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Influence of electron correlation on surface states of nonmetallic solids
Solid State Communications, Vol. 16, pp. 541—544, 1975.
Pergamon Press.
Printed in Great Britain
INFLUENCE OF ELECTRON CORRELATION ON SURFACE STATES OF NONMETALLIC SOLIDS* A.B. Kunz Department of Physics and Materials Research Laboratory, University of fflinois at Urbana-Champaign, Urbana, illinois 61801, U.S.A. (Received 22 July 1974 by C. W. McCombie)
In this note the method of Mott and Littleton is adapted to construct the position dependent correlation potential for energy bands in a non-metallic solid. One specifically studies the case of a solid with a surface. The principle result is that the effect of electronic correlation is to inhibit the effect of fonnation of true localized surface states in the region of the band gaps.
have either been primarily concerned with the question of chemisorption of atoms on surfaces7 or with the question of the electronic structure of solid surfaces.8 All of these calculations employ an independent particle approximation and unless they include correlation effects accidentally through fortuitous choice of parameters in those schemes which are parameterized, all correlation is neglected. This is unfortunate in that one of the serious questions to be answered by solid state physics deals with the presence or absence, as well as quantity and type, of surface-localized states in the energy gap of non-metallic solids. By surface. localized states, one means that the states decay cxponentially as they penetrate into the crystal. In this letter, the author adopts the method of Mott and Littleton to study the questions of the influence of correlation on the surface states of a solid.
THE FIRST usable step taken to correlate the quasiparticle spectrum of the insulating solids was the method of Mott and Littleton.’ This model had its initial application to correlating the optical spectral predictions of band theory for perfect insulating solids in the work of Fowler.2 In Fowler’s application the Mott—Littleton model is applied in a semi-classical way. Fowler predicts that correlation reduces the optical band gap of the alkali halides by about 3.0 ± 1.0 eV when compared to a Hartree—Fock calculation, Since the time of the Mott—Littleton contribution a number of supposedly more sophisticated and exact quantum mechanical models for correlation have been proposed. These include the electronic polaron3’4 and the screened exchange plus coulomb hole model.5 Both these models have been used to correlate a substantial number of perfect crystaline band structures with great success.6 Unfortunately the success of these latter two methods has been obscuring the fact that the correlation corrections predicted by Fowler using the Mott—Littleton theory are in very good agreement with the results of the more elaborate methods.
One proceeds using Maxwell’s equations,°to def’me the correlation energy and the correlation potential. It is reasonable to consider an insulating solid as a collection of polarizable pseudo atoms (ions). Let there be N types present and let the polarizability of the ith type be ~,. One must consider the dielectric response of this system of atoms to a positive test charge, Q. If one starts with the Claussius Mossotti equation for the bulk optical dielectric constant K 0
More recently there has been a great deal of interest in surface state calculations. These calculations *
Work supported in part by the National Science Foundation under Grant GH-33634 and by the Aerospace Research Laboratory, Air Force Systems Command, U.S.A.F., Wright-PattersonAFB, Ohio, Contract 33-615-72-C-i 506.
which may serve either to yield K0 or the sums of the y~’sbut is. not essential to the actual derivation, 541
542
ELECTRON CORRELATION OF NONMETALLIC SOLIDS I 4.0
-
I
I
I
I
I
I
I -
K~8.O
The essential problem here is to correctly include the surface boundary conditions in solving Maxwell’s equations. This is most easily accomplished by solving Poisson’s equation by the method of images. In what follows one uses the following convention: the plane of the surface is the xy plane and the test change is
K, - 4.0 3.0
Vol. 16, No.5
1<,- 3.0
K,- 2.0 —
2.0
-
K,- 1.5 1.0
-
on the z axis. One lets +z refer to the vacuum and —z is in the crystal. If the test charge is in the solid, the field in the solid is simply 1 ‘Q~ qxl E~ =-~i+~) I’
__________________________________________ -40 -30 -20 -tO 0 0 20 30 40 50 60 70 oc I
(5) E~ =
d (Atomic Units)
FIG. 1. The correlation potential is shown as a function of distance from the surface as a function of K0 for the [1001 face of a simple cubic solid surface. +d is the vacuum and —d is in the bulk.
