- Email: [email protected]
Surface-state resonances in low-energy electron diffraction
SURFACE
SCIENCE
15 (1969) 345-348 0 North-Holland
SURFACE-STATE
RESONANCES
ELECTRON
Publishing Co., Amsterdam
IN LOW-ENERGY
DIFFRACTION
Received 26 March 1969 In the course of some computations of low-energy electron diffraction (LEED) intensities as a function of electron energy, we have found sharp fluctuations of intensity suggestive of a resonance-scattering effect. An analysis of the computational results indicates that the narrowest resonances coincide with the existence of quasi-stable surface states l). The computational work has been carried through exactly for a real potential of the “muffin tin” form inside the model crystal, terminated by a step potential barrier at the surface. The primary-wave field is taken to be a plane wave. The primary wave propagation vector is denoted by Kt with I1Fr’z1s K. The electron wave field at a point R outside the crystal may be expressed in the following form, which is valid for any two-dimensionally periodic model 2) :
Y(R) = C [a,(K:) ”
exp(iK:.R)
+ b,(K,)
exp(iK,.R)].
The summation extends over an arbitrarily large number points u. The a,‘~ and b,‘s are constant coefficients and
iz of reciprocal-net
(2)
KZ = (K,, + 2rca, + K”)) K, = (K2 - lly,, + 2rcu~*)+,
(positive
(1)
root).
(3)
The entries in parentheses in eq. (2) denote projections on the surface and on the inward surface normal, respectively. The computational program evaluates the matrix M of amplitude reflection coefficients. This is an n-square matrix connecting the column vector of coefficients a, for the incident wave field with the column vector b, for the emitted field: b, = Ma,. All the points special case
(4)
we want to make here can be covered K,z>Q>K,f,
i.e., there is just one diffracted
adequately
(u # 0))
beam (the 00 beam). 345
in the (5)
The amplitude
of this
346
E.G. MCRAE AND P.3. JENNINGS
beam is given with an obvious notation by M(K;K:). The significance of the other diagonal elements of M is that the level of the “two-beam” approximation to diffraction problems, a singularity in M(Ki K:) corresponds to the existence of a stable surface state. This can be shown in the following way. In one possible two-beam description, the wave field in the region of locallyconstant potential just above the topmost atom layer (but inside the barrier) is a superposition of Bloch functions that are individually of the form q,(R) = a(K:)
exp(iK:
lR) +
b(K;) exp(iK,*R);
(61
the symbols on the right have the same meaning as before except for replacement of K2 by KZ + (2mjtr “) U,, U, being the barrier height, in the definition of K,“. If M(K;K:) is singular, a wave with propagation vector K; is emitted in the absence of an incident field, i.e., a bound state exists. Furthermore, matching of the “inside” function cputo the “outside” one exp (3; R) is only possible for /al = jbl. The latter condition can be satisfied only if cpUis an evanescent Bloch function, and this in turn means that the bound state for M(KiKw')singular is a surface state. In higher approximations the singularity is replaced by a Breit-Wigner expression with surface-state energy Es, and width Tss corresponding to a surface-state lifetime tt/rss. The surfacestate energy Es, should lie within the range of the Bragg condition Re (k,) = ng,, where k, is the inward surface normal component of the wave vector of the Bloch fLln~tion cp,,and g1 is a reciprocal-lattice vector normal to the surface. This is the range over which cp,,is evanescent. In the free-electron (kinematical) description, the above Bragg condition reduces to K,,=ng,; it is a tertiary Bragg condition with respect to the 00 reflection, meaning that it contains neither the incident nor the emitted beam. A theoretical treatment 1) based on a generalization of Darwin’s method for diffraction problemsa) indicates at least two types of resonance in LEED: one resonance extends over the entire range of the tertiary Bragg condition Re (k,) =Ing, and thus has a width of the order of 1 eV; the other resonance the surface-state resonance - coincides with the existence of a quasi-stable surface state and has a width of the order of 0.01 eV. In physical terms of the surface-state resonance occurs through capture of an electron in a surface state followed by the decay of the surface state through emission of a propagating outgoing wave. The interference between this contribution to the outgoing wave field and the contribution from the direct process not involving a surface state leads to a sharp intensity fluctuation. The theory leads to an expression for the amplitude in the immediate vicinity of a surfacestate resonance of the form l
(7)
SURFACE-STATE
RESONANCES
where the c’s are relatively
IN LOW-ENERGY
slowly-varying
DIFFRACTION
347
of the electron
energy E.
