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Is there a universal mean-free-path curve for electron inelastic scattering in solids?
Journal of Electron Spectroscopy and Related Phenomena, 23 (1981) 97-102 Elsevler Sclentlfic Pubhshmg Company, Amsterdam - Printed m The Netherlands
IS THERE A UNIVERSAL MEAN-FREE-PATH CURVE FOR ELECTRON INELASTIC SCATTERING IN SOLIDS?
J SZAJMAN,
J LIESEGANG,
J G JENKIN and R C G LECKEY
Physrcs Department and Research Centre for Electron Spectroscopy, Unrverslty, Bundoora, Vactona 3083 (Australta)
La Trobe
(Fust received 9 September 1980, m final form 18 December 1980)
ABSTRACT On the baaIs of a recent modlficatlon to dielectric theory, it is shown that the mean free path for melastlc electron scattenng m a mde range of mateaals m described by a formula gnnng a dependence on electron energy of the form AEQ75 where A IS, however, sample-dependent
INTRODUCTION
Based on the lnmted data then avtiable, a number of authors m the early 1970’s suggested the use of a universal curve to descnbe the melastlc-scattenng mean free path X(E) as a function of electron energy E m the range 10 < E < 2000 eV m all solids [l-5] As additional experimental data became avdable, however, it became apparent that this approxlmatlon was madequate for many apphcatlons, for example, m the quantltatlve use of Auger and X-ray photoelectron spectroscopy. Recently, Seah and Dench [63 have complied a data file contammg all known experunental determmatzons of X(E) for sohds, some 350 such measurements have now been made by means of a vmety of experimental techniques One outcome of this study has been the suggestion by Seah and Dench that, for the elemental sohds, X(E) may reasonably be approximated by an emplrlcal curve of the form X(E) = =
538E -2 + 0 41 (mE)1’2 538mK2
(monolayers)
+ 0 13 (,3E)1’2
(a)
(1)
where the monolayer thickness m IS sven by m3 = 1O24/nN, N being the molecular density and n the number of atoms m each molecule From this definition of m3 It can be shown that rns312 a (p/A)lj2, where p IS the density 036%2048/81/0000-0000/$02
50
o 1981
Elsevler Sclentlfic Pubhshmg Company
98
and A the molecular weight The plasmon energy, m electronvolts, for a solid 1s given by E, = 28 8(pZ/A)1/2, where 2 1s the number of valence electrons Hence E, 0: rne312, and the expression (1) 1s therefore dependent on E, Seah and Dench further proposed that solids should be drstmgmshed as being elemental, inorganic compounds, or organic compounds, and that the followmg expression should be used for morgamc compounds X(E)
=
217OE -’
=
2170mE
+ 0 72 (mE)“2
(monolayers)
-2 + 0 23 (m3E)lj2
(2)
(a),
equation (1) being reserved for the elements As the avdable data on organic compounds are still extremely limited and hotly contested, we do not intend to discuss such compounds here By categonzmg materials mto these groups, Seah and Dench showed that, for E > 150 eV, the P m s scatter value unthm each group was -1 37. The quantitative use of eqns (1) and (2) could then proceeed with a confidence limit set by this factor The purpose of the present paper IS to demonstrate the utility of a recently derived formula for X(E) for all sohds [ 71 The denvatlon of this formula depends on the suggestion that, If any dlvlslon of solids mto classes 1s indeed necessary, then such a dlvlslon should be on the basis of the dlstmctlon between metals and msulators We shall be concerned with primary electron energies greater than 300 eV
OUTLINE
OF THEORY
Qumn [8] used the followmg relation to calculate the mean free path of electrons m bulk Jelhum X(k) =
laE(k)/akl
/
! =(k)
(3)
I
where l?(k) 1s the lmagmary part of the electron self-energy, and (Ifi)[ aE(k)/ ak] is the electron velocity Recently, Penn [9] has used the Lundqvlst [lo] form of the self-energy to calculate numencally the value of r(k) and hence the mean free path m metals SzaJman and Leckey [7] extended Penn’s model to morgamc sohds and produced an analytical expression of the form X,(E)
=
[2 lBE~/E~]/[ln(aE/~)
(4)
+ #]
where QC= #
=
Q =
8Q/[(E(4 3E,(Q EF/&.,
1Q2 + 0 6S)/E,)“2
+ 0 11E;/z2)/4Q2j?