1 /(z_d)Q+(z+d) ,\ K0t~ R~ R~ q
)
E~is obtained from E~by replacing x withy, d is the position of the test charge on the z axis, q’ is the image charge given by /1—K0
K0—14~1~71 K0 + 2 3 ~ 3 is the volume of one of our pseudo atoms (unit acell volume), the polarization field of the system is
q If the electron is in the vacuum, the field is given by ~q x E~ = K 0 R~’
—
F,
given then from the total electric field by the result K0—! = E. (2) 4ir In turn one may use this to calculate the induced dipoles on the atoms on the solid. The dipole on the ith type p~is found to be Pg =
~-~-a~P. N ~ 7i 1=1
(3)
If one knows the lattice structure and the induced dipole moments, p~,one can calculate the correlation potential by summing the contributions of the individual dipoles using the dipole potential, V(r)
=
Nr
IrIs
(4)
This potential is identified as the correlation potential because it is explicitly caused by dipoles induced upon the mean charge distribution by the instantaneous position of the test charge. This polarization “clothes” the test charge and “follows” the test charge around the lattice.
and
(6) ______
E~
iq(z—d) K0 R~
In (5) and (6) Ri is the vector from Q to the field point and R2 is from the image charge to the field point, and 2K0 q l+K0 Q. ‘F
—
The use of equations (5) and (6) along with (2)—(4) permit us to construct the proper correlation potential. In order to obtain a general form for this potential we define a pseudo atomic constant M1 for the ith type as M,
=
a~ —(1—1/K0)~4ir N ~ 7, j=1
(7)
Vol. 16, No.5
ELECTRON CORRELATION OF NONMETALLIC SOLIDS
This then yields the correlation potential
V~(r)as:
(a) For a field point in the solid N fQx,1 + q’x,1\
f
V~(r)=Z tYPS
the solid and outside also. The results of this calculation is shown for hole states in Fig. 1. We see that V, is essentially constant inside the crystal and falls
XU
~) jj~’~
M, ~~
PO~U0fl
of the atom
+
+ qy,1\1 ~ (8) \R~,.1 R2~J~ + (Q(z1i R~ d) + q’(z,1 R~~ + d)\ (z,1 d)1 1 R~
\,
)
—
—
j
‘
the site containing the test charge is omitted from the sum; (b) For a field point in the vacuum N
V~(r)=
“
M, types
1 pod~on
R’h1’
543
(9)
of the ~ In terms of these correlation potentials the self energy of an electron or hole in state ~ is found by letting I Q = electronic charge, e, and = (n I~ e V,~(r) In). (10)
The self energy is positive for creating a hole state (i.e. for occupied bands) and is negative for creating an electron state (i.e. virtual or conduction levels), One anticipates from (8) and (9) that the correlation potential will be relatively constant inside the crystal and fall to zero rapidly outside. Thus one expects that bulk hole (electron) states will be moved up (down) more strongly than will surface states which spend a non-negligible fraction of their time outside the crystal in contrast to bulk state which spend no time outside the semi-infinite solid, To quantize this effect a sample calculation is performed for a simple cubic solid of nearest neighbor distance 4 atomic units as a function ofK 0. It iS assumed the xy plane is a [100] face and the origin is an atom site. ~eV~is computed at atom site inside
rapidly to zero outside. This means that the presence of correlation effects will serve to inhibit the formation of stable surface-localized states in the gap when corn. pared to a Hartree—Fock model. It is also noted in summary that the bulk part of V~,obtained here agrees well with those for bulk solids of like K0 given by 2 Finally these results would be essentially Fowler. correct for K0 a function of position jfK0 varies much more slowly with position than does E, a condition which is likely true. There is one final point to discuss and this concerns the applicability of the linear response, image charge model when the test charge approaches and penetrates the image plane into the solid. Applebaum and Hamann studied this for jellium and found the force on the electron to be poorly given by the image charge resultisfor distances of less 1°That thecharge-jelliurn response of the charge near thethan test 2A. charge is found to saturate at small distances. This problem is avoided in the present case on two accounts. The first is that insulating systems have finite static dielectric constants, and secondly by, the central cell (that atom or group of atoms nearest to the test charge) is explicitly excluded from the pertinent formulae in this paper, equations (8) and (9). Thus the expression given for the correlation energy is too small for the portion below the surface plane, but is correct outside the surface. Therefore, the conclusion reached about correlation inhibiting the formation of surface states is enhanced by this effect. The short range correlation may be included simply using the models of Kunz, Mickish and Collins.6 Recent unpublished calculations of Euwema and Kunz indicate that the response of an atom to an electron remains nearly linear to a separation of about 5 a.u. Thus the size of the excluded central cell need not be too large.
REFERENCES 1.
MOTT N.F. and LITTLETON M.J., Trans. Faraday Soc. 34,485 (1938).
2.
FOWLERW.B.,Phys. Rev. 151,657 (1966).
3.
TOYOZAWA Y., Frog. Theoret. Phys. (Kyoto) 12, 421(1954).
4.
KUNZ A.B.,Phys. Rev. B6, 606 (1972).
544
ELECTRON CORRELATION OF NONMETALLIC SOLIDS
Vol. 16, No. 5
5. 6.
