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 + +

mil ml m2

+ 2 (kJ..l - kO.lI)(k.l2 - kO.l2) + 2 (k ll - kOIl)(kJ..l - koJ..l)

m12 mill

(kll - kOIl )(k.l2 - k0.l2) ] +2 .

m211 (A3)

In deriving (A3) mi/ = mjl has been used. The different masses follow from their definitions

1 ( A -I ) -= ell,m ell ' mil

1 ( A I ) -- = e.lI,.l2, m- e.l1 . .l2 , ml.2

1 (AI) 1 ( AI) -- = ell, m- e.l1 • .l2 , - = e.ll, m- e.l2 . mlll.2 m12

(A4)

Now, Ec(k) is rewritten identically in the form

k j li2 [ ]2 li2 [ A I ] Ec( )=Eg+- (kll-koll)+K.l +- (k-koh,mJ: (k-koh 2m II 2

(AS)

with the two-dimensional transverse tensor

I mil) m12 millmll2

I mil m2 - m~2

(A 6)

and the abbreviation

K.l = mil (kJ..l-koJ..l)+ mil (k.l2- k0.l2). mill mll2

(A 7)

So far we have not specified the direction of the vectors eJ..l and e.l2. We turn the coordinate system until the tensor in J: I becomes diagonal:

. li2 2 Ec(k)=E~+-2 [(kll-koll)+K.l] +Ec . .l,

mil

where Ec . .l denotes

li2 li2 Ec . .l = -2 - (kJ..l - kOJ..l)2 + -2 - (k.l2 - k0.l2)2 .

m.ll m.l2

(A 8)

(A 9)

322 Appendices

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

. ~ 2 nwi (kl1 , q) = E~ + - [kl1 ,11 +qll-kOIl +K(k1.2 -kol.)]

2m II h,2 h,2

+ -2 k~,11 + -2 (k.l1 +ql.l)2 mv mt

(A 14)

(A1S)

(A 16)

h,2 2 h,2 ( 2 2 ) +-2- (k1.2+q1.2- kOl.) +-2 kl. l +k1.2 · (A17)

m1.2 mv

B Evaluation of a Double Integral

In this Appendix we derive the exact solution of

00 00 { t } Q~(k.L,q)~ 1 dk lll dtexp -~[ dU[Iiw,(k",q)-liw'Fllwo] .

(B.1)

Using (A17) and the abbreviations

/]11 = qll - kOIl + K (k1.2 - kOl.), (B.2)

324 Appendices

(B.3)

00 00 t

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

~II ~II

( 3 )3 1 3 2 1 3 8 11 13 +-8 1I fJ --8 II -3 3 r, 8 3 r,lI

(B. 6)

Since 8~/8~,1I = /LII/mll, where /LII is the reduced effective mass in field direc­

tion: /Lil l = mill +m;;l, the double integral becomes

qll (i ) Q'f(k-L,q)=pexp -"h,CiP J(811' 8 v )J(8v , 811) (B.7)

B Evaluation of a Double Integral 325

with

x (i_:IIIIP)+~8Tip3[1-(JLII/mll)2J}). (B.8)

Now, we can extract an Airy function from J according to its definition [App.l, App.2]

which gives for J(E>II' E>v)

2rr . (np28Ti (1- JLII/mll) + Ci) J(811' E>v) = -Ai

E>r,1I n8r,II

( . { Ci JLII 3 3 ( JLII ) JLII x exp -l -p---p 8 11 1-- -n mil mil mil

(B.9)

+ ~p38H 1- (JLII/m ll )2]}) . (B. 10)

Taking the product of J(8 11 , 8 v) and J(8v, (11) the exponential can be further simplified by introducing a new reduced mass JL3

1 1 1 -=-+---JL3 3JLII mil +mv

(B. 11)

Inserting the expressions for p and qll' the double integral takes the form

2 qF Q'f (k.L , q) = (2rr) -2-

nE>r,1I

( (1- ~);:: [qll - kOIl + K(k12 - k01J]2 + Ci)

xAi mil II

n8r, II

X Ai (( 1-~) ~ [qll-kOIl +K(k.l2 -kQ.l)f +Ci)

n8r ,II

X exp {-i 2 n2 [qll - kOIl + K(k12 - ko.l) r} . (B.12) JL3qF

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 are identical. With the defini­tion mil +mv == mE one obtains the final result

C Transmission Probability for a Parabolic Barrier

The matching conditions for the wave functions (wave supposed to be incoming from the metal side)

lfrI(X) = AeikMx + B e-ikMX

'lfrIl(X) = a U~ +,8 V~

'lfrIl [(x) = C eiksx

(metal)

(barrier)

(semiconductor bulk)

at ~ = ~B (x = 0) and ~ = 0 (x = XB) read

A + B = a U~B + ,8 V~B '

I I (' ') -(ikMA-ikMB)=--- aU~B+,8V~B ' mM Am~

ex, Uo + ,8 Vo = C eiksXB ,

-~ (aU~+,8V~) = CikseiksXB, A

since mejf(xB) = me. The transmission probability is defined as

ICl2 vs,x ICl2 K~ T(Ks,KM)=-·-=_·-.

