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Interaction of low-energy electrons with silicon dioxide
Journal of Electron Spectroscopy and Related Phenomena, 24 (1981) 127-148 Elsevler Sclentlflc Pubhshmg Company, Amsterdam - Prmted m The Netherlands
INTERACTION OF LOW-ENERGY SILICON DIOXIDE
J C ASHLEY
ELECTRONS WITH
and V E ANDERSON*
Health and Safety Research Dwwon, Tennessee 3 7830 (U S A )
Oak Radge National Labomtoly,
Oak Rzdge,
(Received 15 April 1981)
ABSTRACT Theoretlcal models and calculations have been combined with expenmental optical data to determine a model energy-loss function for SIOz Sum-rule checks and compansons with experlmental information are made to ensure overall consistency of the model The model energy-loss function IS employed to calculate melastlc inverse mean free paths and stoppmg powers m S102 for electrons with energies
INTRODUCTION
S&on dioxide, m its various solid-state forms, IS a matenal of mdespread mterest and Importance m the electromcs mdustry Studies of its basic physlcal properties are so extensive that entire conferences are devoted to reports of current research [l] For mllltary, space [ 21, and commercial [3] electronics apphcatlons, the performance of devices mvolvmg S102 m a radiation environment may be of critical nnportance No matter what type of mcldent radlatlon 1s mvolved, calculations of energy deposltlon must mclude contributions from electron cascades generated m the material The purpose of this paper 1s to develop a model to describe the melastlc events occurmg as an electron slows down m $10, For the mcldent electron energies considered here (5 10 keV), the mam source of energy loss 1s due to mteractlons with the electrons m the medium The response of a medium to a given energy transfer o and momentum transfer q may be described by a complex dlelectnc function e (q, w). In general, e may be a tensor which depends on the dvectlon of q In this paper it 1s assumed that the medium 1s homogeneous and lsotroplc so that E 1s a scalar quantity which depends on only the magnitude of q and not its dlrectlon If E were known m det4 for * On assignment from Computer Scrences Dlvlslon
O368-2~48/81/0000-0000/$02
50
o 1981
Elsevler Sclentlfic Pubhshmg Company
128
an electron of energy E, the probability of an energy loss o per unit distance traveled could be determmed from [4,5] 7(E,w)
= $
@+‘dq s *_ 4 Im1--1/MLN
(1)
and Im() indicates the lmagmary part of where qr G dz[&? + d(E - a)], () This expression of qk assumes that the energy-momentum-transfer relatlon for a non-relatlvlstlc electron m the solid does not differ appreciably from that for a free electron m vacuum The quantity ~(E,u) 1s also called the dtiferentlal inverse mean free path (DIMFP), smce by mtegratmg it over allowed energy transfers the melastrc inverse mean free path for an electron of energy E 1s obtamed In addition, the quantity c~T(E, w) integrated over allowed values of w gives the energy loss per umt path length or, neglectmg radiation by the electron, the stuppmg power of the medium for an electron of energy E Note that here, and m the followmg, quantities are usually expressed m atomic units (a u ) where 5 = m = e = 1 The results are converted to more conventional units where appropriate The mam task then 1s to fmd a suitable expression for E(~,o) to descnbe the dlelectrlc response of SIOz For the purpose of modelmg, the electrons m the medium are divided mto two groups valence electrons and mner-shell electrons A model-msulator theory 1s used to describe the valence electrons, while generahzed oscillator strengths are used to describe inner-shell lomzabon The theoretical model for e 1s based, m part, on optIcal data for SlOz Vanous sum-rule checks are then made to ensure overall consistency of the model Fmally, electron mean free paths m S102 and the stoppmg power of SlO, for electrons are calculated for E < 10 keV and comparisons made vvlth available expemental data
VALENCE
ELECTRON
MODEL
A model-msulator theory 1s used to describe the valence electrons’ response [6] This model has been employed m several previous calculations [ 7-123 and IS related to that employed by Fry [ 133 m which ground-state wavefunctlons for the valence electrons are described m the tight-bmdmg approxlmatlon, while excited states are represented by orthogonallzed plane waves (OPW) However, m our use of the model the normahzatlon of the OPW excited states was fured by requlrmg that the sum rule
r
dw w Im [e(q,w)] 0
=
2n2 n,
be satlsfled, where n, IS the valence electron density
(2)
129
Fit to optrcal
data
The msulator-model results of ref 12 are improved by carrymg out a more dettied f1t of c2, the 1magmary part of the dlelectnc function for zero momentum transfer, E(O,O) E E(O) = Em + 1e2(0), to the expernnental values of Ed for crystallme S102 measured by Phllllp 114,151 as a function of photon energy H1s expenmental values are plotted 1n F1g 1 as the solid line, and the new f1t to the data obtamed by using eight tight-bmdmg levels and a band gap [ 161 of 8.9 eV ISshown by the dashed lme. The parameters used 1n this f1t are shown 1n Table 1 0 = l-8) The sharp peak at 10.3 eV 1s generally agreed to be due to an exclton transltlon, while the structure at higher ener@es 1s of less certam omgm and 1s described by some workers as due to interband transltlons [17] or, most recently, as bemg due to excltonlc resonances [ 161 The assignment of the processes responsible for these features m e2 may be important m exammmg the detailed energydeposltlon processes m S102 as was done earlier for silicon [18] and water
WV *
The e2 spectrum for the valence electrons IS,however, not complete Usmg the sum rule, eqn (2), m the form do w ~~(0) I=0
= 2r2n0 N&
(3)
with e2 gwen by the msulator model f1t shown m F1g 1,1t 1s found that Ne = 7.87, whereas 16.82 effective valence electrons per S102 molecule
11 6
5
---
EXPERIMENTAL OATA (PHILLIP) INSULATOR MODEL
_
Fig 1 The nnagmary part of the dlelectrw function as a function of photon energy The sold curve USfrom the optlcal data of Phlllrp 1141, the dashed curve 1s the msulatormodel ht
130 TABLE1 PARAMETERS FOR THE INSULATOR MODEL ~=~)DETERMINEDFROMFKTSTOOPTICALDATAFORez
J
1 2 3 4 5 6 7 8 9 10 11 12 13
89
100 109 12 5 13 3 15 0 158 190 210 310 292 310 420
03272 0 3676 04007 04596 0 4908 05515 05809 0 6985 07721 1140 1074 1140 1544
(AS DEFINED
IN REF
6 WITH
cuJ(au I
4
noNj(x
07800 02100 03787 0 5145 04067 06642 05145 06301 09963 2 047 06301 1485 2720
2000 06860 1188 07000 06573 1680 05230 04318 1400 5 290 005840 06100 1612
7 860 2 696 4669 2751 2583 6603 2056 1697 5 502 2079 02295 2 397 6335
103)
are expected In eqn. (3), no IS the number of molecules per umt volume For the crystal density p = 2 65 g cmm3, no IS gwen (m a u ) by no = 3 930 x 10m3. The effective number of valence electrons per molecule 1s determmed from an accountmg of the effectwe number of inner-shell electrons. Table 2 shows the atomic levels of the inner-shell electrons for Sl and 0 with the magnitudes of then bmdmg energies and N,(O), the effectwe number of electrons partlclpatmg m transltlons from the gwen level to the contmuum for zero momentum transfer Q, where &f&J)
=
I0
= da df(q,o)ldw
(4)
In eqn (4), df/dw 1s the generalized oscillator strength (GOS) for a given transltlon. The values of N&(O) were calculated from McGuue’s tables [20] of GOS’s for the K shell of 0, from Manson’s GOS’s [21] for the L shell of Sl, and from GOS’s delnved from hydrogemc wavefunctlons [ 223 for the K shell of Si Thus, of the 30 electrons per SIOZ molecule, 30 - 9 86 2(166) = 16.82 are considered to be “valence electrons” The contnbutlon of the oxygen 2s electrons (bmdmg energy - 29 eV) u mcluded m the e2 spectrum through the relatron (m atomic umts) e2 = (2r2n,/u) (df,/dw) where n, I.S the number of shells of type L per unit volume This contibution to e2, calculated using GOS values from ref 20 and = 2no, 1s shown m Fig. 2. A fit to this curve was made using three terms 122s m the insulator model (parameter sets 11, 12, and 13 m Table 1) Since Jo”dwdf2,/dw = 1.14, this leaves 16 82 - 7 87 - 2(1.14) = 6.67 valence electrons to be accounted for m the e2 spectrum.