+ 0 66Q2 + 0
181
99
and E, and E, are the free-electron values of the Fermi and plasma ener@es respectively. For free-electron-like metals j!? NE,, while for msulators and semxconductors zt has been observed that the energy centrold E of Im(---l/ E(O, 0)), obtamed numencally from optlcal data, may be well approximated by
E=E,+Eg
(5)
where E, 1sthe band gap With this additional approxlmatlon eqn (4) therefore represents a universal curve for X(E) for most solids Although not particularly cumbersome, eqn (4) may be recast mto a form slmllar to that proposed by Seah and Dench (eqns (1) and (2)). The plasma energy 1s usually m the range 5-30 eV, and if an average value of 18 eV 1s assumed, then the denominator m eqn (4) may be slmphfled to ln(E/4) In addltlon, E/ln(E/4) = E 3/4,and consequently
X,(E) = 2 12f?E3'4/E;,
(6)
of the Seah and Dench form but with a shghtly increased energy-dependence It should be noted that if the plasmon energy 1s vaned between 5 and 30 eV, then the term E/ln(4E/E, ) vanes by 15% at most from the average value assumed above If the contrlbutlon of scattenng from core electrons, estimated to be, on average, 15% of the valence-band scattermg [ 41, 1s now mcluded, the total melastlc mean free path 1s finally obtamed as
X(E) = 1 &i?E 3’4 /Es
(ii)
(7)
or
This form of expression for X(E) 1suniversal, m the sense that it applies to a very large group of matenals, but it 1s not universal m the sense mdely used previously, m the present case sample-dependent parameters are clearly included In the follow-mg Section we examme the utility of this expression for the descnptlon of the scattermg of electrons of energy E > 300 eV m a unde vanety of sohds
DISCUSSION
Smce delis of the denvation of eqn. (4) and an exammatlon of its effectiveness m predlctmg X(E) for a number of solids have been sven separately [ 71, we ~11 concentrate here on the success of the more appr0xlmat.e form, eqn (7), for calculatmg A(E) for mdely &ffermg mater&s. We first of all select CdTe, a narrow-band-gap semiconductor (E, = 1 4 eV), which would be characterized as an morgamc compound and treated by Seah and Dench
100
on the basis of eqn (2) As may be seen from Fig 1 and ref 11, the use of eqn (2) senously overestunates X(E) as compared vvlth the experimental values, which, m fact, he close to the predlctlons of eqn (1) (for elemental solids) for this matenal Equation (7) matches experiment well In Fig 2 we have plotted all expenmental mean-free-path values for metals, morgamc msulators and semiconductors from the Seah and Dench data-base m the form h(E) Ei/C, where C now mcludes an element-depen-
l
200
600 ELECTRON
1000 ENERGY
1400 (eV)
Fig 1 Measured mean-free-path values for CdTe as a function of electron energy The various theoretical lines are explained m the text, the Lotz curve IS based on a tlght-bmdmg approxunation
JO,7 ELECTRON
Fig 2 plotted
All expenmental usmg parameters
ENERGY
(eV)
mean-free-path values from the Sesh and Dench compilation, described in the text l , elements, x, morgamc compounds
101
dent correction for core-level scattermg Thus C = 2 12 X,(lOOO)I?-, where X,(1000) 1s the mean free path for core-level scattermg of lOOO-eV electrons, selected as a mid-range correction and calculated usmg the method @ven by Powell [43 outlined m the precedmg paper [ 71 The free-electron plasmon energies were calculated using E, = 28 ~(JLZ/A)'~~and E obtamed from E = E, i-E, using hterature values of the band gap E, where appropriate The solid hne m Fig 2 1s the y = E * 75 dependence of eqn. (7), and the dashed lme (y = 1 73E* 68 ) IS a least-squares frt to the data (above 300 eV), clearly the agreement 1squite reasonable The r m s scatter of the data points around the E* 75 prediction 1s 1 32, which compares favourably with the value (“1 37) obtamed by Seah and Dench, but without the need to separate sohds mto elemental and morgamc compounds The magnitude of the scatter LS, of course, largely dependent on the accuracy of the expemmental procedures used m determmmg the X(E) values, Seah and Dench’s compllatlon mcludes all avdable measurements lrrespectlve of accuracy, whereas the present calculation of the r m s scatter has excluded nme points showmg gross deviations We may conclude therefore that the theory outlined above, even m the approximate form of eqn (7), has been able to impose a degree of order on undely separate expenmental X(E) values, and may be used m quantltatlve apphcations, at least with a confidence level implied by the 1 32 r m s scatter value
Ia
11
1
l
h
0
A%&*
NlzL?