ELECTRON
functions
Correspondingly, the diagonal element M(KiKz) has the Breit-Wigner form indicated in brackets in eq. (7). For illustration of the above description, we show in fig. 1 the results of a computation for copper (100) surface for normal incidences). The top frame is a plot of the computed intensity (the squared modulus of M(K;Kl)) and the bottom frame shows a plot of the real and imaginary parts of ELECTRON ENERGY (0’)
I
g_
0
O
X =
-I
ELECTRON
ENERGY (AU)
Fig. 1. Computational results for copper (100) surface (normal incidence) showing: (a) a resonance in the 00 intensity curve, and (b) the corresponding variation of the relevant diagonal element of the matrix of amplitude reflection coefficients. The insert in (a) serves to locate the energy range of the computation with reference to an intensity curve with the usual proportions. The behavior shown in (b) is evidence- for the existence of a quasi-stable surface state (see text).
348
E.G. MCRAE
AND I’. I. JENNINGS
M(KiK:), where u has components (1 ,O) (the fact that there are four degenerate functions q,, does not affect our discussion). The surface-state resonance extends over an energy range of 0.5 eV centered near 15 eV. Within this energy range the computed variation of M is approximately of the Breit-Wigner form with Ess= 15.02 eV and rs,=O.O6 eV corresponding to a surface-state life-time of the order of lo- l4 sec. The resonance is presumably related to the Bragg condition K,=n/d (d= layer spacing, u= {IO}) which for copper (001) lies near 21 eV (ref. 4). Resonances have been observed in LEED experiments by McRae and Caldwel15). The observed effects have been interpreted in terms of interactions involving a wave travelling parallel to the crystal surface 5,s). A possible connection with surface states has been suggested explicitely by Hirabayashi’). We think that the observed effect is a tertiary Bragg resonance, and that the surface-state resonance remains to be observed in high-resolution experiments. Such an observation would have important implications in view of theoretical indications that the surface-state resonance is exceedingly sensitive to surface structure l). E. G. MCRAE and P. J. JENNINGS* Bell Telephone Laboratories, Inc., Murray Hill, New Jersey 07974, U.S.A. M-
References 1) A detailed treatment of LEED resonances is in preparation and will be submitted for publication in Surface Science. 2) E. G. McRae, Surface Sci. 11 (1968) 479. 3) The computation was carried out for the same potential as that used by Mattheiss in his computation of the band-structure of copper: L. F. Mattheiss, Phys. Rev. 139 (1965) A1893. The barrier height UO was 0.47 au ( = 13 eV). The method of computation was a combination of a generalized Darwin methoda) with a modified KKR method for a single atom layer; K. Kambe, Z. Naturforsch. 22a (1967) 322. A detailed account of these computations is in preparation and will be submitted to Surface Science. 4) In arriving at this figure, an “inner potential” of 13 eV was assumed. 5) E. G. McRae and C. W. Caldwell, Surface Sci. 7 (1967) 41. 6) E. G. McRae, J. Chem. Phys. 45 (1966) 3467; D. S. Boudreaux and V. Heine, Surface Sci. 8 (1967) 426; Y. H. Ohtsuki, J. Phys. Sot. Japan 24 (1968) 1116. 7) K. Hirabayashi, J. Phys. Sot. Japan 25 (1968) 856.