131 TABLE
2
MAGNITUDE OF THE BINDING ENERGY AND EFFECTIVE NUMBER OF ELECTRONS PARTICIPATING IN TRANSITIONS TO THE CONTINUUM FOR INNER-SHELL ELECTRONS EIement
Shell
Sl
wz
( 2sJ2 (2P? W2
0
----
i0
INSULATOR
08
Bmdmg energy (eV)
N&f(O)
1829 152 108 537
155 131 7 00 1 66
030 62
015 0
ul” 06
02
1 t
0' 20
I
I
I
25
30
35
40
hw (eV)
Fig 2 The sohd line shows the Imaginary part of e(w) versus photon energy determmed as described m the text The dashed lme IS the msuIator-model fit The inset shows the contnbutlon to ~2 from lonlzatlon of the L1 shell of oxygen
The e2 spectrum was extended to 40 eV usmg Buechner’s data [ 231 denved from electron energy-loss measurements His values were scaled to fit Phtilp’s data at 20 eV, with the result shown m Fig 2 Two addltlonal msulator-model terms have been included accounting for an additional 6.67 electrons (parameter sets 9 and 10 m Table 1). The total msulator-model fit m this energy region 1s shown as the dashed curve m Fig 2 Sum-rule
checks (q = 0)
To check the overall consistency of the insulator-model fit and its predlctlon of e2 we examme some quantltles denved from e2 and compare them wAh available expenmental data Various sum-rule checks are useful here. If we define
s w
N1 (w)
= (1/27r2n0)
0
dw’ o’ e2 (~3’)
(5)
132
then ,$..=N1 (w) = N, the total number of electrons per molecule, rf e2 describes the response of all the electrons 1n the matenal (valence plus inner shell) If e2 describes the valence electrons only, then eqn (5) represents a partial sum-rule and Jls_ N1 (0) = Nsy as given 1n eqn (3) In F1g 3 N1 1s plotted as a function of energy for the valence electrons 1n S102 usmg our insulator-model f1t and for transitions to the contmuum from the L, and L2,3 subshells of S1 usmg Manson’s GOS’s [ 211 The value N1 = 6.18 at 22 eV 1s 1n good agreement vvlth the value 6 24 calculated directly from Phti1ps’s optical data [14]. The shape of the N1 curve for crystalline S102 (Fig 3) 1s snnllar to that for amorphous S102 given 1n ref 15, but predicts smaller values for a given energy, due 1n part to the different densities For large photon energies NT”1approaches 16 82, as 1t should, since this was built into the fitting procedure. The real part of the dlelectnc response function, E1 (o), 1s related to e2 (w) through the Kramers-Kron1g relation e1(w)
=
1 + (2/n)P Jb; dw’[w’e2(W’)/(W’2
- w2)]
(6)
where P mdlcates prmc1paLvalue mtegral Because of the energy denommator, values of el up to - 20 eV are determined mamly by the valence electrons. Usmg the msulator-model f1t for e2, e1 has been calculated from eqn (6) The results are shown 1n F1g 4 by the dashed line, for comparison with Phlll1p’s results [ 141. At IO eV the Insulator model predicts a value e1 2~ 9, somewhat larger than that gzven by the expenmental curve, due to e2 being set to zero for energres below the assumed optical band gap [ 16,241 of 8.9 eV (see Fig. 1) Othermse the shapes and magnitudes of the two curves are 1n good agreement I
I I IllIll
I
I I IIIIII
I
I I Illlll
iI
15 90, VALENCE 10 -3 s
5
ELECTRONS
I II
1
I II
102 tTo(eVl
Fig 3 Effectwe number of electrons partwlpatmg m energy-loss processes #iw The curves saturate at high energies at the values indicated
up to energy
133
For calculations of DIMFP’s, the energy-loss function Im [-l/e(c~,w)] is required It has been evaluated m the hmlt Q = 0 from e1 and e2 as determmed by the insulator model. The results are shown by the dashed lme in Fig 5, for comparison with the energy-loss function determined by Buechner from electron energy-loss measurements (solid lme) 1231 The structure for energies ~15 eV IS sharper m the dashed curve determmed from the insulator model (or from optical data) than that m the solid curve These peaks occur at nearly the same posltlons m the two curves. This difference in structure 1s also true for e2, as shown by fig 7 m ref 12 The mam peak m our predlcted energy-loss function occurs at 23.9 eV, -1 eV above that measured by
4 w3
0 5
0
(0 hw (eV)
Fig 4 The solid curve shows the variation of the real part of the dlelectrlc function with photon energy from Phllhp’s experlmental data [14] The dashed curve was obtained from the insulator-model ht. to Q through the Kramers-Kromg relation
-
(BUECHNER)
Fig 5 The dashed lme shows the energy-loss function predrcted by the msulator The sohd curve IS from electron energy-loss measurements by Buechner [ 231
model
134
Buechner. This could be accounted for 1f the density of the polycrystalllne quartz fell used m the latter experiment [!23] were -8% less than the value 2 65 g crnm3for the quartz used 1n the optical measurements [ 141 However, no densities were given 1n ref 23 so this pomt cannot be checked On the other hand, If 1t 1s assumed that the shape of the mam peak 1n our curve 1s desclnbed approxunately by Im(-l/e) a rw/[(o’ - w$)’ + r2 o* ], then +iWO N 24.7 eV 1s m excellent agreement with the plasma energy ?iw, = 24.8 eV calculated from p = 2 65 g crnm3and NeE = 16 82 As a final check on the energy-loss function predicted by the insulator model, the energy-loss sum rule
I
mdw w Im(-l/e) 0
=
2n2nON
(7)
was evaluated numerically for the valence electrons The value of N 1s found to be N = 16.83,1n excellent agreement with Nefp= 16 82, As a fmal overall check on the lnformatlon incorporated 1n the model for q = 0 (optical limit), including the inner-shell contrlbutlons, the mean excitat1on energy I for S102 was calculated This important constant 1n stoppmgpower theory 1s defined through the equation [ 251 InI = (1/2~2noN)
lomdw w lnw Im(-l/e)
For 1oruzat1on of the inner shells Im(-l/e) eqn (8) may be written
(8) = e2 = EZ[(2x2 n,/w)(df,/dw)]
,
so
InI = (1/27r2noN) + c I
{(nt/n~N)
JOmdww lnw Im(-l/P) lo-
dw
l+WW,/d~)l~
where n, 1s the number of inner shells of type 1 per unit volume, and N = 30 for S102 The approximate form employed here for the energy loss 1s dlscussed 1n Append= I of this paper The first term m eqn (9) 1s the contrlbutlon from the valence electrons, as indicated by the superscript v on E This term 1s evaluated using the msulator model while the second term 1s evaluated using the GOS’s discussed earlier for the mner shells The mean excitation energy 1s found to be I= 1416 eV, m excellent agreement mth the value I = 139 2 eV determined from proton range measurements [ 261. The checks made 1n this Section on the model that WIU be used to desclnbe the response of the electrons 1n S102 to a aven energy transfer (but small momentum transfer) are quite encouragmg. They examine mamly the overall consistency of the proposed model through the sum rules However, the good agreement of the energy-loss function measured expemmentally
135
with the model predlctlon provides a more detded check for small energy transfers Compmson of the predicted and Yneasured” mean excltatlon energy IS particularly unportant for stoppmg-power calculations A small correction to I 1s discussed m Append= I The “extension” of the energyloss function to non-zero momentum transfers IS discussed m the next Section Extenston to g # 0 With the parameters determined from optical data, the insulator model [6] gives a prescrlptlon for extension of e2 to non-zero momentum transfer, i.e. E2(W) s eZ(O,a) -+ eZ(q,w) On e feature of the insulator model 1s that the effective number of valence electrons, as determined by the e2 sum rule, eqn. (2) or eqn (3), IS the same for any value of q, while lr dwdf,/dw approaches the occupation numbers for the inner shells (e g , two for K shells) for large q. Thus, as the effective number of inner-shell electrons mcreases vvlth mcreasmg q, the effective number of valence electrons must decrease from 16 82 at q = 0 to 16 for large q. The function N&(q), eqn (3), has been determined from GOS’s for the inner shells, and for the oxygen 2s electrons, which were included with the valence electrons The valence electrons are treated as two groups. The oxygen 2s electrons, which were mcluded mth the valence electrons, will be made to increase from 2.24 to 4 effective electrons per SIOZ molecule, based on N,(q) for this group Smce e2(q,u) = 2;:& eJ2(q,w) from the insulator model, the three terms m this sum descrlbmg the oxygen 2s electrons are thus multlphed by the function Nd(q)/NM(0) -g%(q) To a good approxlmatlon this function 1s given by
g2”(q)
=
1+0256q
for05
1 577 + 0.0253 q
for25
( 1754
q L 2 5, (10)
forq L 7.
The remammg valence electrons must thus decrease m number so that the total number of valence electrons becomes 16 for large q The appropriate factor for the other terms m the insulator model 1s found to be l-00445q g”(q)
=
for05
qs
0 9300 -
0 0127 q
for22
0 8488 -
0 00112 q
for7L
i 0.8253
q5
22,
21,and
(11)
for q > 21
The lmagmary part of the dielectric response function for the valence electrons, mcludmg the above modlflcatlons, 1s thus
136
w!L~)
=
,fl4(Q,W + g2”(q) F EJ2(4 ,a)
h?“(4)
I=1 1
(12)
where the summation index corresponds to the parameters 1n Table 1 The total e2 for the system 1s
Ez(qw)
= EZ(q,m) +
c1 ((2~‘,ll,)[df,(q,w)/dwl}
(13)
This function now has the property that the general sum rule Jun_N1 (0) = *N from eqn (5), which 1svahd for any q when e2 1s a function of both q and ~3,yields N = 30 for S102 as described by eqn (13) ENERGY-LOSSFUNCTION The contrlbutlons to e2 due to inner-shell ionization have been discussed 1n the previous Section 1n connection with modlf1catlons of the insulator model The function E~(~,w), eqn (13), may now be used 1n the KramersKyqn1g relation, eqn (61, to determine e1 (q,o) and thus the total energy1 >s function Im [-l/e(q,w)] = e2/(e2, + e$J Since e1 = 1 at energies where “he inner shells begm contributing to e2, and e2+Z i, the energy-loss function may be written, to a good approxlmatlon, as