1
(
I
80
Fig 3 (a) Mean-free-path values at 1OOO-eV electron energy as a function of atomw number for metals, calculated using eqn (5). (b) values from (a) dwded by ms2 - the honzontal line IS the average value, 4 2 A-113
102
We have also thought it of some interest to examme the question of why the maJonty of metals have very sunllar X(E) values On the basis of eqn (5), which we expect to be adequate for the maJonty of elements (for a dlscusslon of the tifflcultles associated mth the transitIon metals see SzaJman and Leckey [ 7 3 ), we have calculated the mean free path at lOOO-eV electron energy as a function of atomic number for most metals (Frg 3(a)) If the monolayer-thickness dependence 1s approxunately removed by dlvldmg the X(E) values by m 3/2, then it 1s found (Fig 3(b)) that, with relatively few exceptions, the values so calculated he vvlthm 30% of 4 2 kli2, and a universal curve for the metals, with an energy-dependence close to E” 75, 1s thereby m&cated
REFERENCES
8 9
10 11
E 3auer, Vacuum, 22 (1972) 539 C R Brundle, J Vat SCI, Technol , ll(l974) 212 I Lmdau and W E Spacer, J Electron Spectrosc Relat Phenom ,3 (1974) 409 C J Powell, Surf SIX ,44 (1974) 29 R C G Leckey, Phys Rev A, 13 (1976) 1043 M P Seah and W A Dench, Surf Interface Anal , l(l979) 2 J Szajman and R C G Leckey, J Electron Spectrosc Relat Phenom , 23 (1981) 83-96 (precedmg paper) J J Qumn, Phys Rev, 126 (1962) 1453 D R Penn, J Electron Spectrosc Relat Phenom , 9 (1976) 29, Phys Rev B, 13 (1976) 6248 B I Lundqvlst, Phys Kondens Mater, 6 {1967) 206 J Szaman, R C G Leckey, J Llesegang and J G Jenkm, J Electron Spectrosc Relat Phenom , 20 (1980) 323
IS THERE A UNIVERSAL MEAN-FREE-PATH CURVE FOR ELECTRON INELASTIC SCATTERING IN SOLIDS?
J SZAJMAN,
J LIESEGANG,
J G JENKIN and R C G LECKEY
Physrcs Department and Research Centre for Electron Spectroscopy, Unrverslty, Bundoora, Vactona 3083 (Australta)
La Trobe
(Fust received 9 September 1980, m final form 18 December 1980)
ABSTRACT On the baaIs of a recent modlficatlon to dielectric theory, it is shown that the mean free path for melastlc electron scattenng m a mde range of mateaals m described by a formula gnnng a dependence on electron energy of the form AEQ75 where A IS, however, sample-dependent
INTRODUCTION
Based on the lnmted data then avtiable, a number of authors m the early 1970’s suggested the use of a universal curve to descnbe the melastlc-scattenng mean free path X(E) as a function of electron energy E m the range 10 < E < 2000 eV m all solids [l-5] As additional experimental data became avdable, however, it became apparent that this approxlmatlon was madequate for many apphcatlons, for example, m the quantltatlve use of Auger and X-ray photoelectron spectroscopy. Recently, Seah and Dench [63 have complied a data file contammg all known experunental determmatzons of X(E) for sohds, some 350 such measurements have now been made by means of a vmety of experimental techniques One outcome of this study has been the suggestion by Seah and Dench that, for the elemental sohds, X(E) may reasonably be approximated by an emplrlcal curve of the form X(E) = =
538E -2 + 0 41 (mE)1’2 538mK2
(monolayers)
+ 0 13 (,3E)1’2
(a)
(1)
where the monolayer thickness m IS sven by m3 = 1O24/nN, N being the molecular density and n the number of atoms m each molecule From this definition of m3 It can be shown that rns312 a (p/A)lj2, where p IS the density 036%2048/81/0000-0000/$02
50
o 1981
Elsevler Sclentlfic Pubhshmg Company
98
and A the molecular weight The plasmon energy, m electronvolts, for a solid 1s given by E, = 28 8(pZ/A)1/2, where 2 1s the number of valence electrons Hence E, 0: rne312, and the expression (1) 1s therefore dependent on E, Seah and Dench further proposed that solids should be drstmgmshed as being elemental, inorganic compounds, or organic compounds, and that the followmg expression should be used for morgamc compounds X(E)
