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Theory of secondary electron emission
Solid State Communications, Vol. 26, pp. 519—521 ©Perganxn Press Ltd. 1978. Printed in Great Britain
0038—1098/78/0522—0519
$02.00/0
THEORY OF SECONDARY ELECTRON EMISSION R. Feder Institut für Festkôrperforschung der KFA Jilhich, Postfach 1913, D—5170 Jfihich, GermarW J.B. Pendry Science Research Council, Daresbury Laboratory, Daresbury, Warrington WA I4AD, U.K. (Received 3 April 1978 by P.H. Dedericha) The fine structure in angle—resolved secondary electron spectra is shown to be related to the total reflectivity in low—energy electron diffraction (LEED). Theoretical results for tungsten are compared with experimental data. For non—normal emission, spin—orbit coupling is predicted to produce spin polarization of the emitted electrons.
Recent angle—resolved measurements of 5cc— ondary electron emission (SEE) spectra frau low— index surfaces of tungsten reveal a wealth of 13. fine structure in thecorrelation 0—20 eV kinetic range A semi—quantitative was found be— tween the data and one—dimensional densities of states obtained frau bulk band structure calculation, and features observed at energies within bulk band gaps were interpreted as sur— face state reasonance. A quantitative interpre— tation of the data is, however, still missing. In this paper we present a theory of angle—resolved SEE, which is capable of explain— ing quantitatively the observed fine structures, and compare numerical results with the data. In particular, our theory incorporates transport of electrons out of the crystal and emission from states which, though not localized on the vacuum side of the surface, have only finite penetra— tion into the crystal. Our theory further takes account of the electron spin, and substantial polarization effects are predicted to occur. We make the observation that the secondary spectrum is relatively insensitive to the orig— inal source of excitation. Because of the corn— plicated cascade processes at work the electrons have limited memory and statistics dictate their
key to our result is had by noticing the rela— tionship between the emissivity of a surface and the reflection reflection coefficient coefficient were of the If the zerosurface. the emissivity would correspond to ‘black body’ emission of electrons and would be given by d31 =
dEd2kN
C (E,i~,T) ~
(2)
.
It is a slow varying function of energy and is independent of i~, as defined, as can be deduced by considering equilibrium with a gas of dee— trons in an equal temperature enclosure. We retain the argument L,~ only to make subsequent working more transparent. If the reflectivity now becomes finite, the ernissivity must change, from considerations of detailed balance. The nun~er of electrons arriving at the surface with energy E and parallel momentum k equals the number departing from the surface with energy E and parallel momentum tN, i.e. C(E,k~,T)
(3)
=
e
-‘~
+
-~
C(E,k
distribution over states. We therefore make the assumption that states close together in energy are in thermodynamic equilibrium appropriate to some effective temperature, T. Then the distri— bution of electrons emitted within a relatively broad range of energies is the same as we would see if the solid was very hot 31 = s(E it ) = C(E it T) (1) dEd2it d ‘ N ‘ fl’ ‘
11,T) + B -~-(E,k1) C0(E,k1+g,T) 1(E,E’,i~g,~) g 0 C f R 0(E’,k~,T) dE’ dk~ .
+
where C is the emissivity of the reflecting sur— face, Re the elastic.current reflection—coef— ficient matrix and Ri the inelastic reflection— coefficient. If we express Ri as a product of function f(E,E’,~), varies slãw]ytimes with aE the secondary emissionwhich current S(E,I~) but can be unknown otherwise, the integral in eq. (3) can be evaluated as
where S is the secondary current and C the ther— isal emissivity of the surface for electrons. Of
+
+
S(E,k 11) x f f(E,E’,k~) C0(E’,k~,T) dE’
course the effective temperature would not be the same at each electron energy, but we assume that T(E) is a relatively slowly varying function and the fine structure superimposed on the broad background we assume is due to density of states effects. Having made this connection the
+
=
a(E) s(E,k1)
+
=
u(E) C(E,k,,T)
,
(14)
where ct(E) is a slowly varying function of en— ergy. Substituting into eq. (3) and using eq. (1) we obtain 519
520
THEORY OF SECONDARY ELECTRON EMISSION
I
I
THEORY ~ ~BARRIER
Vol. 26, No. 8
ization vector ~ of a beam, described by a den— sity matrix p, is obtained as p = tr[~ p] / tr[p] (7)
-
where is a vector comprising the three Pauli spin matrices. From eq. (6) we thus obtain for the spin polarization vector of the SEE cur— 5
/BARRIER
~ (E,it )
{E9-go R~(E,it g rent —
z ~ o
u
) ~(E,it )J/[i
—
~ R~.(E,it )J
polariz~ion ve~tar of the ~th LEED beam.