Im
I-~IWw)I
= Im W/Ww)l
+ 1
{(2~21Z,lW)[df,(q,W)/dWl)
(14)
I
where the superscnpt v indicates the energy-loss function determined from e; (q,o), eqn. (12) This form was used earlier 1n the q = 0 lmut to evaluate the mean excitation energy for S102 The approxrmatlons involved 1n usmg this form for the energy-loss function are discussed 1n Append= I The energy-loss sum rule, eqn. (7), has been evaluated using eqn (14) for several values of q Excellent agreement (well within 1%) was found with the expected value N = 30 In Fig. 6 the valence electron contmbutlon to the energy-loss function 1s shown for several values of qao where a0 = 0.529 a 1sthe Bohr radius Some mformatlon 1s avdable from electron energy-loss measurements for compansons Melxner et al [27] have measured the energy-loss function for amorphous S102 up to *cc, -25 eV for several momentum transfers from 0 to 1 ii-1 Even though our calculations are based on optical data for crystalline rather than amorphous S102, the remarkable snnllanty of the optical data for the two different forms [ 153 permits a general comparison of the energyloss functions Both the model predlctlons shown 1n Fig. 6 and the expenmental data [27] show that the detaled structure present 1n Im(--I/e) at q = 0 1s dissipated rapidly as q increases and only small remnants persist at q = 1 ii-1 (qao = 0.529) As q increases, the position of the mam peak 1n the
137
Fig 6 Energy-loss function for valence electrons predlcted by the insulator model for several values of momentum transfer energy-loss function moves to higher energies In the model calculations the shift from the q = 0 posltlon IS -1 3 eV at q = 0 8 am1 (qao = 0 423) and -1 5 eV at q = 1 a-l, compared to -1 eV at q = 1 ii-l m the expemment (fig 4 of ref 27) Overall, reasonable agreement 1s found between the model predlctlons and the expenmental results For q S 1, the peak m the energy-loss function occurs at G) = q2/ 2, correspondmg to energy transfer to a free electron The mdth of the peak 1s due mamly to the residual mfluence of electron bmdmg [28] For very large q, the contrlbutlon to the energy-loss function from the valence electrons 1s Jomed by that from the inner-shell electrons to form the “Bethe-ridge” feature famlhar from stopping-power theory [ 25,281
EXCHANGE-CORRECTED
DIMFP’S
The effect of exchange between the incident electron and the electrons m the medium 1s included m a simple manner based on the form of the Mott formula (non-relatmlstlc Moller formula) for scattermg of an incident electron by a free electron. The cross+ectlbn for fmdmg-a scattered electron with energy W per urnt energy interval 1s given by [ 29,301 (15) for an mcldent electron of energy E The third term on the r@t m eqn. (15) 1s due to the electron exchange effect The factor A in this term has a value of -1 except for enerses close to W = 0 and W = E. Near those values, A 1s a rapidly osclllatmg function which makes the third term on the nght-hand side of eqn (15) small compared with the first or second term, respectively
138
The widths of the levels from which the valence and Inner-shell electrons are excited are assumed to be quite narrow Thus the DIMFP for production of a secondary electron mth energy E, 1ssven by 7;tE,E,)
=
TJ(E,%,
+
ES1
(16)
where wBJ 1s the magnitude of the bmdmg energy of the jth level. The exchangecorrected DIMFP IStaken as
~Jex=(E,w)= rJ(E,u) x
+E-w)-[~-(~~~]
+ T~(E,w~~
f7J(&u)TJ(&&BJ
+
E-
CJJ)]"~
(17)
Smce 5 0~(Ea2 )-l for large E and w, eqn (17) reduces m this limit to the form given by eqn. (15) The factor I(w,,/E)~'~ reduces the contrlbutlon of the third term m eqn (17) as E + For the valence electrons, q is defined through the equation c+~
T”(E,o)
=
1J
ST%cJ lq+dq {(E;),/[(e;)2 4
TJ(E,w)
+ (e;)21)
(18)
Q-
usmg eqn (1), where (E;)~ 1s the contnbutlon to E; from the Ith tlghtbmdmg level (see Table 1) For a sven inner-shell level, ~J(J%“-')
=
2m4, Ew
Iq-‘+dqlWs,Wd4
(19)
Q
At the higher electron eneraes considered m this work, small relatlvlstlc corrections are expected These are included by replacing the E appearmg exphcltly on the nght-hand side of eqns (18) and (19) by v2/2 = E( 1 + E/2c2 )/(I -I- E/c2)" where v 1s the electron velocity and c = 137 02, and redefmmg the louts on the integrals by qk
=
[2E(l
+ E/2c2 )] 1’2 f [2E(l
+ E/2c2)
-22w(l
+ E/c2) + w2/c2]l/2
Most of the (small) relatlvlstlc correction arises from the replacement of l/E by 2/v2 outside the integrals m eqns (18) and (19) Corrections to the form of E when an electron m the medium attams a relatlvlstlc velocity due to a large-momentum-transfer colhslon are not included [ 311 If we now define the more energetic of the two electrons after colllslon to be the prunary and account for electron exchange through eqn (17), the contibutlon to the inverse mean free path of an electron for excltatlon of electrons from the Ith level 1s given by
pJ(E) = 1" +wBJ)'2dwTexc (E,w) wm
cw
139
Smularly, for the contrlbutlon to the energy loss of an electron per unit path length, or to the stoppmg power of the solid, due to excltatlon of electrons from the Ith level, we have S,(E)
= 1’” + ‘J@‘~ dw o qexc(E,w) (21) WB The total melastlc mverse mean free path and stopping power are gwen by
P(E) =
c P,(E)
(22)
c S,(E)
(23)
and
WV
=
where the sum over 1 covers both the valence and inner-shell electrons Electron energes are measured from the bottom of the conduction band
ELECTRON
MEAN
FREE PATHS
The results for melastlc mverse mean free paths are shown m Table 3. The valence electron contrlbutlon 1s the result of the new formulation presented above while the inner-shell contnbutlons are taken from our earher calculations [ 121 The mner-shell contrlbutlons have been corrected to account approximately for relatrvlstlc effects as discussed above In Fig 7 the melsstic mean free path A E l/p 1s plotted as a function of incident electron energy The mean free paths predicted here are -20% greater than our earlier results [ 121 for E 5 80 eV For comparison, predicted mean free paths for electrons m silicon (p = 2 33 gcmW3 ) are also shown, where the dashed curve 1s from ref 12 and the dot-dash curve 1s from ref 32 The dlfference between these two calculations for silicon 1s 5 3% for E 2 60 eV The curve for SI has approxunately the same energy dependence as the curve for S102 at the hgher ener@es. For very low election eneraes, mean free paths m SIOz are much larger than those m Sl, due to the difference m the band gaps. Experunental measurements of electron mean free paths from three sources are shown m Fig 7. The sources of these data are open cmzles, Fhtsch and Rasder [ 333 , triangles, Klasson et al [34] , and solid circle, Hill et al [35]. Reasonably good agreement 1s found between the theoretIcal result and the experunental measurements. The data of Fhtsch and Rader [33] are consistent with the energy dependence of the model calculation but he 20-25s below our predicted values. Although it 1s beyond the scope of this paper to provide a cntlque of such compansons, at least two pomts
0 587 137 4 18 7 29 110 119 119 10 8 9 65 7 89 6 67 5 14 4 22 3 60 2 14 1,23 0 881 0 694 0 577
Total p(10-2
ii-‘)
0 587 137 4 18 7 29 110 119 119 10 7 9 54 7 71 6 47 4 94 4 03 3 42 2 01 115 0 822 0 646 0 537
VAL( x 10’ )
-
-
-
-
. .._
_
-
to 1-1m 8-l contrlbutlon
0 526 102 158 176 174 164 150 104 0 646 0 481 0 388 0 327
sl(L2,3)(x
Contrlbutlons a to p (8-l)
a For ease of tabulation the mdwldual contrlbutlons column Thus, for example, the oxygen Kshell [O(K)]
20 30 40 60 80 100 150 200 300 400 600 800 1000 2000 4000 6000 8000 10 000
15
WeV)
Electron energy )
0 a22 2 25 2 71 2 72 2 50 2 28 154 0 940 0 692 0 554 0 464
Sl(Ll)(X
104)
0 2 4 5 4 3 3 2
241 71 18 86 88 99 39 95
O(K)(X 105)
0 2 2 2 2
188 32 68 64 50
SI(K)(X 106)
-
.- -
have been multlphed by the power of ten mdlcated in each to 1-1at 800 eV is 2 71 X lo-’ 8-l
lo3
INVERSE MEAN FREE PATHS OF ELECTRONS IN &02 (p = 2 65 g ems)
TABLE 3
141
400
40'
I
I
1111111
I
I
i02
1111111
401
I
,
,,,,,J 404
EleV)
Fig 7 The sohd curve IS the melastlc mean free path as a function of electron energy for SlOz at p = 2 65 gcmm3 Also shown for comparison 18 A for electrons m Si at p = 2 33gcme3, the dashed curve 1s from ref 12 and the dot-dash curve from ref 32 Experimental data are from 0, ref 33, A, ref 34, and l , ref 35
should be kept m mmd. First, the value of the mean free path at a even energy will depend on the density of the material (1 e , whether the matenal IS amorphous, crystallme, etc.) No attempt has been made to account for these differences m making the comparisons m Fig. 7 (frequently the density of the SrOz used m the experiment 1s not reported) Second, the theoretical mean free path 1s due to vtelusttc processes, while the reported data may mclude contrlbutlons from elastic processes as well. An analysis of the role of elastic scattermg m extracting melastlc mean free paths from X-ray photoemlsslon data has been gwen recently by Baschenko and Nefedov 1361. The mcreasmg interest m careful measurements of electron mean free paths and the growmg avsulablllty of theoretical models for predlctmg mean free paths 1s nnportant for more detzuled, future compmsons.
