs temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17 1.4 Comparison of different theoretical BGN models based on calculations of the rigid
shifts of the band edges.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5 Comparison of different empirical BGN models derived at room temperature. 29 1.6 Effective intrinsic density at T = 300K as calculated from different models. 31 1.7 The ratio /LMatth.fI,dor ac- and imp-scattering vs temperature and doping. . . 37 1.8 Electron channel mobility as a function of average interface roughness.. . . . 38 1.9 Inversion-layer peak mobility /Le/f.max and bulk mobility vs substrate doping. 44 1.10 Doping dependence of the majority electron and hole mobilities in silicon. . . 50 1.11 Electron mobilities in As- and P-doped silicon. ................ 50 1.12 Minority electron and hole mobility vs acceptor and donor concentration, respectively. 52 1.13 The partial mobilities /Lac and /Lsr of the Lombardi et al. model vs normal field. . .. 61 1.14 Electron mobility vs doping and normal field according to the models of Hiroki et al.
and Lombardi et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 1.15 Drift velocity saturation at 300K as calculated from different heuristic models. . .. 66 1.16 Temperature dependence of the electron and hole drift velocities. . . . . . . . . . .. 67 1.17 Partial mobility resulting from electron-hole scattering according to the Conwell-
Weisskopf and Brooks-Herring theories. . . . . . . . . . . . . . . . . . . . . . . .. 69 1.18 Comparison of the eh-mobility models for n = p . . . . . . . . . . . . . . . . . .. 69 1.19 Surface recombination velocity of the Si-Si02 interface vs surface doping concentration. 77 1.20 Concentration dependence of carrier lifetimes as reported by different authors. . .. 81 1.21 Electron and hole ionization rates vs field strength as reported by different authors .. 97 1.22 Electron and hole ionization rates vs field strength. .................. 100 1.23 2D doping prQfile and cut along the indicated line through the critical region of the
gated diode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 1.24 2D field profile and cut along the indicated line through the critical region of the gated
diode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 1.25 2D distribution of the SRH rate in a small spot below the gate comer.. . 107 1.26 2D distribution of the impact ionization rate. . . . . . . . . . . . . . . . 107 1.27 Profile of the impact ionization rate along a vertical cut across its peak. . 108 1.28 Source-to-substrate current vs source voltage. . . . . . . . . . . . . . . 108
346 List of Figures
2.1 Normalized distribution function f(E)/n of electrons in silicon ... . 2.2 Electron density of states Dn (E) in silicon. ............. . 2.3 Ratio of the scattering strengths of intravaUey lA-phonon scattering .. 2.4 Intervalley scattering strength vs carrier temperature. ........ . 2.5 Calculated mobility vs dopinf . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Mobility calculated with Iin,:;p Eq. (2.61) and the numerical integral Eq. (2.59). 2.7 Ratio of the scattering integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Comparison of the analytical approximation for J (q, ii) and the numerical integral. 2.9 Effect of dispersive screening on the mobility. . 2.10 Calculated total DOS for large donor densities. 2.11 Effect of perturbed DOS on the mobility ..... 2.12 Electron mobility vs lattice temperature. . . . . 2.13 Electron mobility vs carrier temperature at TL = 300K. 2.14 Average carrier temperature vs electric field from simulations with MC programs and
from the analytical relation (2.97). . . . . . . . . . . . . . 2.15 Electron mobility vs doping calculated with the ideal DOS ......... . 2.16 Hole density of states Dp(E) in silicon. . ................. . 2.17 Calculated hole density of states D p (E) for different band structure models. 2.18 Hole mobility vs lattice temperature .......... . 2.19 Average hole temperature vs electric field ........... . 2.20 Hole mobility vs doping calculated with the ideal DOS. . . . . 2.21 Saturation of the hole drift velocity for different doping levels 2.22 Hole drift velocity saturation calculated with the self-consistent Tp(F)-relation. 2.23 Simulated I (V)-characteristics of an nin-device according to different carrier
temperature dependent mobility models. . . . . . . . . . . . . . . . . . . . 2.24 Drain current vs gate voltage at 0.1 V drain voltage for a 0.5/1-m-MOSFET.