I
-
(8)
=
—
z o 0 ~RIMENT
—
where We R~ notice and that denote the in the slowly varying and function the C /(i+a), which occurs theintensity intensity formula 0 eq. (5), is absent in the polarization formula eq. (8). Since this function is at present not known from first principles, it follows that the predictive value of our theory is higher for the
the spin spin trum. structure polarization Following polarization of theequations intensity thanoffor the (5)the spectrum angle—resolved and intensity (8),as the well specfine SEE as current can be obtained via a relativistic LEED theory. Such a theory has been described elsewhere~,except for a modification required in
10
20
ENERGY RELATIVE TO VACUUM (eV) Figure 1: Secondary emission normal to the tungsten (001) surface. Theory with V .—0.25 eV, experiment after Willis 01
and Feuerbacher1.
S(E,itu)
il
(5)
— — —
the small present at energies context. below Since the absorption surface plasmon is very threshold, the finite—crystal method employed 5 is likely to give rise to conin earlierproblems. work vergence We have therefore replaced it by a Bloch wave method, which can be viewed as the spin—dependent generalization of a nodification of a Bloch wave method described in ref. 9. In applying our theory to tungsten, we make the following specific model assumptions. Ion—core phase shifts are obtained from a muffin— tin potential due to Christensen10, and Debye cor— rected for room temperature11 using the
~ R:~(E,~)]C 0(E)/1+a(E))
-
C can be factored out of the summation because i is independent of . The factor in square brackets produces all ~he rapid variation of 5, i.e. the SEE fine structure is determined by the LEED reflection coefficient~. The reduction of the secondary emission intensity problem to a LEED intensity problem suggests that spin polarization effects should 5’6 and occur in SEE, since such effects have experimen— recently tally7~8 in both LEED.theoretically The spin—dependent generaliza— been found tion of eq. (5) is i~rittenas
it
=
p(E,kM) .4
[p(E,kN)
-~ —
‘
~ p~(E,k
11)JC0/(1+a) (6) where p p and p+are (2x2) density matrices, which cYjara8terize~thesecondary current, an unpolarized normalized primary LEED beam and the diffracted LEED beams, respectively. Since C and a do not depend on itt, as was mentioned 0 above, they also do not depend on spin, as can be shown by symmetry arguments. The spin polar—
temperature of 380 K. Phase shifts up to t=5 are included. Surface reciprocal lattice vectors (beams) are selected by the computer program such as to ensure convergence. Typically, 21 beams are found to be adequate. The real inner potential is found fran the alignment of normal incidence results with experiment to be 16.5 eV for W(oOi). The imaginary absorption potential is assumed to be —0.25 eV. For the surface bar— rier, two models are employed: firstly a re— fracting no—reflection barrier and secondly a nature above the crystal. smooth located function atof 1.2 an essentially exponential As a check for the correctness of the corn— puter program, the band structure normal to w(ooi) was calculated using the new Bloch wave routine. It was found to be identical with Christensen’s Relativistic Argumented Plane 3. The secondary emission current Wavecalculated results was according to eq. (5) with the slowly varying function C /(1+a) chosen to be unity. 0 In figure 1 we present results for normal emission from the tungsten (001) surface. The two theoretical curves demonstrate the sensitivity of SEE calculations to the sur— face barrier. This is particularly marked near
Vol. 26, No. 8
THEORY OF SECONDARY ELECTRON EMISSION
the threshold of emergence of a new set of beams near 15 eV (shown in fig.1 by an arrow), where the reflecting barrier gives rise to a surface state reasonance (cf. ref. 12). Comparison with the data of ref. 1 shows overall agreement ex— cept for the surface reasonance peak near 15 eV, which is absent in the data and could be an artifact of our particular barrier model. Our choice of —0.25 eV for V ~ gives about the right amount of broadening at 0 low energies, but should be larger above about 15 eV. Further, it is important to note that the experimental curve still contains the slowly varying function of energy C/(1+a) (cf. eq. (5)), i.e. the ‘cascade background’, which was taken to be unity in our calculation. A fully quantitative comparison be—
tween theoretical and experimental SEE fine structures will require careful correction of the rew data for the background. For normal emission, we find the SEE current to be unpolar— ized, which is plausible on the grounds of sym— metry. Preliminary results for non—normal emis— sion, however, show pronounced spin polarization features. We conclude that we have found a simple relation between the fine structure of angle— resolved SEE spectra and LEED intensities. The agreement of our theoretical results with ex— periment is encouraging, but refinements in the crystal model assumptions and more extensive calculations are required as well as a study of the ‘cascade background’ problem.