STOPPING POWER FOR ELECTRONS
The results for the stopping power of SIOz at a density p = 2.65 g cme3 are shown m Table 4. The valence electron contrlbutlon 1s calculated from the new formulation Inner-shell contrlbutlons are taken from our earlier calculations [ 123 and corrected shghtly to account for relatlvlstlc effects. For the smaller electron enerses, the stopping power 1s determined entirely by mteractlons with the valence electrons. As the mcldent electron energy mcreases, lomzatlon of the mner shells becomes mcreasmgly Important m the stoppmg process and accounts for -30% of the total stopping power at 10 keV In Fig. 8 the total stoppmg power 1s.displayed as S’ = S/p m MeV cm2 g-l
142
IO3 E (eV)
Fig 8 The stoppmg power S’ = S/p of SlO2 curve 18 the Bethe theory result for I = 143 eV
for an electron
of energy
E
The dashed
Also shown 1s the total inner-shell contnbutlon. For E 2 10 keV the Bethe theory calculations of Pages et al. 1373, recalculated with I = 143 eV, are gwen by the dashed curve. The two theoretical calculations differ by - 1 5% at 10 keV If the Bethe theory calculation 1s extended to lower enerses, the difference between the two theoretical predlctlons mcreases to -5% at 1000 eV There appear to be no expernnental data avdable for comparison
CONCLUSIONS
A model for the response of the valence electrons m SIOz to energy and momentum transfers has been combmed with generalized oscillator strengths for mner-shell lonlzatlon to determine the energy-loss function for SIOz Various sum-rules have been employed to constram and check the overall behavior of this function Differential mverse mean free paths for melastlc processes were calculated from the energy-loss function and used to calculate electron melastlc mean free paths m SIOz and the stopping power of SIOz for electrons. It has been assumed nnphcltly that the medium 1s sufficiently extended that an “average” property such as stopping power 1s useful m descnbmg the electron energy loss For studies of the dlstnbutlon of energy deposited m the medium, the differential mverse mean free paths form part of the Input data for Monte Carlo calculations as described earlier for J&O [19,31] and Sl [ 18] The results of the calculations described m this paper wrll be mcorporated m future studies of the detds of energy deposltlon 111 thm layers of Si02
ACKNOWLEDGEMENTS
This research was sponsored Jointly by the Deputy for Electronic Tech-
0 0636 0 176 0 693 146 277 3 39 3.63 373 3 63 3 36 3.09 2 64 2 32 2 07 140 0 887 0 666 0 541 0 461
Total S (eV a-’
)
0 0636 0 176 0 693 146 2 77 3 39 3 63 3 67 3 48 3 08 2 74 2 25 191 167 106 0 640 0 470 0 375 0 316
VAL( x 10’ )
0 606 134 2 37 2 89 3 23 3 27 3 14 2 44 162 124 102 0 874
w&,3)(X
Contnbutlonsa to S (eV A-’ )
FOR ELECTRONS
10’
)
135 424 5 62 6.44 6 44 6 22 4 98 3 43 2 68 2 22 191
Sl(LI
)(X
lo2 )
0 134 163 2 68 4 61 4 54 3 92 3 44 3 07
O(K)( x lo2 )
0 354 5 34 634 7 38 7 38
Sl(K)(X lo3 )
a For ease of tabulation the mdlvldual contnbutlons to S m eV A-’ have been multiplied by the power of ten mdlcated m each column Thus, for example, the contnbutlon of the slhcon &-shell [Sl (L1 )] to S at 600 eV 1s 6 44 X 10e2 eVA_’
15 20 30 40 60 80 100 150 200 300 400 600 800 1000 2000 4000 6000 8000 10 000
E(eV)
Electron energy
STOPPING POWER OF &Oz (p = ‘2 65 gcnC3)
TABLE 4
145
I
-I
I
I
I
I
0
I
I
I
III1
5
l
I 15
i0 w(au)
F1g Al
Contrlbutlons
to eI (w)
from the L,
and
LZ,J
shells
of S11n S102
value of AeI 1s -0 029, at an energy correspondmg to the bmdmg energy of the L2,3 shell At w = 0, Ael = 0 0082, and for very large W, Ael N-O& / w2 , where wPL 1s the plasma energy correspondmg to the density of L-shell electrons. The function C(w) can be approximated to wlthm a few percent by The result for C(w) calculated from this approxlC(w) ~2e~AeJe~1~ mate form 1s displayed m Fig. A2 Some small fine-structure m the curve m the region 0.4 < c3 < 0.6 and at w = 5 5 has been omitted Usmg this function, differences between the exact and approximate forms of the energyloss function can be exammed. Near the peak m the energy-loss function at o N 0.9 (see Fig 5), C(w) = 0 014, so the exact form 1s -1% smaller. For o > 0.9, the difference mcreases from -22% to -6% near the first mnershell contrlbutlon at o = 3 98. This mcrease occurs 111a region where the energy-loss function IS decreasmg rapidly and should not produce much error m quantities mvolvmg mtegratlon over the approximate energy-loss w&
IIll
1
III1
1
III
I
6
Fig A2 The function C(w) the energy-loss function
used in dlscusslon
of “exact”
and “approximate”
forms
for
146
function At w = 4, Just above the L2,3 threshold, l/(1 + C) 11:0 94 and l/[(l -t C) 1 E” 1 2] - 1 03. At this energy, Im[-l/e”] == $, so the value of the exact form of the energy-loss function 1s -3% smaller than the approxlmate form. For w = 8, C has 1ts largest negative value and results 1n an -3% mcrease 1n the exact form of the energy-loss function over the approximate form This difference decreases as w mcreases The role of C(w) may be examined m another way The valence electrons may be thought of as moving 1n a polanzable background formed by the mner-shell electrons and thus their response to a given energy transfer 1s modified as shown 1n eqn (A5) The effective dielectric constant of this polanzable background, Ed, may be obtamed from certam “finite-energy” sum rules. Smith and Shales [ 381 showed that
E;
w Im [-~/E(O)]
N
(A71
where ti 1s a value of w two or three times the plasma energy but below the onset for L-shell ionization For S10,, with cj = 3.8, our model predicts fb = 1 03. This 1s close to the value 1 04 found for aluminium 138,393, due to the smulanty of the mner-shell structures of Al and S1 Since the interest here 1s 1n integrals contsumng the energy-loss function as part of the mtegrand, we examine the effect of these correction terms on the energy-loss sum rule and the mean excitation energy At energres where the mner-shell terms begm to contribute to the energy-loss function, ev II. 1 -&v/w2 With C(w) -22~;Ae,/I$’ 12, eqn. (A5) can be approximated by Im(-l/e)
= Im(-l/e’)
+ e? +
c [Im(-l/e)]
w
-CIm(-l/e’)
+ D(0)
+ 2e~[(o&,/02)
-
Ael1 WI
The relative size of the correction term m this equation, D(w), has been discussed above. The difference between the energy-loss sum rules using dw w D (w). Thrs xntegral 1s Im(---l/e) and CIm(-l/~)l approx1s given by _rb” found to be essentially zero for S102, due to cancellation from different portions of the spectrum, so that the energy-loss sum rules usmg the exact form and the approxnnate form differ by < 1% The correction to lnl, eqn. (9), 1s mven by [JF dw wD(w)lnw] /(27r2noN). This correction increases the value of llzr calculated from eqn. (9) by -0.6% and leads to I = 143 3 eV versus 1416 eV found earlier usmg the approxlmate form for Im(-l/e) Changes 1n lnl of this size lead to snnllarly small changes m stopping power according to the Bethe stopping-power formula The function C(o) shown 1n F1g. A2 for Q = 0 1s expected to produce even smaller corrections 1n the energy-loss function for g > 0. Given the
147
uncertamtles m the mput data (optical data and GOS’s), the approximate form for the energy-loss function employed m this paper should be quite adequate for calculations of stopping powers and mean free paths for S102 From the practical standpomt, the amount of data handling required 1s reduced dramatically when the approximate form can be used.
REFERENCES 1
Sokrates T Pantehdes (Ed ), The Physics of SlOz and Its Interfaces Proceedmgs of the Internatlonal Top~al Conference, Pergamon, New York, 1978 2 See for example, J C Plckel and J T Blandford, Jr, Inst Electr Electron Eng , Trans Nucl Scl , NS-25 (1978) 1166 3 J F Ziegler and W A Landford, Science, 206 (1979) 776 4 J Lmdhard, K Dan Vldensk Selsk Mat Fys Medd , 28 (1954) No 8 5 R H Rltchle, Phys Rev, 114 (1959) 644 6 C J Tung, R H Rltchle, J C Ashley and V E Anderson, Inelastlc Interactions of Swift Electrons m Solids, Oak Ridge Natlonal Laboratory Rep ORNL/TM-5188 (1976), available from Nat1 Tech Inf Serv , U S Department of Commerce, Sprmgfield, VA 22161 7 R H Rltchle, C J Tung, V E Anderson and J C Ashley, Radlat Res , 64 (1975) 181 8 J C Ashley, C J Tung and R H Rltchle, Inst Electr Electron Eng , Trans Nucl Scl , NS-22 (1975) 2533 9 C J Tung, J C Ashley, R D Blrkhoff, R H Rltchle and L C Emerson, Phys Rev B, 16 (1977) 3049 10 J C Ashley, C J Tung and R H Rltchle, Inst Electr Electron Eng , Trans Null Scl , NS-25 (1978) 1566 11 L R Painter, E T Arakawa, M W Wllhams and J C Ashley, Radlat Res , 83 (1980) 1 12 C J Tung, J C Ashley, V E Anderson and R H Rltchle, Inverse Mean Free Path, Stopping Power, CSDA Range, and Stragglmg m Slhcon and Sllrcon Dloxlde for Electrons of Energy
148 21 22 23 24 25 26 27
28 29 30 31
32 33 34 35 36 37 38 39