IAI2 VM,x IAI2 KM

Resolving the system (C.4)-{e.7) and inserting the amplitudes gives

T(KS,KM) =! ;~ I(V~+iK~VO) (U~B +iKAiu~B) - (U~+iK~UO)(V~B+iKAiV'B)I-2 ,

(C.I) (C.2)

(C.3)

(C.4)

(C.S)

(C.6)

(C.7)

(C.S)

(e.9)

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

4 s

(2;rr) 1/41 ~~ - K21-1/4 1/6 V~~ = 4 s (~ISI) Bi[(3S/2)2/3], (D.7)

Jr(~ +K~) 2

· (2;rr)1/4 Ir(l +K~) 3 -1/6 U/ Ai = V 2 (-lSI) Ai/ [(3S/2)2/3] (D.8) ~B le2 1-1/ 4 2 ' ~_K2

4 S

· (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):

Ai(-Y) ----+ 1l'-1/2Y-l/4Sin(~y3/2+~) , (F.1)

Ai'(-Y) ----+ _1l'-1/2Yl/4cos(~y3/2+~) , (F. 2)

Bi(-Y) ----+ 1l'-1/2Y-l/4cos(~y3/2+~) , (F.3)

Bi'(-Y) ----+ 1l'-1/2Yl/4Sin(~y3/2+ :) . (F.4)

Then it follows

IAi(Y e-i~hl = ~J Ai2(y) + Bi2(y) ~ ~1l'-1/2IYI-l/4 , (F.5)

IAi'(Y e-i~)1 = ~J Ai,2(y) + Bi,2(y) ~ ~1l'-1/2IYll/4 . (F.6)

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

where the matrix elements follow from

Ml= ( rr[iklroBi(~I)+Bi'(~I)]

rr [iklroAi(~z) +Ai'(~l)]

-rr [ikzro Bi(~l) - Bi' (~l)]

-rr [ikzroAi(~z) -Ai'(~z)]

rr [Ai' (~t,s )Bi(~t,s+) - Ai(~t,s )Bi' (~t,s+)]

-rr [Ai' (~t,s)Ai(~t,s+) - Ai(~t,s)Ai' (~t,s+)]

rr [Bi' (~t,s )Bi(~t,s+) - Bi(~t,s )Bi' (~t,s+)] ).

-rr [Bi' (~t,s)Ai(~t,s+) - Bi(~t,s)Ai' (~t,s+)] (G.6)

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

1 _ '- ( iklro 1/4) Sl m12 - -y 1l' 1~111/4 -1~iI e, (G.9)

ml = - ,.j1i (iklro + It: 11/4) e-Sl (G. 10) 22 2 1~111/4 sl ,

where Sv denotes the action Sv = ~ I~v 13/ 2• The position and width of the reso­nance are determined by Mr, the elements of which read

m/ = !e±(S/,r+-S/,I) [cos (St,r _ St,/+) (I ~/,I~/,r 11/4 + 1 ~/,I+~/,r+ 11/4) ~2 ~/.I+~/,r+ ~/,l~/,r

± 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

Tres(E,x) = 16<1>t me,rkl (1+ me,1 EI )-11;1-1/2 me,lkr me,o CPt +qFx

References 335

x [",-2 (Ai; + ':" Ai~) (Bi;+ n:'"B~)

( lie )2]-! +'T,.-24e4Sr Ai;+ ;roAi~ 8[Et(x)-E] ,

(G.29)

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

Adjustable parameters, 1, 35, 41, 85, 92, 201,219,287,289

Affinity, 10 Alpha particles, 83 Arrhenius plot, 309 Asperity height, 41 Auger coefficient, 78, 80, 82, 103, 104 Auger recombination

band-to-band, 75, 77, 223 excitonic, 215 phonon-assisted, 77, 198 rate of, 78,103 trap-assisted, 74, 215, 223

Average field in inversion layer, 39, 40, 54, 55,58,63

Bandgap in silicon, 1, 15, 17, 21, 104, 174, 178,

181,182,222 in Si02, 287, 297 temperature dependence, 13, 223, 225

Band gap narrowing, 18, 21, 28, 30, 222, 223

apparent, 24, 26, 29, 30 plasma-induced, 24, 25

Band multiplicity, 23, 24, 133 Band offset, 25 Band structure, 5, 51, 78, 127, 129, 132, 135,

158,166,177,180,256,281,284 pseudopotential, 79, 90-92, 133, 152,

173 Band tails, 18, 19, 148,222,295 Bandstructure mismatch, 282, 287, 288 Bargmann potential, 22 Base current, 72, 171, 198 Binding energy

of dopant, 149, 155, 236, 243, 245 Bipolar transistor, 24, 25, 27-29, 53, 72, 80,

84,171,198,220 Bloch factor, 32,157,178,191 Bohr radius, 18, 20, 35, 36 Boltzmann equation, 4, 45, 84, 86, 91, 127-