REFERENCES 1. 2. 3. it. 5.
6. 7.
8.
9. 10. 11. 12.
521
WIlLIS, R.F. and FEIJERBACHER, B., Surf. Sci. ~, 11~1~(1975). WILLIS, R.F., FEUERBACHER, B. and CERISTENSEN, N.E., Phys. Rev. Lett. 38, 1087 (1977). WILLIS, R.F. and CHEISTENSEN, N.E., Phys. Rev., in press. C1~ISTENSEN,N.E. and WILLIS, R.F., Phys. Rev., in press. A relation between SEE and LEED, which differs fran ours, was derived by M0DINC~,A., J. Phys. ~, 3867 (1976). 1t); FEDER, R., phys. stat. sol. (b) ~, 699 (1972); phys. stat. sol. (b) 62, 135 (197 Phys. Rev. Lett. 36, 598 (1976). FEDER, H., Surf. Sci. 63, 283 (1977). O’NEILL, M.R., KALISVA.~RT, M., DiJNNING, F.B. and WALTERS, O.K., Phys. Rev. Lett. 31t, 1167 (1975). MULLER, N. and WOLF, D., Bull. Am. Thys. Soc. II 21, 9)414 (1976). PENDRY, J.B., “Low Energy Electron Diffraction” (Academic, New York, 1971t). CERISTENSEN, N.E., private communication (1975). The actual crystal temperature is not related to the fictitious electronic temperature T introduced above. MCRAE, E.G., Surf. Sci. 25, 1491 (1971).
0038—1098/78/0522—0519
$02.00/0
THEORY OF SECONDARY ELECTRON EMISSION R. Feder Institut für Festkôrperforschung der KFA Jilhich, Postfach 1913, D—5170 Jfihich, GermarW J.B. Pendry Science Research Council, Daresbury Laboratory, Daresbury, Warrington WA I4AD, U.K. (Received 3 April 1978 by P.H. Dedericha) The fine structure in angle—resolved secondary electron spectra is shown to be related to the total reflectivity in low—energy electron diffraction (LEED). Theoretical results for tungsten are compared with experimental data. For non—normal emission, spin—orbit coupling is predicted to produce spin polarization of the emitted electrons.
Recent angle—resolved measurements of 5cc— ondary electron emission (SEE) spectra frau low— index surfaces of tungsten reveal a wealth of 13. fine structure in thecorrelation 0—20 eV kinetic range A semi—quantitative was found be— tween the data and one—dimensional densities of states obtained frau bulk band structure calculation, and features observed at energies within bulk band gaps were interpreted as sur— face state reasonance. A quantitative interpre— tation of the data is, however, still missing. In this paper we present a theory of angle—resolved SEE, which is capable of explain— ing quantitatively the observed fine structures, and compare numerical results with the data. In particular, our theory incorporates transport of electrons out of the crystal and emission from states which, though not localized on the vacuum side of the surface, have only finite penetra— tion into the crystal. Our theory further takes account of the electron spin, and substantial polarization effects are predicted to occur. We make the observation that the secondary spectrum is relatively insensitive to the orig— inal source of excitation. Because of the corn— plicated cascade processes at work the electrons have limited memory and statistics dictate their
key to our result is had by noticing the rela— tionship between the emissivity of a surface and the reflection reflection coefficient coefficient were of the If the zerosurface. the emissivity would correspond to ‘black body’ emission of electrons and would be given by d31 =
dEd2kN
C (E,i~,T) ~
(2)
.