HEDIN L.,Phys. Rev. 139, A796 (1965);ArkivFysik 30, 19 (1965). BRINKMAN W.F. and GOODMAN’B., Phys. Rev. 149, 597 (1966); INOUE M., MAHUTTE C.K. and WANG S., Phys. Rev. B2, 539 (1970); LIPARI N.O. and FOWLER W.B., Solid State Commun. 8, 1395 (1970); KUNZ A.B. and MICKISH D.J.,Phys. Rev. B8, 779 (1973); MICKISH D.J., KUNZ A.B. and COLLINS T.C., Phys. Rev. B9,4461 (1974).
7.
BENNETT AJ., MCCARROLL B. and MESSMER R.P.,Surf Sci., 24, 191 (l971);Phys. Rev. B3, 1397 (1971); KUNZ A.B., MICIUSH D.J. and DEUTSCH P.W., Solid State Commun. 13,35 (1973).
8.
APPLEBAUM J.A. and HAMANN D.R.,Phys. Rev. B6, 2166 (1972); Phys. Rev. Lett. 31, 106 (1973); VAN DOREN V.E. and KUNZ A.B., Phys. Rev. (to be published); KASOWSKI R.V., Phys. Rev. Lett. 33,83 (1974);
9.
JACKSON J.D., Classical Electrodynamics, John Wiley, New York (1962).
10.
APPLEBAUM J.A. and HAMANN D.R.,Phys. Rev. B6, 1122 (1972).
Pergamon Press.
Printed in Great Britain
INFLUENCE OF ELECTRON CORRELATION ON SURFACE STATES OF NONMETALLIC SOLIDS* A.B. Kunz Department of Physics and Materials Research Laboratory, University of fflinois at Urbana-Champaign, Urbana, illinois 61801, U.S.A. (Received 22 July 1974 by C. W. McCombie)
In this note the method of Mott and Littleton is adapted to construct the position dependent correlation potential for energy bands in a non-metallic solid. One specifically studies the case of a solid with a surface. The principle result is that the effect of electronic correlation is to inhibit the effect of fonnation of true localized surface states in the region of the band gaps.
have either been primarily concerned with the question of chemisorption of atoms on surfaces7 or with the question of the electronic structure of solid surfaces.8 All of these calculations employ an independent particle approximation and unless they include correlation effects accidentally through fortuitous choice of parameters in those schemes which are parameterized, all correlation is neglected. This is unfortunate in that one of the serious questions to be answered by solid state physics deals with the presence or absence, as well as quantity and type, of surface-localized states in the energy gap of non-metallic solids. By surface. localized states, one means that the states decay cxponentially as they penetrate into the crystal. In this letter, the author adopts the method of Mott and Littleton to study the questions of the influence of correlation on the surface states of a solid.
THE FIRST usable step taken to correlate the quasiparticle spectrum of the insulating solids was the method of Mott and Littleton.’ This model had its initial application to correlating the optical spectral predictions of band theory for perfect insulating solids in the work of Fowler.2 In Fowler’s application the Mott—Littleton model is applied in a semi-classical way. Fowler predicts that correlation reduces the optical band gap of the alkali halides by about 3.0 ± 1.0 eV when compared to a Hartree—Fock calculation, Since the time of the Mott—Littleton contribution a number of supposedly more sophisticated and exact quantum mechanical models for correlation have been proposed. These include the electronic polaron3’4 and the screened exchange plus coulomb hole model.5 Both these models have been used to correlate a substantial number of perfect crystaline band structures with great success.6 Unfortunately the success of these latter two methods has been obscuring the fact that the correlation corrections predicted by Fowler using the Mott—Littleton theory are in very good agreement with the results of the more elaborate methods.
One proceeds using Maxwell’s equations,°to def’me the correlation energy and the correlation potential. It is reasonable to consider an insulating solid as a collection of polarizable pseudo atoms (ions). Let there be N types present and let the polarizability of the ith type be ~,. One must consider the dielectric response of this system of atoms to a positive test charge, Q. If one starts with the Claussius Mossotti equation for the bulk optical dielectric constant K 0
More recently there has been a great deal of interest in surface state calculations. These calculations *
Work supported in part by the National Science Foundation under Grant GH-33634 and by the Aerospace Research Laboratory, Air Force Systems Command, U.S.A.F., Wright-PattersonAFB, Ohio, Contract 33-615-72-C-i 506.