131,134,136,154,256 Boltzmann statistics, 30, 71, 131, 136, 141,

142, 145, 160, 198, 199, 203, 221,227,228,235,269

Born approximation, 32, 34, 36, 44, 51, 53, 69,133,135,148,166

Bose function, 33,134,137,182,202 Boundary, 1,72,267,270 Boundary conditions, 267, 270, 274

Dirichlet, 269

Subject Index

for Boltzmann equation, 256 for electrostatic potential, 252, 270,

273,274 ideal Ohmic, 252, 273 ideal Schottky, 252, 273 ideal versus realistic, 276

Breakdown avalanche, 109, 164, 170 electrical, 83, 170, 271 hard, 171 of oxide, 292 soft, 171, 292 tunneling, 184, 192, 221, 227

Breakdown voltage, 84, 100, 106, 170, 171, 177,305

Brillouin zone, 13, 91, 93, 103, 176, 180, 256

Brooks-Herring theory, 33-35, 49, 51, 68, 135, 156

Built-in potential, 274

Capture inter-level, 231 thermal, 72,198,200,214,244,295

Capture coefficient, 71, 82, 200, 215, 216, 228,234

Capture cross section, 70, 76, 215, 216, 220, 228

Capture-emission two-step, 282, 301, 310, 334

Capture rate, 70, 73, 200, 233, 239-241, 297 Capture time, 75, 296 Chaotic motion, 7 Charge in inversion layer, 45 Charge loss in EPROMs, 281, 282, 299,

305-310 Charge loss measurements, 305 Charge multiplication, 83, 84, 94, 96, 97,

171 Chemical energy, 4 Chemical potential, 9, 21 Chynoweth's law, 85, 94-96, 98, 99, 104,

171 local, 100 modified, 98 nonlocal, 100

Clustering of impurities, 18,35,51,53,148, 150

Coherence length, 2 Collector current, 27, 80 Collector saturation current, 27 Collision

electron-phonon, 173, 181 hole-phonon, 174, 181, 184

Collision broadening, 87 Collision operator, 129-131 Collision rate, 36 Collision term, 5, 7, 8, 164 Conductance, 4, 41,171 Conductive heat flow, 7

337

Conductivity, 8, 16, 131, 172, 174,301,307 measurements, 17,39 negative differential, 170 tensor of, 173 tunnel, 174, 178, 183-185, 187

Confinement, 2, 290 Conservation law, 6, 136, 183, 230, 254,

256,257,284,287,288 Continuity equation, 6, 34, 70, 229, 291 Convective energy transport, 11 Conwell-Weisskopf theory, 33, 49 Correlation, 21, 42, 216 Correlation function, 173, 175-177, 181 Correlation length, 41, 43 Coulomb blockade, 2 Coulomb potential, 36, 135, 155, 199 Critical field, 58, 95, 192-194, 220, 286 Cross section, 36, 86, 87, 202, 301, 302 Current

breakdown, 221 bulk-limited, 282, 307 contact, 230, 255, 257, 260, 262, 264 diffusion, 52, 244 drift-diffusion, 252, 268, 269, 273, 291 electrode-limited, 306 emission, 253, 256, 257, 262, 265,

267,269,270,273 excess, 170-172, 197,231,238,243-

245 Fowler-Nordheim, 281, 286-288, 290,

293,306 generation, 226 leakage, 172, 196, 198,281-283,293,

307,309 recombination, 244, 269, 271 resonant tunnel, 300-302 tunnel, 171, 172, 179, 180, 187, 189,

192, 197, 222, 225, 271, 281, 283, 285, 287, 288, 291-294, 299

Current density, 1, 6, 9-11, 84, 96, 108, 172, 175, 244, 257, 262, 263, 267-269, 277, 278, 288, 296, 298, 304,306,308,310

338

SRH, 73, 244 vector of, 54, 100, 108, 173

Current gain, 20, 25, 172 CV measurements, 94, 95, 288 Cyclotron resonance measurements, 13, 14

Dead space (dark space), 94, 96 De Broglie wavelength, 252 Debye-Conwell formula, 48 Debyelength,26,59,140,254,292 Debye shift, 25 Debye sphere, 19 Deformation potential, 32, 42, 127, 133-

135, 138, 152, 153, 155, 159, 160,163,165,191,239

Degeneracy of bands, 183, 203 of carrier gas, 18,27,30,32, 135, 166,

262 Degeneracy factor, 71, 76, 199,234 Degradation

of mobility, 96, 98 of MOSFET, 281

Density matrix, 3, 173 Density of states, 19, 20, 22, 24, 90, 92,

133, 141, 148-150, 157, 158, 166,170,201,203

effective, 13, 14, 19,31, 129 field-dependent, 228, 239 ideal, 19,20,28,30,32,141,142,148-

150, 157, 164, 166 real, 15, 20, 24, 127, 133, 135, 148,

150,157,166,222 two-dimensional, 98, 290 zero-field, 216

Density operator, 173, 177 Detrapping, 70 Device simulation, 1, 22, 51, 72, 104, 106,