It is a slow varying function of energy and is independent of i~, as defined, as can be deduced by considering equilibrium with a gas of dee— trons in an equal temperature enclosure. We retain the argument L,~ only to make subsequent working more transparent. If the reflectivity now becomes finite, the ernissivity must change, from considerations of detailed balance. The nun~er of electrons arriving at the surface with energy E and parallel momentum k equals the number departing from the surface with energy E and parallel momentum tN, i.e. C(E,k~,T)
(3)
=
e
-‘~
+
-~
C(E,k
distribution over states. We therefore make the assumption that states close together in energy are in thermodynamic equilibrium appropriate to some effective temperature, T. Then the distri— bution of electrons emitted within a relatively broad range of energies is the same as we would see if the solid was very hot 31 = s(E it ) = C(E it T) (1) dEd2it d ‘ N ‘ fl’ ‘
11,T) + B -~-(E,k1) C0(E,k1+g,T) 1(E,E’,i~g,~) g 0 C f R 0(E’,k~,T) dE’ dk~ .
+
where C is the emissivity of the reflecting sur— face, Re the elastic.current reflection—coef— ficient matrix and Ri the inelastic reflection— coefficient. If we express Ri as a product of function f(E,E’,~), varies slãw]ytimes with aE the secondary emissionwhich current S(E,I~) but can be unknown otherwise, the integral in eq. (3) can be evaluated as
where S is the secondary current and C the ther— isal emissivity of the surface for electrons. Of
+
+
S(E,k 11) x f f(E,E’,k~) C0(E’,k~,T) dE’
course the effective temperature would not be the same at each electron energy, but we assume that T(E) is a relatively slowly varying function and the fine structure superimposed on the broad background we assume is due to density of states effects. Having made this connection the
+
=
a(E) s(E,k1)
+
=
u(E) C(E,k,,T)
,
(14)
where ct(E) is a slowly varying function of en— ergy. Substituting into eq. (3) and using eq. (1) we obtain 519
520
THEORY OF SECONDARY ELECTRON EMISSION
I
I
THEORY ~ ~BARRIER
Vol. 26, No. 8
ization vector ~ of a beam, described by a den— sity matrix p, is obtained as p = tr[~ p] / tr[p] (7)
-
where is a vector comprising the three Pauli spin matrices. From eq. (6) we thus obtain for the spin polarization vector of the SEE cur— 5
/BARRIER
~ (E,it )
{E9-go R~(E,it g rent —
z ~ o
u
) ~(E,it )J/[i
—
~ R~.(E,it )J
polariz~ion ve~tar of the ~th LEED beam.
I
-
(8)
=
—
z o 0 ~RIMENT
—
where We R~ notice and that denote the in the slowly varying and function the C /(i+a), which occurs theintensity intensity formula 0 eq. (5), is absent in the polarization formula eq. (8). Since this function is at present not known from first principles, it follows that the predictive value of our theory is higher for the
the spin spin trum. structure polarization Following polarization of theequations intensity thanoffor the (5)the spectrum angle—resolved and intensity (8),as the well specfine SEE as current can be obtained via a relativistic LEED theory. Such a theory has been described elsewhere~,except for a modification required in
10
20
ENERGY RELATIVE TO VACUUM (eV) Figure 1: Secondary emission normal to the tungsten (001) surface. Theory with V .—0.25 eV, experiment after Willis 01
and Feuerbacher1.