S T Manson, personal commumcatlon E Merzbacher and H W Lems, m S Flugge (Ed ), Handbuch der Physlk, Sprmger, Berlin, 1958, pp 166-192 U Buechner, J Phys C, 8 (1975) 2781 D L GrLscom, J Non-Cryst Sohds, 24 (1977) 155 U Fano, Ann Rev Nucl SIX , 13 (1963) 1 C Tschalar and Hans Blchsel, Phys Rev, 175 (1968) 476 A E Melxner, P M Platzman and M Schluter, m Sokrates T Pantehdes (Ed ), The Physics of SIO~ and Its Interfaces Proceedmgs of the International Toplcal Conference, Pergamon, New York, 1978, pp 85-88 Mltlo Inokutl, Rev Mod Phys ,43 (1971) 297 H A Bethe and Juhus Ashkm, in E Segre (Ed ), Experlmental Nuclear Physics, Vol 1, Wiley, New York, 1953, pp 166-357 A S Davydov, Quantum Mechanms, Addison-Wesley, Readmg, MA, 1968, pp 404-406 R H Rltchle, R N Hamm, J E Turner and H A Wnght, m J Booz and H G Ebert (Eds ), Proc 6th Symp on Mlcrodoslmetry, Brussels, Belgmm, May 22-26, 1978, Harwood Academic Pubhshers, London, 1978, P 345 J C Ashley, C J Tung, R H Rltchle and V E Anderson, Inst Electr Electron Eng , Trans Nucl Scl , NS-23 (1976) 1833 R Fhtsch and S I Raider, J Vat SC] Technol , 12 (1975) 306 M Klasson, A Berndtsson, J Hedman, R N&son, R Nyholm and C Nordhng, J Electron Spectrosc Relat Phenom , 3 (1974) 427 J M Hill, D G Royce, C S Fadley, L F Wagner and F J Grunthaver, Chem Phys Lett ,44 (1976) 225 0 A Baschenko and V I Nefedov, J Electron Spectrosc Relat Phenom ,17 (1979) 406,21(1980) 153 L Pages, E Bertel, H Joffre and L Sklavemtls, At Data, 4 (1972) 1 D Y South and E Shdes, Phys Rev B, 17 (1978) 4689 E Shdes, Talzo Sasakl, Mltlo Inokutl and D Y Smith, Phys Rev B, 22 (1980) 1612
INTERACTION OF LOW-ENERGY SILICON DIOXIDE
J C ASHLEY
ELECTRONS WITH
and V E ANDERSON*
Health and Safety Research Dwwon, Tennessee 3 7830 (U S A )
Oak Radge National Labomtoly,
Oak Rzdge,
(Received 15 April 1981)
ABSTRACT Theoretlcal models and calculations have been combined with expenmental optical data to determine a model energy-loss function for SIOz Sum-rule checks and compansons with experlmental information are made to ensure overall consistency of the model The model energy-loss function IS employed to calculate melastlc inverse mean free paths and stoppmg powers m S102 for electrons with energies
INTRODUCTION
S&on dioxide, m its various solid-state forms, IS a matenal of mdespread mterest and Importance m the electromcs mdustry Studies of its basic physlcal properties are so extensive that entire conferences are devoted to reports of current research [l] For mllltary, space [ 21, and commercial [3] electronics apphcatlons, the performance of devices mvolvmg S102 m a radiation environment may be of critical nnportance No matter what type of mcldent radlatlon 1s mvolved, calculations of energy deposltlon must mclude contributions from electron cascades generated m the material The purpose of this paper 1s to develop a model to describe the melastlc events occurmg as an electron slows down m $10, For the mcldent electron energies considered here (5 10 keV), the mam source of energy loss 1s due to mteractlons with the electrons m the medium The response of a medium to a given energy transfer o and momentum transfer q may be described by a complex dlelectnc function e (q, w). In general, e may be a tensor which depends on the dvectlon of q In this paper it 1s assumed that the medium 1s homogeneous and lsotroplc so that E 1s a scalar quantity which depends on only the magnitude of q and not its dlrectlon If E were known m det4 for * On assignment from Computer Scrences Dlvlslon
O368-2~48/81/0000-0000/$02
50
o 1981
Elsevler Sclentlfic Pubhshmg Company
128
an electron of energy E, the probability of an energy loss o per unit distance traveled could be determmed from [4,5] 7(E,w)
= $
@+‘dq s *_ 4 Im1--1/MLN
(1)
and Im() indicates the lmagmary part of where qr G dz[&? + d(E - a)], () This expression of qk assumes that the energy-momentum-transfer relatlon for a non-relatlvlstlc electron m the solid does not differ appreciably from that for a free electron m vacuum The quantity ~(E,u) 1s also called the dtiferentlal inverse mean free path (DIMFP), smce by mtegratmg it over allowed energy transfers the melastrc inverse mean free path for an electron of energy E 1s obtamed In addition, the quantity c~T(E, w) integrated over allowed values of w gives the energy loss per umt path length or, neglectmg radiation by the electron, the stuppmg power of the medium for an electron of energy E Note that here, and m the followmg, quantities are usually expressed m atomic units (a u ) where 5 = m = e = 1 The results are converted to more conventional units where appropriate The mam task then 1s to fmd a suitable expression for E(~,o) to descnbe the dlelectrlc response of SIOz For the purpose of modelmg, the electrons m the medium are divided mto two groups valence electrons and mner-shell electrons A model-msulator theory 1s used to describe the valence electrons, while generahzed oscillator strengths are used to describe inner-shell lomzabon The theoretical model for e 1s based, m part, on optIcal data for SlOz Vanous sum-rule checks are then made to ensure overall consistency of the model Fmally, electron mean free paths m S102 and the stoppmg power of SlO, for electrons are calculated for E < 10 keV and comparisons made vvlth available expemental data
VALENCE
ELECTRON
MODEL
A model-msulator theory 1s used to describe the valence electrons’ response [6] This model has been employed m several previous calculations [ 7-123 and IS related to that employed by Fry [ 133 m which ground-state wavefunctlons for the valence electrons are described m the tight-bmdmg approxlmatlon, while excited states are represented by orthogonallzed plane waves (OPW) However, m our use of the model the normahzatlon of the OPW excited states was fured by requlrmg that the sum rule
r
dw w Im [e(q,w)] 0
=
2n2 n,
be satlsfled, where n, IS the valence electron density
(2)
129
Fit to optrcal
data
The msulator-model results of ref 12 are improved by carrymg out a more dettied f1t of c2, the 1magmary part of the dlelectnc function for zero momentum transfer, E(O,O) E E(O) = Em + 1e2(0), to the expernnental values of Ed for crystallme S102 measured by Phllllp 114,151 as a function of photon energy H1s expenmental values are plotted 1n F1g 1 as the solid line, and the new f1t to the data obtamed by using eight tight-bmdmg levels and a band gap [ 161 of 8.9 eV ISshown by the dashed lme. The parameters used 1n this f1t are shown 1n Table 1 0 = l-8) The sharp peak at 10.3 eV 1s generally agreed to be due to an exclton transltlon, while the structure at higher ener@es 1s of less certam omgm and 1s described by some workers as due to interband transltlons [17] or, most recently, as bemg due to excltonlc resonances [ 161 The assignment of the processes responsible for these features m e2 may be important m exammmg the detailed energydeposltlon processes m S102 as was done earlier for silicon [18] and water
WV *
The e2 spectrum for the valence electrons IS,however, not complete Usmg the sum rule, eqn (2), m the form do w ~~(0) I=0
= 2r2n0 N&
(3)
with e2 gwen by the msulator model f1t shown m F1g 1,1t 1s found that Ne = 7.87, whereas 16.82 effective valence electrons per S102 molecule
11 6
5
---
EXPERIMENTAL OATA (PHILLIP) INSULATOR MODEL
_
Fig 1 The nnagmary part of the dlelectrw function as a function of photon energy The sold curve USfrom the optlcal data of Phlllrp 1141, the dashed curve 1s the msulatormodel ht
130 TABLE1 PARAMETERS FOR THE INSULATOR MODEL ~=~)DETERMINEDFROMFKTSTOOPTICALDATAFORez
J
1 2 3 4 5 6 7 8 9 10 11 12 13
89
100 109 12 5 13 3 15 0 158 190 210 310 292 310 420
03272 0 3676 04007 04596 0 4908 05515 05809 0 6985 07721 1140 1074 1140 1544
(AS DEFINED
IN REF
6 WITH
cuJ(au I
4
noNj(x
07800 02100 03787 0 5145 04067 06642 05145 06301 09963 2 047 06301 1485 2720
2000 06860 1188 07000 06573 1680 05230 04318 1400 5 290 005840 06100 1612
7 860 2 696 4669 2751 2583 6603 2056 1697 5 502 2079 02295 2 397 6335
103)
are expected In eqn. (3), no IS the number of molecules per umt volume For the crystal density p = 2 65 g cmm3, no IS gwen (m a u ) by no = 3 930 x 10m3. The effective number of valence electrons per molecule 1s determmed from an accountmg of the effectwe number of inner-shell electrons. Table 2 shows the atomic levels of the inner-shell electrons for Sl and 0 with the magnitudes of then bmdmg energies and N,(O), the effectwe number of electrons partlclpatmg m transltlons from the gwen level to the contmuum for zero momentum transfer Q, where &f&J)
=
I0
= da df(q,o)ldw
(4)
In eqn (4), df/dw 1s the generalized oscillator strength (GOS) for a given transltlon. The values of N&(O) were calculated from McGuue’s tables [20] of GOS’s for the K shell of 0, from Manson’s GOS’s [21] for the L shell of Sl, and from GOS’s delnved from hydrogemc wavefunctlons [ 223 for the K shell of Si Thus, of the 30 electrons per SIOZ molecule, 30 - 9 86 2(166) = 16.82 are considered to be “valence electrons” The contnbutlon of the oxygen 2s electrons (bmdmg energy - 29 eV) u mcluded m the e2 spectrum through the relatron (m atomic umts) e2 = (2r2n,/u) (df,/dw) where n, I.S the number of shells of type L per unit volume This contibution to e2, calculated using GOS values from ref 20 and = 2no, 1s shown m Fig. 2. A fit to this curve was made using three terms 122s m the insulator model (parameter sets 11, 12, and 13 m Table 1) Since Jo”dwdf2,/dw = 1.14, this leaves 16 82 - 7 87 - 2(1.14) = 6.67 valence electrons to be accounted for m the e2 spectrum.