109, 127, 128, 133, 141, 146, 151, 157, 172, 191-193, 197, 200, 207, 221, 228, 231, 242, 252, 257, 267, 270, 273, 282, 288,293,298,307

Dielectric constant, 285, 286 optical, 285 static, 59, 90, 135, 152

Dielectric function, 21,79, 145 Diffusion approximation, 128, 145 Diffusion coefficient, 52 Diffusion current measurements, 51 Diffusion length, 27, 29,52 Diffusion potential, 222

SUbject Index

Diode, 72,223,229,231,241,244 Esaki, 17(}-172, 197,221,226,229,

238 gated, 2, 104, 106, 109,220,221,223 IMPATT,101 merged pin-Schottky, 273, 274, 277,

278 mesa, 95 MOS, 199, 291 nin-Schottky, 274 photo, 27, 52 pin, 231, 277, 278 power, 25, 215, 273 resonant tunnel, 3, 302 Schottky, 94, 253, 270,274 Zener, 170

Dispersive screening, 21, 92, 93, 140, 142, 145, 147, 157, 164

Distribution function, 5, 38, 45, 84, 86, 87, 90, 91, 93, 101, 103, 128, 129, 132, 142, 145, 150, 156, 179, 184, 193, 194, 201, 213, 227, 254,257,262,281,283,291

Doping concentration, 1, 18, 19, 21, 26, 27, 34,39,43,51,53,55,74,78,81, 99,106,107,132,148-150,152, 198,265,267,268,271,274

Drain current, 40, 41, 53, 54, 58, 88, 164, 220,293,294,305,307

Drift-diffusion model, 9, 13, 71, 106, 226, 267, 273

Drift velocity, 45, 54, 64, 66, 83, 129, 154, 161-163

anisotropy of, 46 Driving force, 6, 67, 193, 221, 226 Dynamic screening, 93

EBIC measurements, 51, 80 Effective channel width, 55 Effective intrinsic density, 9, 13, 30, 31, 104,

106 Effective mass, 13, 14, 22, 23, 28, 34, 41,

67,144,183,189,194,195,221, 239, 253-255, 260, 284, 308, 329

density of states, 13, 14, 18, 42, 132, 133,144,152,157,163,221

longitudinal, 13, 144,323 reduced, 25,34, 68, 178, 180, 191, 194 temperature dependence, 13 tensor of, 320 transverse, 13, 144, 152, 157, 185,323

Subject Index

Effective mass approximation, 44, 46, 87, 172,174,178,190,191

Effective Rydberg energy, 20, 22, 33, 36 Einstein model, 201, 202, 297 Einstein relation, 9 Electric quantum limit, 40-44, 55 Electro-chemical potential, 9, 11 Electron beam lithography, 3 Electron density, 6, 7, 25, 28, 30, 34, 71, 82,

129,142,238,269 Electron-hole drag, 11, 53 Electron-hole pair

generation, 83-85, 95, 109,221,226 secondary, 98

Electron temperature, 7, 8, 11, 101-103, 127, 129, 132, 136, 138, 139, 144,146,147,152,154,156,164

Electrooptical energy, 190, 205, 218, 219, 284,297

Electrooptical frequency, 178, 202 Electrooptical function, 202, 205 Electrostatic potential, 8, 30, 70, 190, 267 Emission rate, 70, 233 Emitter, 27, 29, 171, 273 Emitter-base junction, 25, 171, 172, 198,

220 Empirical formulas, 1, 27, 29, 43, 46, 51,

84,87,96,215 Energy balance model, 8 Energy density, 7 Energy flux density, 7, 8, 11, 12 Energy generation rate, 10 Energy loss rate, 89 Energy relaxation time, 163 Energy transport model, 8 Entropy current density, 10 Entropy density, 9 EPROM, 281-283, 295, 305, 307 Equipartition approximation, 137 Excess charge density, 9 Excess current, 245 Exchange, 23,42 Exchange-correlation energy, 21, 22 Exchange-correlation hole, 284, 285 Exciton, 25, 216

bound,80 Exciton density, 25 Exponential integral function, 33, 141 External forces, 4, 34, 178

Feature size, 2

339

Fermi energy, 18, 20, 24, 28, 70, 141, 142, 148, 150, 215, 253, 257, 267, 270, 271, 274, 285, 291, 293, 294,297,302

Fermi integral, 4, 30, 141, 142 approximation, 31 inverse of, 31, 142

Fermi level pinning of, 70

Fermi statistics, 30, 32, 129, 141, 145, 147, 156,162,164,166,199,235,267

Field effect band state, 73, 197, 201, 216-218,

220,231,239,243,244 bound state, 73, 199

Field emission, 256, 300 Field enhancement factor, 72, 73, 198, 199,

201, 203-205, 207-212, 216, 220,225,228

Fluctuations of potential, 18, 24, 26, 41, 252

Flux, 6,10 Fredkin-Wannier operator, 172 Free-boundary model, 42 Fuchs scattering factor, 55, 58