3.1 Indirect band-to-band tunneling in silicon .......... . 3.2 Tunneling length and band diagram. . . . . . . . . . . . . . 3.3 Spherical coordinates of the field vector in the [100]-system. 3.4 Calculated band-to-band tunneling rate in silicon .... . . 3.5 Band-to-band tunneling rate in silicon for different directions of the electric field 3.6 Band-to-band tunneling rate in silicon for three values of the effective hole mass. 3.7 Comparison of band-to-band tunneling rate and field-dependent SRH rate. 3.8 Change of the most probable transition path with electric field strength. 3.9 Lowering of the activation energy in a strong electric field. .. . . . . 3.10 Transition energy vs electric field. . .................. . 3.11 Field enhancement factor vs field strength in different approximations .. 3.12 Temperature dependence of the zero-field electron lifetime in the low-temperature
approximation. .................. . . . . . . . . . . . . . . . 3.13 Comparison of the different approximations for the thermal weight function .. 3.14 Electron lifetime vs electric field for different field orientations ........ . 3.15 Electron lifetime vs electric field in (l11)-direction for different temperatures. 3.16 Electron lifetime vs electric field in (l11)-direction for different Huang-Rhys factors. 3.17 Electron lifetime vs electric field in (111)-direction for different lattice relaxation
energies .................................... . 3.18 Distribution of the BBT rate beneath the gate oxide of a MOS-gated diode .. 3.19 BBT and DXf current-voltage characteristics of the gated diode. . .... . 3.20 Simulation of the 298K I (V)-characteristic of a silicon tunnel diode ... . 3.21 Reverse I(V)-characteristics for the individual generation-recombination processes
of l/1-m x l/1-m diodes. . . . . . . . . . . . 3.22 I (V)-characteristics of a steep pn-junction. . .................... .
3.23 Impact of the variation of ER on defect-assisted tunneling.. . . . . . . . . . . . 225 3.24 Band-to-band tunneling and defect-assisted tunneling at different temperatures. 226 3.25 Reverse-bias j (V)-curves of a p+n+ -diode in comparison with measured data. 229 3.26 Electron-hole pair generation by band-to-band tunneling. . . . . . . . . . . . . 230 3.27 Notation for all capture and emission processes via two coupled defect levels. 232 3.28 Tunnel-assisted electron and hole capture into sublevels of a two-defect system. 232 3.29 Direct tunneling into a shallow donor-like state. . . . . . . . . . . . . . . . . . 237 3.30 Net doping profile and electric field distribution of the n + p-junction used in the device
simulation ....................................... 242 3.31 Simulated and measured j (V)- and n(V)-characteristics of the diode in Fig. 3.30. . . 242 3.32 Simulated and measured j (V)- and n(V)-characteristics of the diode in Fig. 3.30
under changed conditions. ................... 243
4.1 Schematic band diagram of the metal-semiconductor interface 253 4.2 Comparison of the transmission probabilities using parabolic cylinder and WKB wave
functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 4.3 Approximation of the Airy function by a Gaussian. . . . . . . . . . . . . . . . . . . 261 4.4 Comparison of the transmission probabilities using parabolic cylinder functions and
the Gaussian approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 4.5 Calculated j (V)-characteristics of an Al/n-Si contact for various donor concentrations.266 4.6 Illustration of the energies <l>T and -ql/>n (XT) at the reverse-biased contact. . . . . . 268 4.7 Energy <l>T vs applied voltage for different doping levels ................ 269 4.8 Electron quasi-Fermi energy -ql/>n at the boundary XT vs applied voltage for different
doping levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 4.9 Comparison of the analytical model with a measured I (V)-characteristic of Ti/n-Si. . 272 4.10 Schematic band diagram of a whole device under non-equilibrium conditions. 272 4.11 nin structures for varying surface doping and j (V)-characteristics ........... 275 4.12 Schematic and doping of the MPS diode used for Fig. 4.13 and Fig. 4.14. . ..... 275 4.13 I (V)-characteristics of the unit cell shown in Fig. 4.12a for different peak values of
the n-doped region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 4.14 I (V)-characteristics of the unit cell shown in Fig. 4.12a for different n-contact models. 276 4.15 Static forward j (V)-characteristics of the MPS diode and the conventional pin-rectifier.277 4.16 Lateral distributions of electron and hole current in the MPS diode. . . . . 278 4.17 Switching performance of the MPS diode and the conventional pin-diode . . . . . . 278
5.1 Image-force effect on an idealized potential barrier due to an oxide of 1 nm thickness. 284 5.2 Calculated transmission probabilities for a MOS structure with 1 nm oxide thickness. 286 5.3 Illustration of the hypothetical bandstructure mismatch at the Si-Si02 interface ... 287 5.4 Calculated transmission probabilities for oxides of 0.5 nm and 1 nm thickness, respec-