131 TABLE
2
MAGNITUDE OF THE BINDING ENERGY AND EFFECTIVE NUMBER OF ELECTRONS PARTICIPATING IN TRANSITIONS TO THE CONTINUUM FOR INNER-SHELL ELECTRONS EIement
Shell
Sl
wz
( 2sJ2 (2P? W2
0
----
i0
INSULATOR
08
Bmdmg energy (eV)
N&f(O)
1829 152 108 537
155 131 7 00 1 66
030 62
015 0
ul” 06
02
1 t
0' 20
I
I
I
25
30
35
40
hw (eV)
Fig 2 The sohd line shows the Imaginary part of e(w) versus photon energy determmed as described m the text The dashed lme IS the msuIator-model fit The inset shows the contnbutlon to ~2 from lonlzatlon of the L1 shell of oxygen
The e2 spectrum was extended to 40 eV usmg Buechner’s data [ 231 denved from electron energy-loss measurements His values were scaled to fit Phtilp’s data at 20 eV, with the result shown m Fig 2 Two addltlonal msulator-model terms have been included accounting for an additional 6.67 electrons (parameter sets 9 and 10 m Table 1). The total msulator-model fit m this energy region 1s shown as the dashed curve m Fig 2 Sum-rule
checks (q = 0)
To check the overall consistency of the insulator-model fit and its predlctlon of e2 we examme some quantltles denved from e2 and compare them wAh available expenmental data Various sum-rule checks are useful here. If we define
s w
N1 (w)
= (1/27r2n0)
0
dw’ o’ e2 (~3’)
(5)
132
then ,$..=N1 (w) = N, the total number of electrons per molecule, rf e2 describes the response of all the electrons 1n the matenal (valence plus inner shell) If e2 describes the valence electrons only, then eqn (5) represents a partial sum-rule and Jls_ N1 (0) = Nsy as given 1n eqn (3) In F1g 3 N1 1s plotted as a function of energy for the valence electrons 1n S102 usmg our insulator-model f1t and for transitions to the contmuum from the L, and L2,3 subshells of S1 usmg Manson’s GOS’s [ 211 The value N1 = 6.18 at 22 eV 1s 1n good agreement vvlth the value 6 24 calculated directly from Phti1ps’s optical data [14]. The shape of the N1 curve for crystalline S102 (Fig 3) 1s snnllar to that for amorphous S102 given 1n ref 15, but predicts smaller values for a given energy, due 1n part to the different densities For large photon energies NT”1approaches 16 82, as 1t should, since this was built into the fitting procedure. The real part of the dlelectnc response function, E1 (o), 1s related to e2 (w) through the Kramers-Kron1g relation e1(w)
=
1 + (2/n)P Jb; dw’[w’e2(W’)/(W’2
- w2)]
(6)
where P mdlcates prmc1paLvalue mtegral Because of the energy denommator, values of el up to - 20 eV are determined mamly by the valence electrons. Usmg the msulator-model f1t for e2, e1 has been calculated from eqn (6) The results are shown 1n F1g 4 by the dashed line, for comparison with Phlll1p’s results [ 141. At IO eV the Insulator model predicts a value e1 2~ 9, somewhat larger than that gzven by the expenmental curve, due to e2 being set to zero for energres below the assumed optical band gap [ 16,241 of 8.9 eV (see Fig. 1) Othermse the shapes and magnitudes of the two curves are 1n good agreement I
I I IllIll
I
I I IIIIII
I
I I Illlll
iI
15 90, VALENCE 10 -3 s
5
ELECTRONS
I II
1
I II
102 tTo(eVl
Fig 3 Effectwe number of electrons partwlpatmg m energy-loss processes #iw The curves saturate at high energies at the values indicated
up to energy
133
For calculations of DIMFP’s, the energy-loss function Im [-l/e(c~,w)] is required It has been evaluated m the hmlt Q = 0 from e1 and e2 as determmed by the insulator model. The results are shown by the dashed lme in Fig 5, for comparison with the energy-loss function determined by Buechner from electron energy-loss measurements (solid lme) 1231 The structure for energies ~15 eV IS sharper m the dashed curve determmed from the insulator model (or from optical data) than that m the solid curve These peaks occur at nearly the same posltlons m the two curves. This difference in structure 1s also true for e2, as shown by fig 7 m ref 12 The mam peak m our predlcted energy-loss function occurs at 23.9 eV, -1 eV above that measured by
4 w3
0 5
0
(0 hw (eV)
Fig 4 The solid curve shows the variation of the real part of the dlelectrlc function with photon energy from Phllhp’s experlmental data [14] The dashed curve was obtained from the insulator-model ht. to Q through the Kramers-Kromg relation
-
(BUECHNER)
Fig 5 The dashed lme shows the energy-loss function predrcted by the msulator The sohd curve IS from electron energy-loss measurements by Buechner [ 231
model
134
Buechner. This could be accounted for 1f the density of the polycrystalllne quartz fell used m the latter experiment [!23] were -8% less than the value 2 65 g crnm3for the quartz used 1n the optical measurements [ 141 However, no densities were given 1n ref 23 so this pomt cannot be checked On the other hand, If 1t 1s assumed that the shape of the mam peak 1n our curve 1s desclnbed approxunately by Im(-l/e) a rw/[(o’ - w$)’ + r2 o* ], then +iWO N 24.7 eV 1s m excellent agreement with the plasma energy ?iw, = 24.8 eV calculated from p = 2 65 g crnm3and NeE = 16 82 As a final check on the energy-loss function predicted by the insulator model, the energy-loss sum rule
I
mdw w Im(-l/e) 0
=
2n2nON
(7)
was evaluated numerically for the valence electrons The value of N 1s found to be N = 16.83,1n excellent agreement with Nefp= 16 82, As a fmal overall check on the lnformatlon incorporated 1n the model for q = 0 (optical limit), including the inner-shell contrlbutlons, the mean excitat1on energy I for S102 was calculated This important constant 1n stoppmgpower theory 1s defined through the equation [ 251 InI = (1/2~2noN)
lomdw w lnw Im(-l/e)
For 1oruzat1on of the inner shells Im(-l/e) eqn (8) may be written
(8) = e2 = EZ[(2x2 n,/w)(df,/dw)]
,
so
InI = (1/27r2noN) + c I
{(nt/n~N)
JOmdww lnw Im(-l/P) lo-
dw
l+WW,/d~)l~
where n, 1s the number of inner shells of type 1 per unit volume, and N = 30 for S102 The approximate form employed here for the energy loss 1s dlscussed 1n Append= I of this paper The first term m eqn (9) 1s the contrlbutlon from the valence electrons, as indicated by the superscript v on E This term 1s evaluated using the msulator model while the second term 1s evaluated using the GOS’s discussed earlier for the mner shells The mean excitation energy 1s found to be I= 1416 eV, m excellent agreement mth the value I = 139 2 eV determined from proton range measurements [ 261. The checks made 1n this Section on the model that WIU be used to desclnbe the response of the electrons 1n S102 to a aven energy transfer (but small momentum transfer) are quite encouragmg. They examine mamly the overall consistency of the proposed model through the sum rules However, the good agreement of the energy-loss function measured expemmentally
135
with the model predlctlon provides a more detded check for small energy transfers Compmson of the predicted and Yneasured” mean excltatlon energy IS particularly unportant for stoppmg-power calculations A small correction to I 1s discussed m Append= I The “extension” of the energyloss function to non-zero momentum transfers IS discussed m the next Section Extenston to g # 0 With the parameters determined from optical data, the insulator model [6] gives a prescrlptlon for extension of e2 to non-zero momentum transfer, i.e. E2(W) s eZ(O,a) -+ eZ(q,w) On e feature of the insulator model 1s that the effective number of valence electrons, as determined by the e2 sum rule, eqn. (2) or eqn (3), IS the same for any value of q, while lr dwdf,/dw approaches the occupation numbers for the inner shells (e g , two for K shells) for large q. Thus, as the effective number of inner-shell electrons mcreases vvlth mcreasmg q, the effective number of valence electrons must decrease from 16 82 at q = 0 to 16 for large q. The function N&(q), eqn (3), has been determined from GOS’s for the inner shells, and for the oxygen 2s electrons, which were included with the valence electrons The valence electrons are treated as two groups. The oxygen 2s electrons, which were mcluded mth the valence electrons, will be made to increase from 2.24 to 4 effective electrons per SIOZ molecule, based on N,(q) for this group Smce e2(q,u) = 2;:& eJ2(q,w) from the insulator model, the three terms m this sum descrlbmg the oxygen 2s electrons are thus multlphed by the function Nd(q)/NM(0) -g%(q) To a good approxlmatlon this function 1s given by
g2”(q)
=
1+0256q
for05
1 577 + 0.0253 q
for25
( 1754
q L 2 5, (10)
forq L 7.
The remammg valence electrons must thus decrease m number so that the total number of valence electrons becomes 16 for large q The appropriate factor for the other terms m the insulator model 1s found to be l-00445q g”(q)
=
for05
qs
0 9300 -
0 0127 q
for22
0 8488 -
0 00112 q
for7L
i 0.8253
q5
22,
21,and
(11)
for q > 21
The lmagmary part of the dielectric response function for the valence electrons, mcludmg the above modlflcatlons, 1s thus
136
w!L~)
=
,fl4(Q,W + g2”(q) F EJ2(4 ,a)
h?“(4)
I=1 1
(12)
where the summation index corresponds to the parameters 1n Table 1 The total e2 for the system 1s
Ez(qw)
= EZ(q,m) +
c1 ((2~‘,ll,)[df,(q,w)/dwl}
(13)
This function now has the property that the general sum rule Jun_N1 (0) = *N from eqn (5), which 1svahd for any q when e2 1s a function of both q and ~3,yields N = 30 for S102 as described by eqn (13) ENERGY-LOSSFUNCTION The contrlbutlons to e2 due to inner-shell ionization have been discussed 1n the previous Section 1n connection with modlf1catlons of the insulator model The function E~(~,w), eqn (13), may now be used 1n the KramersKyqn1g relation, eqn (61, to determine e1 (q,o) and thus the total energy1 >s function Im [-l/e(q,w)] = e2/(e2, + e$J Since e1 = 1 at energies where “he inner shells begm contributing to e2, and e2+Z i, the energy-loss function may be written, to a good approxlmatlon, as