Gate current, 281, 282, 293, 294 Generation

avalanche, 83, 88, 96, 108, 109, 155, 164,170,171,223

tunnel, 172, 190, 193, 198, 220 Generation lifetime, 72 Generation-recombination, 7, 70, 170, 223,

269 Golden Rule, 87, 180 Green's function technique, 172 Group velocity, 5, 130, 137

Hall factor, 16, 49, 78 Hall mobility, 16 Harmonic potential, 4 Hartree-Fock approximation, 21 Heat conduction, 11 Heat flow vector, 7 High-density limit, 20, 156 Hole density, 20, 238 Hole temperature, 157, 158, 161 Hot carriers, 9, 12, 91, 101, 154, 164, 252,

291 injection of, 102, 172 temperature of, 101 thermalization of, 88

340

Houston approximation, 172, 177, 179, 181 Huang-Rhys factor, 109,202,216,219,295,

303,308 Hydrodynamic model, 8, 102, 127-129,

133, 164

Ideality factor, 72, 198,231,238,244,245 Image-force effect, 94, 252, 273, 282, 284,

286,293 Impact ionization, 83, 88, 96, 102, 108, 109,

155,164,170,171,223 Impurity band, 18, 148 Independent particle approximation, 283 Interface roughness, 38, 41, 77, 252, 288 Interface states, 3,43,58, 59, 70, 76, 77,108,

223,252,273,292,297,304 Interference effects, 2, 290 Intermediate state, 173, 181, 183 Internal energy, 9 Interpoly dielectric, 283, 305, 307 Intrinsic carrier density, 1, 15-17, 197, 198,

220 Intrinsic level, 9, 30, 221, 274 Inversion

strong, 38 Inv~rsion layer, 38-42, 55, 58, 98, 290 IOnIzation coefficient, 83--85, 88--90, 92-94,

104,108 apparent, 96

Ionization rate, 85--87, 89, 90, 92, 94, 95, 98, 100-102, 108

electron, 93 hole, 93 ~ carrier temperature model, 101 mgases,85 in MOS transistors, 97, 98 in nonlocal model, 96, 99 isotropy of, 87 pseudolocal form, 96, 98 saturation of, 95 significant, 90

Irvin's curve, 34, 46, 49, 78 Iso-energy surface, 132, 183 I(V)-characteristics, 61, 109, 164, 170, 197,

198, 220-223, 225, 229, 238, 242, 243, 266, 271, 273-277, 281,290,291,302

I(V) measurements, 28, 49, 65, 80, 231, 306

Jelliurn model, 24

Kinetic equation, 6

Subject Index

Kohler variational method, 5, 128, 134 k·p-theory, 132 Kubo formula, 172, 173

Landau damping, 156 Landauer formula, 3 Laplace equation, 291 Lattice mismatch, 41 Lattice relaxation energy, 196, 205, 206,

208, 210, 218, 220, 223, 224, 295,298,309

Lattice temperature, 8, 12, 127, 130, 132, 136,152,153,160,162,164,304

Leg~ndre polynomial expansion, 86, 93 Levmson's rule, 19 Lifetime

doping-dependent, 73-75, 106, 223, 226

field-dependent, 72, 106, 198--200, 212,216,217,219,220,295

injection-dependent, 75 minority carrier, 29, 72-74, 78, 109,

197,199,215,234,292 of intermediate state, 183 temperature dependence, 213-215,

218,225 Lindhard dielectric function, 25 Linear response theory, 10, 22, 24, 134 Line shape function, 206 Liouville equation, 3, 4 Local density approximation, 285 Localization radius

of trapped electron, 35, 239, 297, 308 Low-dimensional structures, 2 Lucky drift model, 88, 98 Lucky electron model, 87, 97, 99 Lucovski model, 35

Majority band, 22, 23,148,253 Many-body effects, 18 Mass action law, 16, 25, 30 Material parameters, 1, 42, 266, 273 292 Matthiessen rule, 36-38 42 46 47 '51 55 , , , , , ,

57,60,69 Maxwell average, 37 Maxwell equations, 9 Maxwellian, 45, 86, 128, 135

heated, 101-103, 128--130, 132, 136 Mean free path, 2, 45, 83--89, 91, 93-99

104,282 ' Mean thermal energy, 8

Subject Index

Metal-semiconductor interface, 252, 253, 271,273,275

Method of moments, 6, 127 Minority band, 22, 23 Mobility, 32, 127, 132

anisotropy of, 41, 143, 145 channel, 38,39, 41-43, 53, 54,56, 62,

109 effective, 38, 40, 41, 44, 55, 56, 58,60,

63,68 electron, 8, 27, 128, 132, 153, 155, 166 field effect, 41 high-field, 45, 64, 66, 154, 162, 270 hole, 157, 160, 166 in heavily doped silicon, 50, 68, 148,

150, 162 lattice, 35, 46-49, 57, 60, 70, 152, 153 low-field bulk, 46, 53, 62, 64, 67, 160,