* Present address: Flinders University, Dept. of Chemistry, Bedford Park, South Australia.
SCIENCE
15 (1969) 345-348 0 North-Holland
SURFACE-STATE
RESONANCES
ELECTRON
Publishing Co., Amsterdam
IN LOW-ENERGY
DIFFRACTION
Received 26 March 1969 In the course of some computations of low-energy electron diffraction (LEED) intensities as a function of electron energy, we have found sharp fluctuations of intensity suggestive of a resonance-scattering effect. An analysis of the computational results indicates that the narrowest resonances coincide with the existence of quasi-stable surface states l). The computational work has been carried through exactly for a real potential of the “muffin tin” form inside the model crystal, terminated by a step potential barrier at the surface. The primary-wave field is taken to be a plane wave. The primary wave propagation vector is denoted by Kt with I1Fr’z1s K. The electron wave field at a point R outside the crystal may be expressed in the following form, which is valid for any two-dimensionally periodic model 2) :
Y(R) = C [a,(K:) ”
exp(iK:.R)
+ b,(K,)
exp(iK,.R)].
The summation extends over an arbitrarily large number points u. The a,‘~ and b,‘s are constant coefficients and
iz of reciprocal-net
(2)
KZ = (K,, + 2rca, + K”)) K, = (K2 - lly,, + 2rcu~*)+,
(positive
(1)
root).
(3)
The entries in parentheses in eq. (2) denote projections on the surface and on the inward surface normal, respectively. The computational program evaluates the matrix M of amplitude reflection coefficients. This is an n-square matrix connecting the column vector of coefficients a, for the incident wave field with the column vector b, for the emitted field: b, = Ma,. All the points special case
(4)
we want to make here can be covered K,z>Q>K,f,
i.e., there is just one diffracted
adequately
(u # 0))
beam (the 00 beam). 345
in the (5)
The amplitude
of this
346
E.G. MCRAE AND P.3. JENNINGS
beam is given with an obvious notation by M(K;K:). The significance of the other diagonal elements of M is that the level of the “two-beam” approximation to diffraction problems, a singularity in M(Ki K:) corresponds to the existence of a stable surface state. This can be shown in the following way. In one possible two-beam description, the wave field in the region of locallyconstant potential just above the topmost atom layer (but inside the barrier) is a superposition of Bloch functions that are individually of the form q,(R) = a(K:)
exp(iK:
lR) +
b(K;) exp(iK,*R);
(61
the symbols on the right have the same meaning as before except for replacement of K2 by KZ + (2mjtr “) U,, U, being the barrier height, in the definition of K,“. If M(K;K:) is singular, a wave with propagation vector K; is emitted in the absence of an incident field, i.e., a bound state exists. Furthermore, matching of the “inside” function cputo the “outside” one exp (3; R) is only possible for /al = jbl. The latter condition can be satisfied only if cpUis an evanescent Bloch function, and this in turn means that the bound state for M(KiKw')singular is a surface state. In higher approximations the singularity is replaced by a Breit-Wigner expression with surface-state energy Es, and width Tss corresponding to a surface-state lifetime tt/rss. The surfacestate energy Es, should lie within the range of the Bragg condition Re (k,) = ng,, where k, is the inward surface normal component of the wave vector of the Bloch fLln~tion cp,,and g1 is a reciprocal-lattice vector normal to the surface. This is the range over which cp,,is evanescent. In the free-electron (kinematical) description, the above Bragg condition reduces to K,,=ng,; it is a tertiary Bragg condition with respect to the 00 reflection, meaning that it contains neither the incident nor the emitted beam. A theoretical treatment 1) based on a generalization of Darwin’s method for diffraction problemsa) indicates at least two types of resonance in LEED: one resonance extends over the entire range of the tertiary Bragg condition Re (k,) =Ing, and thus has a width of the order of 1 eV; the other resonance the surface-state resonance - coincides with the existence of a quasi-stable surface state and has a width of the order of 0.01 eV. In physical terms of the surface-state resonance occurs through capture of an electron in a surface state followed by the decay of the surface state through emission of a propagating outgoing wave. The interference between this contribution to the outgoing wave field and the contribution from the direct process not involving a surface state leads to a sharp intensity fluctuation. The theory leads to an expression for the amplitude in the immediate vicinity of a surfacestate resonance of the form l
(7)
SURFACE-STATE
RESONANCES
where the c’s are relatively
IN LOW-ENERGY
slowly-varying
DIFFRACTION
347
of the electron
energy E.