Im
I-~IWw)I
= Im W/Ww)l
+ 1
{(2~21Z,lW)[df,(q,W)/dWl)
(14)
I
where the superscnpt v indicates the energy-loss function determined from e; (q,o), eqn. (12) This form was used earlier 1n the q = 0 lmut to evaluate the mean excitation energy for S102 The approxrmatlons involved 1n usmg this form for the energy-loss function are discussed 1n Append= I The energy-loss sum rule, eqn. (7), has been evaluated using eqn (14) for several values of q Excellent agreement (well within 1%) was found with the expected value N = 30 In Fig. 6 the valence electron contmbutlon to the energy-loss function 1s shown for several values of qao where a0 = 0.529 a 1sthe Bohr radius Some mformatlon 1s avdable from electron energy-loss measurements for compansons Melxner et al [27] have measured the energy-loss function for amorphous S102 up to *cc, -25 eV for several momentum transfers from 0 to 1 ii-1 Even though our calculations are based on optical data for crystalline rather than amorphous S102, the remarkable snnllanty of the optical data for the two different forms [ 153 permits a general comparison of the energyloss functions Both the model predlctlons shown 1n Fig. 6 and the expenmental data [27] show that the detaled structure present 1n Im(--I/e) at q = 0 1s dissipated rapidly as q increases and only small remnants persist at q = 1 ii-1 (qao = 0.529) As q increases, the position of the mam peak 1n the
137
Fig 6 Energy-loss function for valence electrons predlcted by the insulator model for several values of momentum transfer energy-loss function moves to higher energies In the model calculations the shift from the q = 0 posltlon IS -1 3 eV at q = 0 8 am1 (qao = 0 423) and -1 5 eV at q = 1 a-l, compared to -1 eV at q = 1 ii-l m the expemment (fig 4 of ref 27) Overall, reasonable agreement 1s found between the model predlctlons and the expenmental results For q S 1, the peak m the energy-loss function occurs at G) = q2/ 2, correspondmg to energy transfer to a free electron The mdth of the peak 1s due mamly to the residual mfluence of electron bmdmg [28] For very large q, the contrlbutlon to the energy-loss function from the valence electrons 1s Jomed by that from the inner-shell electrons to form the “Bethe-ridge” feature famlhar from stopping-power theory [ 25,281
EXCHANGE-CORRECTED
DIMFP’S
The effect of exchange between the incident electron and the electrons m the medium 1s included m a simple manner based on the form of the Mott formula (non-relatmlstlc Moller formula) for scattermg of an incident electron by a free electron. The cross+ectlbn for fmdmg-a scattered electron with energy W per urnt energy interval 1s given by [ 29,301 (15) for an mcldent electron of energy E The third term on the r@t m eqn. (15) 1s due to the electron exchange effect The factor A in this term has a value of -1 except for enerses close to W = 0 and W = E. Near those values, A 1s a rapidly osclllatmg function which makes the third term on the nght-hand side of eqn (15) small compared with the first or second term, respectively
138
The widths of the levels from which the valence and Inner-shell electrons are excited are assumed to be quite narrow Thus the DIMFP for production of a secondary electron mth energy E, 1ssven by 7;tE,E,)
=
TJ(E,%,
+
ES1
(16)
where wBJ 1s the magnitude of the bmdmg energy of the jth level. The exchangecorrected DIMFP IStaken as
~Jex=(E,w)= rJ(E,u) x
+E-w)-[~-(~~~]
+ T~(E,w~~
f7J(&u)TJ(&&BJ
+
E-
CJJ)]"~
(17)
Smce 5 0~(Ea2 )-l for large E and w, eqn (17) reduces m this limit to the form given by eqn. (15) The factor I(w,,/E)~'~ reduces the contrlbutlon of the third term m eqn (17) as E + For the valence electrons, q is defined through the equation c+~
T”(E,o)
=
1J
ST%cJ lq+dq {(E;),/[(e;)2 4
TJ(E,w)
+ (e;)21)
(18)
Q-
usmg eqn (1), where (E;)~ 1s the contnbutlon to E; from the Ith tlghtbmdmg level (see Table 1) For a sven inner-shell level, ~J(J%“-')
=
2m4, Ew
Iq-‘+dqlWs,Wd4
(19)
Q
At the higher electron eneraes considered m this work, small relatlvlstlc corrections are expected These are included by replacing the E appearmg exphcltly on the nght-hand side of eqns (18) and (19) by v2/2 = E( 1 + E/2c2 )/(I -I- E/c2)" where v 1s the electron velocity and c = 137 02, and redefmmg the louts on the integrals by qk
=
[2E(l
+ E/2c2 )] 1’2 f [2E(l
+ E/2c2)
-22w(l
+ E/c2) + w2/c2]l/2
Most of the (small) relatlvlstlc correction arises from the replacement of l/E by 2/v2 outside the integrals m eqns (18) and (19) Corrections to the form of E when an electron m the medium attams a relatlvlstlc velocity due to a large-momentum-transfer colhslon are not included [ 311 If we now define the more energetic of the two electrons after colllslon to be the prunary and account for electron exchange through eqn (17), the contibutlon to the inverse mean free path of an electron for excltatlon of electrons from the Ith level 1s given by
pJ(E) = 1" +wBJ)'2dwTexc (E,w) wm
cw
139
Smularly, for the contrlbutlon to the energy loss of an electron per unit path length, or to the stoppmg power of the solid, due to excltatlon of electrons from the Ith level, we have S,(E)
= 1’” + ‘J@‘~ dw o qexc(E,w) (21) WB The total melastlc mverse mean free path and stopping power are gwen by
P(E) =
c P,(E)
(22)
c S,(E)
(23)
and
WV
=
where the sum over 1 covers both the valence and inner-shell electrons Electron energes are measured from the bottom of the conduction band
ELECTRON
MEAN
FREE PATHS
The results for melastlc mverse mean free paths are shown m Table 3. The valence electron contrlbutlon 1s the result of the new formulation presented above while the inner-shell contnbutlons are taken from our earher calculations [ 121 The mner-shell contrlbutlons have been corrected to account approximately for relatrvlstlc effects as discussed above In Fig 7 the melsstic mean free path A E l/p 1s plotted as a function of incident electron energy The mean free paths predicted here are -20% greater than our earlier results [ 121 for E 5 80 eV For comparison, predicted mean free paths for electrons m silicon (p = 2 33 gcmW3 ) are also shown, where the dashed curve 1s from ref 12 and the dot-dash curve 1s from ref 32 The dlfference between these two calculations for silicon 1s 5 3% for E 2 60 eV The curve for SI has approxunately the same energy dependence as the curve for S102 at the hgher ener@es. For very low election eneraes, mean free paths m SIOz are much larger than those m Sl, due to the difference m the band gaps. Experunental measurements of electron mean free paths from three sources are shown m Fig 7. The sources of these data are open cmzles, Fhtsch and Rasder [ 333 , triangles, Klasson et al [34] , and solid circle, Hill et al [35]. Reasonably good agreement 1s found between the theoretIcal result and the experunental measurements. The data of Fhtsch and Rader [33] are consistent with the energy dependence of the model calculation but he 20-25s below our predicted values. Although it 1s beyond the scope of this paper to provide a cntlque of such compansons, at least two pomts
0 587 137 4 18 7 29 110 119 119 10 8 9 65 7 89 6 67 5 14 4 22 3 60 2 14 1,23 0 881 0 694 0 577
Total p(10-2
ii-‘)
0 587 137 4 18 7 29 110 119 119 10 7 9 54 7 71 6 47 4 94 4 03 3 42 2 01 115 0 822 0 646 0 537
VAL( x 10’ )
-
-
-
-
. .._
_
-
to 1-1m 8-l contrlbutlon
0 526 102 158 176 174 164 150 104 0 646 0 481 0 388 0 327
sl(L2,3)(x
Contrlbutlons a to p (8-l)
a For ease of tabulation the mdwldual contrlbutlons column Thus, for example, the oxygen Kshell [O(K)]
20 30 40 60 80 100 150 200 300 400 600 800 1000 2000 4000 6000 8000 10 000
15
WeV)
Electron energy )
0 a22 2 25 2 71 2 72 2 50 2 28 154 0 940 0 692 0 554 0 464
Sl(Ll)(X
104)
0 2 4 5 4 3 3 2
241 71 18 86 88 99 39 95
O(K)(X 105)
0 2 2 2 2
188 32 68 64 50
SI(K)(X 106)
-
.- -
have been multlphed by the power of ten mdlcated in each to 1-1at 800 eV is 2 71 X lo-’ 8-l
lo3
INVERSE MEAN FREE PATHS OF ELECTRONS IN &02 (p = 2 65 g ems)
TABLE 3
141
400
40'
I
I
1111111
I
I
i02
1111111
401
I
,
,,,,,J 404
EleV)
Fig 7 The sohd curve IS the melastlc mean free path as a function of electron energy for SlOz at p = 2 65 gcmm3 Also shown for comparison 18 A for electrons m Si at p = 2 33gcme3, the dashed curve 1s from ref 12 and the dot-dash curve from ref 32 Experimental data are from 0, ref 33, A, ref 34, and l , ref 35
should be kept m mmd. First, the value of the mean free path at a even energy will depend on the density of the material (1 e , whether the matenal IS amorphous, crystallme, etc.) No attempt has been made to account for these differences m making the comparisons m Fig. 7 (frequently the density of the SrOz used m the experiment 1s not reported) Second, the theoretical mean free path 1s due to vtelusttc processes, while the reported data may mclude contrlbutlons from elastic processes as well. An analysis of the role of elastic scattermg m extracting melastlc mean free paths from X-ray photoemlsslon data has been gwen recently by Baschenko and Nefedov 1361. The mcreasmg interest m careful measurements of electron mean free paths and the growmg avsulablllty of theoretical models for predlctmg mean free paths 1s nnportant for more detzuled, future compmsons.