164 majority carrier, 36, 51, 68, 155, 156 minority carrier, 27, 36, 50, 51, 53, 68 roll-off, 40, 44 total, 16, 144 universal behavior, 40, 42, 44, 64

Momentum transfer, 35, 173 Monte Carlo, 5, 44, 46,87,91-93,102,104,

127, 129, 133, 135, 152, 154, 155,162,165,166,191,291

MOSFET, 3, 43, 53, 61, 64, 88, 98, 109, 128, 164, 172, 196, 198, 199, 220,281,284,293,294

ballistic, 3 deep-subrnicron, 9, 129 LDD, 100, 102, 104

MOSFET channel, 32, 34, 38, 42 Mott density, 26 Mott transition, 25,26 Multi-stage process, 85, 99 Multiplication factor, 84, 98

Nano-electronics, 3 n+nn+ -structure, 164 Nonparabolic bands, 14, 20, 46, 93, 102,

127, 132, 133, 136, 138, 140, 145, 157, 158, 164

Nonparabolicity parameter, 128, 129, 132, 137,152,155,158,160,166

n(V)-characteristics, 242, 243

Occupation probability of band states, 79,193,209,301 of defect level, 71, 199, 232-234, 296

Onsagers's theorem, 10 Oscillator strength, 179 Oxide charge, 43, 44, 58, 59, 308

Pao-Sah model, 58

341

Parabolic bands, 11, 13, 32, 45, 89, 90, 103, 141, 146, 157, 178, 183, 239, 253,283,287,297

Parabolic (Schottky) barrier, 252, 253, 257, 263,265,267

Partial wave analysis, 22, 36,51,155 Pattern-dependent oxidation, 3 Pauli principle, 5 Phase coherence, 3 Phase-shift analysis, 22, 155 Phase space, 4 Phonon dispersion, 134, 135, 182 Phonon drag, 11,84 Phonons

bulk, 38, 138, 171, 174 oxide, 295, 308, 309 surface, 38

Photo-emission measurements, 284-288, 310

Photoluminescence measurements, 22, 23, 28,245

Pinhole, 281 Plasma

electron-hole, 18, 21, 24, 25 Plasma frequency, 284 Plasmon pole approximation, 25 pn-junction, 83, 84, 94, 99, 109, 170, 171,

197, 198, 216, 221, 223, 226, 244,270,276,277,282,292

pn-product, 16,26,27,29,226 Poisson equation, 9, 164, 252 Poly-silicon, 3, 295, 297 Poole-Frenkel effect, 72, 73, 199, 239, 245,

308,309 Power devices, 34, 74, 215, 273 Pre-breakdown, 209, 220, 223, 224 Pre-breakdown range, 177, 197, 200, 204,

206,210,221,226 Principle of detailed balance, 130, 233 Process simulation, 106, 109,229,271 Pseudobarrier method, 288, 291

Quantum broadening, 18 Quantum dot, 2 Quantum oscillations in tunnel I(V)-curve,

281,289,291,298 Quantum transport, 2, 3

342

Quantum wave guide, 3, 4 Quantum well, 3 Quantum yield, 85, 92, 104

measurements, 92 Quasi-Fermi energy, 9, 26, 39, 60,71,108,

129, 164, 193, 199, 227, 228, 234, 253, 291

Quasi-Fermi potential, 9, 30, 64, 227, 269, 270

Quasi-particle shift, 21, 25

Randomization, 34, 85, 91, 133, 156 Random-k approximation, 98 Random phase approximation, 21, 25,145,

182 Random telegraph noise, 3 Recombination

coupled defect-level, 231, 232, 235, 238,244

donor-acceptor pair, 243-245 multiphonon, 198-200, 221, 244 tunnel, 193, 198,220

Recombination center, 70, 71, 196--198, 200, 210, 215, 219, 220, 223, 225,238,244

Recombination lifetime, 24, 26, 72, 215 Recombination path, 72, 199, 205 Recombination radiation

in rnicroplasma, 85 measurements, 78, 88

Recombination velocity, 76, 77, 97,108 Rectangular potential barrier, 300 Rectifying contact, 267, 273, 274, 277, 278 Relaxation time, 7, 34, 35, 37, 38, 41, 75,

84,89,90,102,134 . energy, 8, 88, 101, 153, 162, 164

momentum, 8, 32, 34, 88, 127, 136, 145, 154

total,5, 134 Relaxation time approximation, 5, 36, 93,

128, 133, 134, 154, 156 Reservoir, 4 Resistance

bulk series, 253, 270 negative differential, 222 sheet, 27 shunt, 242 spreading, 241

Resistivity measurements, 17, 78 Reverse modeling, 221 Rigid shift, 21, 24, 28 Root-mean square velocity, 129

Subject Index

Saturated velocity, 45, 53, 54, 64, 66, 67, 104, 162-164

Saturation of drift velocity, 53, 56, 64, 67, 127,

133, 152, 154, 160-162, 166 Scaling rules, 53, 64 Scattering

acoustic deformation-potential, 32, 37, 47, 57, 60, 128, 133, 134, 152, 154, 191

attractive versus repulsive, 53 carrier-carrier, 32, 51 charged interface states, 38, 43, 44, 57,