=
217OE -’
=
2170mE
+ 0 72 (mE)“2
(monolayers)
-2 + 0 23 (m3E)lj2
(2)
(a),
equation (1) being reserved for the elements As the avdable data on organic compounds are still extremely limited and hotly contested, we do not intend to discuss such compounds here By categonzmg materials mto these groups, Seah and Dench showed that, for E > 150 eV, the P m s scatter value unthm each group was -1 37. The quantitative use of eqns (1) and (2) could then proceeed with a confidence limit set by this factor The purpose of the present paper IS to demonstrate the utility of a recently derived formula for X(E) for all sohds [ 71 The denvatlon of this formula depends on the suggestion that, If any dlvlslon of solids mto classes 1s indeed necessary, then such a dlvlslon should be on the basis of the dlstmctlon between metals and msulators We shall be concerned with primary electron energies greater than 300 eV
OUTLINE
OF THEORY
Qumn [8] used the followmg relation to calculate the mean free path of electrons m bulk Jelhum X(k) =
laE(k)/akl
/
! =(k)
(3)
I
where l?(k) 1s the lmagmary part of the electron self-energy, and (Ifi)[ aE(k)/ ak] is the electron velocity Recently, Penn [9] has used the Lundqvlst [lo] form of the self-energy to calculate numencally the value of r(k) and hence the mean free path m metals SzaJman and Leckey [7] extended Penn’s model to morgamc sohds and produced an analytical expression of the form X,(E)
=
[2 lBE~/E~]/[ln(aE/~)
(4)
+ #]
where QC= #
=
Q =
8Q/[(E(4 3E,(Q EF/&.,
1Q2 + 0 6S)/E,)“2
+ 0 11E;/z2)/4Q2j?
+ 0 66Q2 + 0
181
99
and E, and E, are the free-electron values of the Fermi and plasma ener@es respectively. For free-electron-like metals j!? NE,, while for msulators and semxconductors zt has been observed that the energy centrold E of Im(---l/ E(O, 0)), obtamed numencally from optlcal data, may be well approximated by
E=E,+Eg
(5)
where E, 1sthe band gap With this additional approxlmatlon eqn (4) therefore represents a universal curve for X(E) for most solids Although not particularly cumbersome, eqn (4) may be recast mto a form slmllar to that proposed by Seah and Dench (eqns (1) and (2)). The plasma energy 1s usually m the range 5-30 eV, and if an average value of 18 eV 1s assumed, then the denominator m eqn (4) may be slmphfled to ln(E/4) In addltlon, E/ln(E/4) = E 3/4,and consequently
X,(E) = 2 12f?E3'4/E;,
(6)
of the Seah and Dench form but with a shghtly increased energy-dependence It should be noted that if the plasmon energy 1s vaned between 5 and 30 eV, then the term E/ln(4E/E, ) vanes by 15% at most from the average value assumed above If the contrlbutlon of scattenng from core electrons, estimated to be, on average, 15% of the valence-band scattermg [ 41, 1s now mcluded, the total melastlc mean free path 1s finally obtamed as
X(E) = 1 &i?E 3’4 /Es
(ii)
(7)
or
This form of expression for X(E) 1suniversal, m the sense that it applies to a very large group of matenals, but it 1s not universal m the sense mdely used previously, m the present case sample-dependent parameters are clearly included In the follow-mg Section we examme the utility of this expression for the descnptlon of the scattermg of electrons of energy E > 300 eV m a unde vanety of sohds
DISCUSSION
Smce delis of the denvation of eqn. (4) and an exammatlon of its effectiveness m predlctmg X(E) for a number of solids have been sven separately [ 71, we ~11 concentrate here on the success of the more appr0xlmat.e form, eqn (7), for calculatmg A(E) for mdely &ffermg mater&s. We first of all select CdTe, a narrow-band-gap semiconductor (E, = 1 4 eV), which would be characterized as an morgamc compound and treated by Seah and Dench
100