STOPPING POWER FOR ELECTRONS
The results for the stopping power of SIOz at a density p = 2.65 g cme3 are shown m Table 4. The valence electron contrlbutlon 1s calculated from the new formulation Inner-shell contrlbutlons are taken from our earlier calculations [ 123 and corrected shghtly to account for relatlvlstlc effects. For the smaller electron enerses, the stopping power 1s determined entirely by mteractlons with the valence electrons. As the mcldent electron energy mcreases, lomzatlon of the mner shells becomes mcreasmgly Important m the stoppmg process and accounts for -30% of the total stopping power at 10 keV In Fig. 8 the total stoppmg power 1s.displayed as S’ = S/p m MeV cm2 g-l
142
IO3 E (eV)
Fig 8 The stoppmg power S’ = S/p of SlO2 curve 18 the Bethe theory result for I = 143 eV
for an electron
of energy
E
The dashed
Also shown 1s the total inner-shell contnbutlon. For E 2 10 keV the Bethe theory calculations of Pages et al. 1373, recalculated with I = 143 eV, are gwen by the dashed curve. The two theoretical calculations differ by - 1 5% at 10 keV If the Bethe theory calculation 1s extended to lower enerses, the difference between the two theoretical predlctlons mcreases to -5% at 1000 eV There appear to be no expernnental data avdable for comparison
CONCLUSIONS
A model for the response of the valence electrons m SIOz to energy and momentum transfers has been combmed with generalized oscillator strengths for mner-shell lonlzatlon to determine the energy-loss function for SIOz Various sum-rules have been employed to constram and check the overall behavior of this function Differential mverse mean free paths for melastlc processes were calculated from the energy-loss function and used to calculate electron melastlc mean free paths m SIOz and the stopping power of SIOz for electrons. It has been assumed nnphcltly that the medium 1s sufficiently extended that an “average” property such as stopping power 1s useful m descnbmg the electron energy loss For studies of the dlstnbutlon of energy deposited m the medium, the differential mverse mean free paths form part of the Input data for Monte Carlo calculations as described earlier for J&O [19,31] and Sl [ 18] The results of the calculations described m this paper wrll be mcorporated m future studies of the detds of energy deposltlon 111 thm layers of Si02
ACKNOWLEDGEMENTS
This research was sponsored Jointly by the Deputy for Electronic Tech-
0 0636 0 176 0 693 146 277 3 39 3.63 373 3 63 3 36 3.09 2 64 2 32 2 07 140 0 887 0 666 0 541 0 461
Total S (eV a-’
)
0 0636 0 176 0 693 146 2 77 3 39 3 63 3 67 3 48 3 08 2 74 2 25 191 167 106 0 640 0 470 0 375 0 316
VAL( x 10’ )
0 606 134 2 37 2 89 3 23 3 27 3 14 2 44 162 124 102 0 874
w&,3)(X
Contnbutlonsa to S (eV A-’ )
FOR ELECTRONS
10’
)
135 424 5 62 6.44 6 44 6 22 4 98 3 43 2 68 2 22 191
Sl(LI
)(X
lo2 )
0 134 163 2 68 4 61 4 54 3 92 3 44 3 07
O(K)( x lo2 )
0 354 5 34 634 7 38 7 38
Sl(K)(X lo3 )
a For ease of tabulation the mdlvldual contnbutlons to S m eV A-’ have been multiplied by the power of ten mdlcated m each column Thus, for example, the contnbutlon of the slhcon &-shell [Sl (L1 )] to S at 600 eV 1s 6 44 X 10e2 eVA_’
15 20 30 40 60 80 100 150 200 300 400 600 800 1000 2000 4000 6000 8000 10 000
E(eV)
Electron energy
STOPPING POWER OF &Oz (p = ‘2 65 gcnC3)
TABLE 4
145
I
-I
I
I
I
I
0
I
I
I
III1
5
l
I 15
i0 w(au)
F1g Al
Contrlbutlons
to eI (w)
from the L,
and
LZ,J
shells
of S11n S102
value of AeI 1s -0 029, at an energy correspondmg to the bmdmg energy of the L2,3 shell At w = 0, Ael = 0 0082, and for very large W, Ael N-O& / w2 , where wPL 1s the plasma energy correspondmg to the density of L-shell electrons. The function C(w) can be approximated to wlthm a few percent by The result for C(w) calculated from this approxlC(w) ~2e~AeJe~1~ mate form 1s displayed m Fig. A2 Some small fine-structure m the curve m the region 0.4 < c3 < 0.6 and at w = 5 5 has been omitted Usmg this function, differences between the exact and approximate forms of the energyloss function can be exammed. Near the peak m the energy-loss function at o N 0.9 (see Fig 5), C(w) = 0 014, so the exact form 1s -1% smaller. For o > 0.9, the difference mcreases from -22% to -6% near the first mnershell contrlbutlon at o = 3 98. This mcrease occurs 111a region where the energy-loss function IS decreasmg rapidly and should not produce much error m quantities mvolvmg mtegratlon over the approximate energy-loss w&
IIll
1
III1
1
III
I
6
Fig A2 The function C(w) the energy-loss function
used in dlscusslon
of “exact”
and “approximate”
forms
for
146
function At w = 4, Just above the L2,3 threshold, l/(1 + C) 11:0 94 and l/[(l -t C) 1 E” 1 2] - 1 03. At this energy, Im[-l/e”] == $, so the value of the exact form of the energy-loss function 1s -3% smaller than the approxlmate form. For w = 8, C has 1ts largest negative value and results 1n an -3% mcrease 1n the exact form of the energy-loss function over the approximate form This difference decreases as w mcreases The role of C(w) may be examined m another way The valence electrons may be thought of as moving 1n a polanzable background formed by the mner-shell electrons and thus their response to a given energy transfer 1s modified as shown 1n eqn (A5) The effective dielectric constant of this polanzable background, Ed, may be obtamed from certam “finite-energy” sum rules. Smith and Shales [ 381 showed that
E;
w Im [-~/E(O)]
N
(A71
where ti 1s a value of w two or three times the plasma energy but below the onset for L-shell ionization For S10,, with cj = 3.8, our model predicts fb = 1 03. This 1s close to the value 1 04 found for aluminium 138,393, due to the smulanty of the mner-shell structures of Al and S1 Since the interest here 1s 1n integrals contsumng the energy-loss function as part of the mtegrand, we examine the effect of these correction terms on the energy-loss sum rule and the mean excitation energy At energres where the mner-shell terms begm to contribute to the energy-loss function, ev II. 1 -&v/w2 With C(w) -22~;Ae,/I$’ 12, eqn. (A5) can be approximated by Im(-l/e)
= Im(-l/e’)
+ e? +
c [Im(-l/e)]
w
-CIm(-l/e’)
+ D(0)
+ 2e~[(o&,/02)
-
Ael1 WI
The relative size of the correction term m this equation, D(w), has been discussed above. The difference between the energy-loss sum rules using dw w D (w). Thrs xntegral 1s Im(---l/e) and CIm(-l/~)l approx1s given by _rb” found to be essentially zero for S102, due to cancellation from different portions of the spectrum, so that the energy-loss sum rules usmg the exact form and the approxnnate form differ by < 1% The correction to lnl, eqn. (9), 1s mven by [JF dw wD(w)lnw] /(27r2noN). This correction increases the value of llzr calculated from eqn. (9) by -0.6% and leads to I = 143 3 eV versus 1416 eV found earlier usmg the approxlmate form for Im(-l/e) Changes 1n lnl of this size lead to snnllarly small changes m stopping power according to the Bethe stopping-power formula The function C(o) shown 1n F1g. A2 for Q = 0 1s expected to produce even smaller corrections 1n the energy-loss function for g > 0. Given the
147
uncertamtles m the mput data (optical data and GOS’s), the approximate form for the energy-loss function employed m this paper should be quite adequate for calculations of stopping powers and mean free paths for S102 From the practical standpomt, the amount of data handling required 1s reduced dramatically when the approximate form can be used.
REFERENCES 1
Sokrates T Pantehdes (Ed ), The Physics of SlOz and Its Interfaces Proceedmgs of the Internatlonal Top~al Conference, Pergamon, New York, 1978 2 See for example, J C Plckel and J T Blandford, Jr, Inst Electr Electron Eng , Trans Nucl Scl , NS-25 (1978) 1166 3 J F Ziegler and W A Landford, Science, 206 (1979) 776 4 J Lmdhard, K Dan Vldensk Selsk Mat Fys Medd , 28 (1954) No 8 5 R H Rltchle, Phys Rev, 114 (1959) 644 6 C J Tung, R H Rltchle, J C Ashley and V E Anderson, Inelastlc Interactions of Swift Electrons m Solids, Oak Ridge Natlonal Laboratory Rep ORNL/TM-5188 (1976), available from Nat1 Tech Inf Serv , U S Department of Commerce, Sprmgfield, VA 22161 7 R H Rltchle, C J Tung, V E Anderson and J C Ashley, Radlat Res , 64 (1975) 181 8 J C Ashley, C J Tung and R H Rltchle, Inst Electr Electron Eng , Trans Nucl Scl , NS-22 (1975) 2533 9 C J Tung, J C Ashley, R D Blrkhoff, R H Rltchle and L C Emerson, Phys Rev B, 16 (1977) 3049 10 J C Ashley, C J Tung and R H Rltchle, Inst Electr Electron Eng , Trans Null Scl , NS-25 (1978) 1566 11 L R Painter, E T Arakawa, M W Wllhams and J C Ashley, Radlat Res , 83 (1980) 1 12 C J Tung, J C Ashley, V E Anderson and R H Rltchle, Inverse Mean Free Path, Stopping Power, CSDA Range, and Stragglmg m Slhcon and Sllrcon Dloxlde for Electrons of Energy
148 21 22 23 24 25 26 27
28 29 30 31
32 33 34 35 36 37 38 39
S T Manson, personal commumcatlon E Merzbacher and H W Lems, m S Flugge (Ed ), Handbuch der Physlk, Sprmger, Berlin, 1958, pp 166-192 U Buechner, J Phys C, 8 (1975) 2781 D L GrLscom, J Non-Cryst Sohds, 24 (1977) 155 U Fano, Ann Rev Nucl SIX , 13 (1963) 1 C Tschalar and Hans Blchsel, Phys Rev, 175 (1968) 476 A E Melxner, P M Platzman and M Schluter, m Sokrates T Pantehdes (Ed ), The Physics of SIO~ and Its Interfaces Proceedmgs of the International Toplcal Conference, Pergamon, New York, 1978, pp 85-88 Mltlo Inokutl, Rev Mod Phys ,43 (1971) 297 H A Bethe and Juhus Ashkm, in E Segre (Ed ), Experlmental Nuclear Physics, Vol 1, Wiley, New York, 1953, pp 166-357 A S Davydov, Quantum Mechanms, Addison-Wesley, Readmg, MA, 1968, pp 404-406 R H Rltchle, R N Hamm, J E Turner and H A Wnght, m J Booz and H G Ebert (Eds ), Proc 6th Symp on Mlcrodoslmetry, Brussels, Belgmm, May 22-26, 1978, Harwood Academic Pubhshers, London, 1978, P 345 J C Ashley, C J Tung, R H Rltchle and V E Anderson, Inst Electr Electron Eng , Trans Nucl Scl , NS-23 (1976) 1833 R Fhtsch and S I Raider, J Vat SC] Technol , 12 (1975) 306 M Klasson, A Berndtsson, J Hedman, R N&son, R Nyholm and C Nordhng, J Electron Spectrosc Relat Phenom , 3 (1974) 427 J M Hill, D G Royce, C S Fadley, L F Wagner and F J Grunthaver, Chem Phys Lett ,44 (1976) 225 0 A Baschenko and V I Nefedov, J Electron Spectrosc Relat Phenom ,17 (1979) 406,21(1980) 153 L Pages, E Bertel, H Joffre and L Sklavemtls, At Data, 4 (1972) 1 D Y South and E Shdes, Phys Rev B, 17 (1978) 4689 E Shdes, Talzo Sasakl, Mltlo Inokutl and D Y Smith, Phys Rev B, 22 (1980) 1612