61 Coulomb, 38, 41, 44, 57 difibsesurface, 38,55,58 disorder, 32, 35 elastic, 2, 128, 133, 134, 137, 139 electron-electron, 28, 32, 34, 35, 49,

156 electron-hole, 32, 34, 48, 49, 53, 68,

69 electron-impurity, 21 electron-plasmon, 32, 53 fixed oxide charges, 38, 43, 57, 61 f-type, 32, 135 g-type, 32, 135 impurity, 32,34,36,37,40,47,48,51,

53, 57, 69, 128, 129, 133, 145, 152,153,155,164,166

inelastic, 45, 128, 134, 136, 138, 252 intersubband, 4, 42 intervalley, 24, 32, 42, 57, 133-135,

138, 139, 152, 154 multiple-potential, 157 multivalley, 24 neutral center, 32, 35 nonpolar optical phonon, 32, 33, 47,

89,93,95,98,134,160,192 polar optical phonon, 42 shallow non-ionized impurity, 32, 35 surface roughness, 32, 38, 41, 43, 44,

53,57,60,98 Scattering rate, 38, 53, 87, 90-92, 133

Born approximation, 32, 155 impact ionization, 93, 102 inverse, 9 phase-shift, 155 phonon, 87 total, 5, 128 two-dimensional, 61

Schottky approximation, 27, 252, 254

Subject Index

Schrodinger equation, 3, 254, 258, 286, 288, 291,299

Screening length, 19,33,36,49, 135, 140-142, 144, 148, 149, 156, 164

Second drop of mobility, 150

Secondary ionization, 85 Self-energy, 20-22 Self-heating, 12 Signal-to-noise ratio, 52 SINFET,273 Single-electron transistor, 3 Si-Si02 interface, 38, 39, 53, 55, 58, 77,

96,106,108,221,281,282,287, 297,308,310

Solar cell, 24, 74, 231, 241, 281, 284 Sound velocity

longitudinal, 32, 46, 134, 137, 152, 191

transverse, 152 Space charge, 70 Space charge region, 72, 73, 83, 84, 98, 267,

270 Spherical-harmonics expansion, 127 Spin-orbit energy, 157 Split-off band, 14, 157 Spurious velocity overshoot, 8 Square well potential, 35, 299, 301 SRH recombination, 70-72, 75, 76, 171,

196, 198, 199, 223, 231, 235, 238,242,244

Statistical screening theory, 32 Subband structure, 38 Substrate current, 88, 97,102,172 Superlattice, 3 Supply function, 301 Surface recombination, 72, 76-78, 97, 171,

269 Surface roughness, 38, 41, 77, 252, 288 Surfon,42 Switching behavior of MPS diode, 277

Tail states, 19, 170, 304 Temperature-field relation, 101, 102

electrons, 154 holes, 162, 163

Temperature tensor, 7 Thermal binding energy, 196,199,205,209,

295 Thermal conductivity

of electrons, 7, 11 Thermal velocity, 67, 215

Thermionic emission, 256, 260, 290 Thermionic field emission, 252, 256 Thermodynamic model, 9,129 Thermoelectric power, 11 Third-body exclusion principle, 35 Thomas-Fermi approximation, 18, 140 Threshold energy

effective, 87, 90-92, 98, 104

343

impact ionization, 83-85, 89-96, 98, 99,103

Threshold energy surface, 91, 92 Threshold field, 83, 192 Threshold voltage, 39, 281, 305 Thyristor, 74, 273 Time-of-flight measurements, 51, 65 Transfer matrix method, 286, 300 Transition energy, 72, 180, 193, 205-208,

211,218,221,228,240 Transition probability, 5, 32, 36, 89, 128,

132,134,136,159,201,205,216 Transitions

bound-to-band, 70, 198, 201, 294 direct, 80, 90-92,174,176-178,180,

181, 184, 194 eeh-Auger, 78--80, 82 first-order, 78, 92, 102, 128 indirect, 28, 78, 88, 176 interband, 159, 172, 198, 218 inter-level, 231, 232, 236-238, 243,

244 intervalley, 32 intraband, 28, 159 intravalley, 32 multiphonon, 70, 72, 198, 200-202,

205,216,241,295,296 phonon-assisted, 28, 79, 90, 173, 191,

288 phononless, 79, 90, 91 tunnel, 70, 193, 194, 196, 198, 202,

218,244,304 Transmission

of barrier, 229 Transmission coefficient, 4, 172, 253, 258,

262,283 Transmission matrix, 4 Transmission probability, 172, 252, 254,

256-258, 261, 262, 281, 282, 286,289,299-301,326,332

Transport across thin dielectric layer, 281, 282 ballistic, 2, 86, 252, 253 coherent -ballistic, 2