S(E,itu)
il
(5)
— — —
the small present at energies context. below Since the absorption surface plasmon is very threshold, the finite—crystal method employed 5 is likely to give rise to conin earlierproblems. work vergence We have therefore replaced it by a Bloch wave method, which can be viewed as the spin—dependent generalization of a nodification of a Bloch wave method described in ref. 9. In applying our theory to tungsten, we make the following specific model assumptions. Ion—core phase shifts are obtained from a muffin— tin potential due to Christensen10, and Debye cor— rected for room temperature11 using the
~ R:~(E,~)]C 0(E)/1+a(E))
-
C can be factored out of the summation because i is independent of . The factor in square brackets produces all ~he rapid variation of 5, i.e. the SEE fine structure is determined by the LEED reflection coefficient~. The reduction of the secondary emission intensity problem to a LEED intensity problem suggests that spin polarization effects should 5’6 and occur in SEE, since such effects have experimen— recently tally7~8 in both LEED.theoretically The spin—dependent generaliza— been found tion of eq. (5) is i~rittenas
it
=
p(E,kM) .4
[p(E,kN)
-~ —
‘
~ p~(E,k
11)JC0/(1+a) (6) where p p and p+are (2x2) density matrices, which cYjara8terize~thesecondary current, an unpolarized normalized primary LEED beam and the diffracted LEED beams, respectively. Since C and a do not depend on itt, as was mentioned 0 above, they also do not depend on spin, as can be shown by symmetry arguments. The spin polar—
temperature of 380 K. Phase shifts up to t=5 are included. Surface reciprocal lattice vectors (beams) are selected by the computer program such as to ensure convergence. Typically, 21 beams are found to be adequate. The real inner potential is found fran the alignment of normal incidence results with experiment to be 16.5 eV for W(oOi). The imaginary absorption potential is assumed to be —0.25 eV. For the surface bar— rier, two models are employed: firstly a re— fracting no—reflection barrier and secondly a nature above the crystal. smooth located function atof 1.2 an essentially exponential As a check for the correctness of the corn— puter program, the band structure normal to w(ooi) was calculated using the new Bloch wave routine. It was found to be identical with Christensen’s Relativistic Argumented Plane 3. The secondary emission current Wavecalculated results was according to eq. (5) with the slowly varying function C /(1+a) chosen to be unity. 0 In figure 1 we present results for normal emission from the tungsten (001) surface. The two theoretical curves demonstrate the sensitivity of SEE calculations to the sur— face barrier. This is particularly marked near
Vol. 26, No. 8
THEORY OF SECONDARY ELECTRON EMISSION
the threshold of emergence of a new set of beams near 15 eV (shown in fig.1 by an arrow), where the reflecting barrier gives rise to a surface state reasonance (cf. ref. 12). Comparison with the data of ref. 1 shows overall agreement ex— cept for the surface reasonance peak near 15 eV, which is absent in the data and could be an artifact of our particular barrier model. Our choice of —0.25 eV for V ~ gives about the right amount of broadening at 0 low energies, but should be larger above about 15 eV. Further, it is important to note that the experimental curve still contains the slowly varying function of energy C/(1+a) (cf. eq. (5)), i.e. the ‘cascade background’, which was taken to be unity in our calculation. A fully quantitative comparison be—
tween theoretical and experimental SEE fine structures will require careful correction of the rew data for the background. For normal emission, we find the SEE current to be unpolar— ized, which is plausible on the grounds of sym— metry. Preliminary results for non—normal emis— sion, however, show pronounced spin polarization features. We conclude that we have found a simple relation between the fine structure of angle— resolved SEE spectra and LEED intensities. The agreement of our theoretical results with ex— periment is encouraging, but refinements in the crystal model assumptions and more extensive calculations are required as well as a study of the ‘cascade background’ problem.
REFERENCES 1. 2. 3. it. 5.
6. 7.
8.
9. 10. 11. 12.
521
WIlLIS, R.F. and FEIJERBACHER, B., Surf. Sci. ~, 11~1~(1975). WILLIS, R.F., FEUERBACHER, B. and CERISTENSEN, N.E., Phys. Rev. Lett. 38, 1087 (1977). WILLIS, R.F. and CHEISTENSEN, N.E., Phys. Rev., in press. C1~ISTENSEN,N.E. and WILLIS, R.F., Phys. Rev., in press. A relation between SEE and LEED, which differs fran ours, was derived by M0DINC~,A., J. Phys. ~, 3867 (1976). 1t); FEDER, R., phys. stat. sol. (b) ~, 699 (1972); phys. stat. sol. (b) 62, 135 (197 Phys. Rev. Lett. 36, 598 (1976). FEDER, H., Surf. Sci. 63, 283 (1977). O’NEILL, M.R., KALISVA.~RT, M., DiJNNING, F.B. and WALTERS, O.K., Phys. Rev. Lett. 31t, 1167 (1975). MULLER, N. and WOLF, D., Bull. Am. Thys. Soc. II 21, 9)414 (1976). PENDRY, J.B., “Low Energy Electron Diffraction” (Academic, New York, 1971t). CERISTENSEN, N.E., private communication (1975). The actual crystal temperature is not related to the fictitious electronic temperature T introduced above. MCRAE, E.G., Surf. Sci. 25, 1491 (1971).