ELECTRON
functions
Correspondingly, the diagonal element M(KiKz) has the Breit-Wigner form indicated in brackets in eq. (7). For illustration of the above description, we show in fig. 1 the results of a computation for copper (100) surface for normal incidences). The top frame is a plot of the computed intensity (the squared modulus of M(K;Kl)) and the bottom frame shows a plot of the real and imaginary parts of ELECTRON ENERGY (0’)
I
g_
0
O
X =
-I
ELECTRON
ENERGY (AU)
Fig. 1. Computational results for copper (100) surface (normal incidence) showing: (a) a resonance in the 00 intensity curve, and (b) the corresponding variation of the relevant diagonal element of the matrix of amplitude reflection coefficients. The insert in (a) serves to locate the energy range of the computation with reference to an intensity curve with the usual proportions. The behavior shown in (b) is evidence- for the existence of a quasi-stable surface state (see text).
348
E.G. MCRAE
AND I’. I. JENNINGS
M(KiK:), where u has components (1 ,O) (the fact that there are four degenerate functions q,, does not affect our discussion). The surface-state resonance extends over an energy range of 0.5 eV centered near 15 eV. Within this energy range the computed variation of M is approximately of the Breit-Wigner form with Ess= 15.02 eV and rs,=O.O6 eV corresponding to a surface-state life-time of the order of lo- l4 sec. The resonance is presumably related to the Bragg condition K,=n/d (d= layer spacing, u= {IO}) which for copper (001) lies near 21 eV (ref. 4). Resonances have been observed in LEED experiments by McRae and Caldwel15). The observed effects have been interpreted in terms of interactions involving a wave travelling parallel to the crystal surface 5,s). A possible connection with surface states has been suggested explicitely by Hirabayashi’). We think that the observed effect is a tertiary Bragg resonance, and that the surface-state resonance remains to be observed in high-resolution experiments. Such an observation would have important implications in view of theoretical indications that the surface-state resonance is exceedingly sensitive to surface structure l). E. G. MCRAE and P. J. JENNINGS* Bell Telephone Laboratories, Inc., Murray Hill, New Jersey 07974, U.S.A. M-
References 1) A detailed treatment of LEED resonances is in preparation and will be submitted for publication in Surface Science. 2) E. G. McRae, Surface Sci. 11 (1968) 479. 3) The computation was carried out for the same potential as that used by Mattheiss in his computation of the band-structure of copper: L. F. Mattheiss, Phys. Rev. 139 (1965) A1893. The barrier height UO was 0.47 au ( = 13 eV). The method of computation was a combination of a generalized Darwin methoda) with a modified KKR method for a single atom layer; K. Kambe, Z. Naturforsch. 22a (1967) 322. A detailed account of these computations is in preparation and will be submitted to Surface Science. 4) In arriving at this figure, an “inner potential” of 13 eV was assumed. 5) E. G. McRae and C. W. Caldwell, Surface Sci. 7 (1967) 41. 6) E. G. McRae, J. Chem. Phys. 45 (1966) 3467; D. S. Boudreaux and V. Heine, Surface Sci. 8 (1967) 426; Y. H. Ohtsuki, J. Phys. Sot. Japan 24 (1968) 1116. 7) K. Hirabayashi, J. Phys. Sot. Japan 25 (1968) 856.
* Present address: Flinders University, Dept. of Chemistry, Bedford Park, South Australia.