344

diffusive, 86, 268 dissipative, 2 mesoscopic, 2 quantum-ballistic, 2

Transport models, 1,2,7,135 Trap, 71,172,206,282,295-304,307,308,

310, 311, 330 electron, 70, 294, 295, 308 hole, 70

Trapezoidal potential barrier, 281, 288, 289 Trench DRAM cell, 171, 172, 196, 198, 220 Triangular potential barrier, 42, 209, 239,

245,258 Tunneling

band-to-band, 83, 109,170--173,183, 189, 193-197, 202, 220, 225, 227,229

bound-to-band, 72, 198-200, 202, 218 defect-assisted, 72, 106, 109, 171, 172,

196-198, 200, 202, 220, 221, 225-228, 231, 233, 238, 241-243,252,282,292

direct band-to-band, 180, 191, 194 direct barrier, 281, 282, 287, 293, 311 Fowler-Nordheim, 282, 298, 306, 311 in inhomogeneous field, 174, 190 phonon-assisted, 171, 172, 174, 181,

189,192,194,221,288 resonant, 282, 293,299, 300, 302-304,

310,330,334 two-step multiphonon, 282, 294, 296,

303,310 Tunneling rate

local versus nonlocal, 226 Tunnel length, 180, 190,192,193,200,227,

228,230,293,299 Tunnel mass

in Schottky barrier, 255, 260, 271, 273 in Si, 183, 189, 195, 202, 203, 217,

219,221,239 in Si02, 286, 293, 297

Tunnel time, 178,241,243,244,284,303, 308

Two-band model, 78, 157, 181, 191, 287 Two-level system, 231, 238, 244

Ultra-pure silicon, 17 Umklapp process, 32, 91 Unified mobility model, 70, 151, 159 Unified recombination model, 83

Warped bands, 157, 183

Subject Index

Wentzel-Kramers-Brillouin (WKB) ap­proximation, 172, 174, 190, 193, 205, 206, 252, 258, 261, 262, 285, 286, 288, 289, 297, 298, 302,329

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. . .................... .

130 133 138 140 143 143 145 147 148 149 150 153 154

155 156 158 158 160 161 161 162 163

165 165

174 180 188 195 195 196 196 206 209 211 212

214 217 217 218 219

219 222 222 223

224 224

List of Figures 347

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

348 List of Figures

5.12 Calculated transmission probability for a 100A thick oxide containing a centered square well. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

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

width ........................................... 310

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.

~ SpringerWienNewYork Sacru.enplatz4-6, P.O.Box 89, A-120l Wlen, Fax +43-1-3302426

e-mllii: order®springer.at. Internet: hllp:lJ_w.8pringer.at

New York, NY 10010,175 Fifth Avenue- [)"14197 Berlin. Heiddberger PIaIz 3

Tokyo 113,3-13, Hongo 3-chome, Bunkyo-ku

SpringerEngineering

Computational Microelectronics

D. Schroeder Modelling of Interface Carrier Transport for Device Simulation 1994. 69 figures. XI, 221 pages.

Cloth DM 204,-, fiS 1428,-. ISBN 3-211-82539·8

\'. D. t\ rora

MOSFET Models for VLSI Circuit Simulation Theory and Practice

1993. 270 figures. XXII, 605 pages.

Cloth DM 328,-, fiS 2295,-. ISBN 3-211-82395·6

W. Joppieh, S. Mijalklnie Multigrid Methods for Process Simulation

1993. 126 figures. XVII, 309 pages.

Cloth DM 198,-, fiS 1386,-. ISBN 3-211-82404·9

W. HiluO'ch The Drift Diffusion Equation and Its Applications

in MOSFET Modeling 1991. 95 figures. XII, 271 pages.

Cloth DM 164,-, fiS 1148,-. ISBN 3-211-82222·4

C. Jaeoboui, P. Lugli The Monte Carlo Method for Semiconductor Device Simulation 1989. 228 figures. X, 356 pages.

Cloth DM 204,-, fiS 1430,-. ISBN 3-211-82110·4

P. A. l\1arkowieh The Stationary Semiconductor Device Equations 1986. 40 figures. IX, 193 pages.

Cloth DM 119,-, fiS 836,-. ISBN 3-211-81892-8

~ SpringerWienNewYork Sachsenplall!: 4-6, P.O.Box 89, A-120l Wien. Fax +43-1-330 24 26

e-mail: order@spnnger.al.lntemet:hup:ll_w.springer.aI

New York, NY 10010,175 Fdih Avenue· 0-14197 Bedln, HeidelbergerPlatz 3

Tokyo 113, 3-13, Hongo 3-chome, Bunkyo-ku

Spnnger-Verlag and the Environment

WE AT SPRINGER-VERLAG FIRMLY BELIEVE THAT AN

international science publisher has a special obliga­tion to the environment, and our corporate policies consistently reflect this conviction.

WE ALSO EXPECT OUR BUSINESS PARTNERS- PRINTERS,

paper mills, packaging manufacturers, etc. - to commit themselves to using environmentally friendly mate­rials and production processes.

THE PAPER IN THIS BOOK IS MADE FROM NO-CHLORINE

pulp and is acid free, in conformance with inter­national standards for paper permanency.