tively. . ......................................... 290 5.5 Calculated transmission probabilities for a MOS structure with 5 nm oxide thickness. 290 5.6 I (V)-characteristics of an Al-Si02-Si(n) diode with 2.5 nm oxide thickness for dif
ferent temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 5.7 Simulated vs measured currents of MOS capacitors with different oxide thicknesses. 292 5.8 Simulated drain, gate, and source currents of an n-channel MOSFETwith tunnel gate
oxide ........................................... 293 5.9 Lateral distribution of the gate tunnel current along the interface of the n-channel
MOSFET of Fig. 5.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 5.10 Energy band diagram illustrating a) resonant tunneling and b) multiphonon-assisted
tunneling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 5.11 Transition rate R as function of trap position and different lattice relaxation energies. 298
5.13 Dependence of the resonance peak of a repulsive trap on the position within a 100 A thick oxide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
5.14 Resonant tunnel I (V)-curves for an MOS capacitor with 42A gate oxide. . ..... 302 5.15 Transition rate R as function of trap position and different trap depths <l>t. • ..... 304 5.16 Current density as function of trap depth <l>t for different temperatures and field
strengths. ................................. . . . . . . . 305 5.17 Structure of the interpoly dielectric in the measured devices. ............. 306 5.18 Current density vs voltage for FN tunneling, thermionic emission, and thermionic
field emission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . " 306 5.19 Drain-source current of ONO devices vs time with bake temperature as parameter.. 307 5.20 Current density vs oxide field: measurement and calculation ............ 309 5.21 Arrhenius plot of the current density vs temperature.. . . . . . . . . . . . . . . .. 309 5.22 Current density vs oxide field in the low-field range for a single oxide layer of 10 nm
A.l Field vector, valley vector, and coordinate system in modeling band-to-band tunneling. 320 A.2 Coordinate transformation in modeling band-to-band tunneling. ........... 322
List of Tables
1.1 Coefficients for the hole DOS mass. .. . . . . . . . . . . . . . . . . . . 14 1.2 Coefficients for the temperature dependent gap. . . . . . . . . . . . . . . 15 1.3 Parameters for the theoretical gap narrowing in Si after Jain and Roulston 23 1.4 Parameters of the mobility model by Arora et al. .. 47 1.5 Parameters of the mobility model by Dorkel/Leturcq .. 49 1.6 Parameters of the mobility model by Masetti et al. . . . . 51 1. 7 Parameters for the mobility model by Soppa/Wagemann 59 1.8 Parameters for the mobility model by Lombardi et al. 60 1.9 Parameters for the mobility model by Hiroki et at. . . . . 62 1.10 Parameters for the mobility model by Selberherr . . . . . 63 1.11 Parameters for the high-field mobility model by Scharfetter/Gummel and Thornber 65 1.12 Temperature dependence of the parameters in the Caughey/Thomas high-field mobil-
ity model as determined by Canali et al. . . . . . . . . . . . . . . . . . . . . . . .. 65 1.13 Auger coefficients in silicon at different temperatures as measured by Dziewior and
Schmid ......................................... 78 1.14 Calculated threshold energies for phononless impact ionization in different crystallo-
graphic directions after Anderson and Crowell . . . . . . . . . . . 91 1.15 Impact ionization data measured by Moll and van Overstraeten . . . . . 93 1.16 Impact ionization data measured by van Overstraeten and de Man ... 94 1.17 Impact ionization data obtained from Schottky contacts by Woods et al. 95 1.18 Impact ionization data after Grant .................... 96 1.19 Impact ionization parameters of the empirical model by Okuto and Crowell . 97 1.20 Impact ionization parameters of the modified Chynoweth's model by Lackner 99
3.1 Electron tunneling masses for different orientations of the electric field. . . . 218
Springer Engineering
Arokia Nathan, Henry Baltes
Microtransducer CAD
Physical and Computational Aspects
1998. With figures. Approx. 300 pages.
Cloth approx. DM 180,-, oS 1260,
ISBN 3-211-83103-7
Computational Microelectronics
Due June 1998
Computer-aided-design (CAD) of semiconductor microtransducers is relatively new in contrast to their counterparts in the integrated circuit world. Integrated silicon microtransducers are realized using microfabrication techniques similar to those for standard integrated circuits (ICs). Unlike IC devices, however, microtransducers must interact with their environment, so their numerical simulation is considerably more complex. While the design of ICs aims at suppressing "parasitic" effects, microtransducers thrive on optimizing the one or the other such effect. The challenging quest for physical models and simulation tools enabling microtransducer CAD is the topic of this book. The book is intended as a text for graduate students in Electrical Engineering and Physics and as a reference for CAD engineers in the microsystems industry.