on the basis of eqn (2) As may be seen from Fig 1 and ref 11, the use of eqn (2) senously overestunates X(E) as compared vvlth the experimental values, which, m fact, he close to the predlctlons of eqn (1) (for elemental solids) for this matenal Equation (7) matches experiment well In Fig 2 we have plotted all expenmental mean-free-path values for metals, morgamc msulators and semiconductors from the Seah and Dench data-base m the form h(E) Ei/C, where C now mcludes an element-depen-
l
200
600 ELECTRON
1000 ENERGY
1400 (eV)
Fig 1 Measured mean-free-path values for CdTe as a function of electron energy The various theoretical lines are explained m the text, the Lotz curve IS based on a tlght-bmdmg approxunation
JO,7 ELECTRON
Fig 2 plotted
All expenmental usmg parameters
ENERGY
(eV)
mean-free-path values from the Sesh and Dench compilation, described in the text l , elements, x, morgamc compounds
101
dent correction for core-level scattermg Thus C = 2 12 X,(lOOO)I?-, where X,(1000) 1s the mean free path for core-level scattermg of lOOO-eV electrons, selected as a mid-range correction and calculated usmg the method @ven by Powell [43 outlined m the precedmg paper [ 71 The free-electron plasmon energies were calculated using E, = 28 ~(JLZ/A)'~~and E obtamed from E = E, i-E, using hterature values of the band gap E, where appropriate The solid hne m Fig 2 1s the y = E * 75 dependence of eqn. (7), and the dashed lme (y = 1 73E* 68 ) IS a least-squares frt to the data (above 300 eV), clearly the agreement 1squite reasonable The r m s scatter of the data points around the E* 75 prediction 1s 1 32, which compares favourably with the value (“1 37) obtamed by Seah and Dench, but without the need to separate sohds mto elemental and morgamc compounds The magnitude of the scatter LS, of course, largely dependent on the accuracy of the expemmental procedures used m determmmg the X(E) values, Seah and Dench’s compllatlon mcludes all avdable measurements lrrespectlve of accuracy, whereas the present calculation of the r m s scatter has excluded nme points showmg gross deviations We may conclude therefore that the theory outlined above, even m the approximate form of eqn (7), has been able to impose a degree of order on undely separate expenmental X(E) values, and may be used m quantltatlve apphcations, at least with a confidence level implied by the 1 32 r m s scatter value
Ia
11
1
l
h
0
A%&*
NlzL?
1
(
I
80
Fig 3 (a) Mean-free-path values at 1OOO-eV electron energy as a function of atomw number for metals, calculated using eqn (5). (b) values from (a) dwded by ms2 - the honzontal line IS the average value, 4 2 A-113
102
We have also thought it of some interest to examme the question of why the maJonty of metals have very sunllar X(E) values On the basis of eqn (5), which we expect to be adequate for the maJonty of elements (for a dlscusslon of the tifflcultles associated mth the transitIon metals see SzaJman and Leckey [ 7 3 ), we have calculated the mean free path at lOOO-eV electron energy as a function of atomic number for most metals (Frg 3(a)) If the monolayer-thickness dependence 1s approxunately removed by dlvldmg the X(E) values by m 3/2, then it 1s found (Fig 3(b)) that, with relatively few exceptions, the values so calculated he vvlthm 30% of 4 2 kli2, and a universal curve for the metals, with an energy-dependence close to E” 75, 1s thereby m&cated
REFERENCES
8 9
10 11
E 3auer, Vacuum, 22 (1972) 539 C R Brundle, J Vat SCI, Technol , ll(l974) 212 I Lmdau and W E Spacer, J Electron Spectrosc Relat Phenom ,3 (1974) 409 C J Powell, Surf SIX ,44 (1974) 29 R C G Leckey, Phys Rev A, 13 (1976) 1043 M P Seah and W A Dench, Surf Interface Anal , l(l979) 2 J Szajman and R C G Leckey, J Electron Spectrosc Relat Phenom , 23 (1981) 83-96 (precedmg paper) J J Qumn, Phys Rev, 126 (1962) 1453 D R Penn, J Electron Spectrosc Relat Phenom , 9 (1976) 29, Phys Rev B, 13 (1976) 6248 B I Lundqvlst, Phys Kondens Mater, 6 {1967) 206 J Szaman, R C G Leckey, J Llesegang and J G Jenkm, J Electron Spectrosc Relat Phenom , 20 (1980) 323