which may serve either to yield K0 or the sums of the y~’sbut is. not essential to the actual derivation, 541
542
ELECTRON CORRELATION OF NONMETALLIC SOLIDS I 4.0
-
I
I
I
I
I
I
I -
K~8.O
The essential problem here is to correctly include the surface boundary conditions in solving Maxwell’s equations. This is most easily accomplished by solving Poisson’s equation by the method of images. In what follows one uses the following convention: the plane of the surface is the xy plane and the test change is
K, - 4.0 3.0
Vol. 16, No.5
1<,- 3.0
K,- 2.0 —
2.0
-
K,- 1.5 1.0
-
on the z axis. One lets +z refer to the vacuum and —z is in the crystal. If the test charge is in the solid, the field in the solid is simply 1 ‘Q~ qxl E~ =-~i+~) I’
__________________________________________ -40 -30 -20 -tO 0 0 20 30 40 50 60 70 oc I
(5) E~ =
d (Atomic Units)
FIG. 1. The correlation potential is shown as a function of distance from the surface as a function of K0 for the [1001 face of a simple cubic solid surface. +d is the vacuum and —d is in the bulk.
1 /(z_d)Q+(z+d) ,\ K0t~ R~ R~ q
)
E~is obtained from E~by replacing x withy, d is the position of the test charge on the z axis, q’ is the image charge given by /1—K0
K0—14~1~71 K0 + 2 3 ~ 3 is the volume of one of our pseudo atoms (unit acell volume), the polarization field of the system is
q If the electron is in the vacuum, the field is given by ~q x E~ = K 0 R~’
—
F,
given then from the total electric field by the result K0—! = E. (2) 4ir In turn one may use this to calculate the induced dipoles on the atoms on the solid. The dipole on the ith type p~is found to be Pg =
~-~-a~P. N ~ 7i 1=1
(3)
If one knows the lattice structure and the induced dipole moments, p~,one can calculate the correlation potential by summing the contributions of the individual dipoles using the dipole potential, V(r)
=
Nr
IrIs
(4)
This potential is identified as the correlation potential because it is explicitly caused by dipoles induced upon the mean charge distribution by the instantaneous position of the test charge. This polarization “clothes” the test charge and “follows” the test charge around the lattice.
and
(6) ______
E~
iq(z—d) K0 R~
In (5) and (6) Ri is the vector from Q to the field point and R2 is from the image charge to the field point, and 2K0 q l+K0 Q. ‘F
—
The use of equations (5) and (6) along with (2)—(4) permit us to construct the proper correlation potential. In order to obtain a general form for this potential we define a pseudo atomic constant M1 for the ith type as M,
=
a~ —(1—1/K0)~4ir N ~ 7, j=1
(7)
Vol. 16, No.5
ELECTRON CORRELATION OF NONMETALLIC SOLIDS
This then yields the correlation potential
V~(r)as:
(a) For a field point in the solid N fQx,1 + q’x,1\
f
V~(r)=Z tYPS
the solid and outside also. The results of this calculation is shown for hole states in Fig. 1. We see that V, is essentially constant inside the crystal and falls
XU
~) jj~’~
M, ~~
PO~U0fl
of the atom
+
+ qy,1\1 ~ (8) \R~,.1 R2~J~ + (Q(z1i R~ d) + q’(z,1 R~~ + d)\ (z,1 d)1 1 R~
\,
)
—
—
j
‘
the site containing the test charge is omitted from the sum; (b) For a field point in the vacuum N
V~(r)=
“
M, types
1 pod~on
R’h1’
543
(9)
of the ~ In terms of these correlation potentials the self energy of an electron or hole in state ~ is found by letting I Q = electronic charge, e, and = (n I~ e V,~(r) In). (10)
The self energy is positive for creating a hole state (i.e. for occupied bands) and is negative for creating an electron state (i.e. virtual or conduction levels), One anticipates from (8) and (9) that the correlation potential will be relatively constant inside the crystal and fall to zero rapidly outside. Thus one expects that bulk hole (electron) states will be moved up (down) more strongly than will surface states which spend a non-negligible fraction of their time outside the crystal in contrast to bulk state which spend no time outside the semi-infinite solid, To quantize this effect a sample calculation is performed for a simple cubic solid of nearest neighbor distance 4 atomic units as a function ofK 0. It iS assumed the xy plane is a [100] face and the origin is an atom site. ~eV~is computed at atom site inside
rapidly to zero outside. This means that the presence of correlation effects will serve to inhibit the formation of stable surface-localized states in the gap when corn. pared to a Hartree—Fock model. It is also noted in summary that the bulk part of V~,obtained here agrees well with those for bulk solids of like K0 given by 2 Finally these results would be essentially Fowler. correct for K0 a function of position jfK0 varies much more slowly with position than does E, a condition which is likely true. There is one final point to discuss and this concerns the applicability of the linear response, image charge model when the test charge approaches and penetrates the image plane into the solid. Applebaum and Hamann studied this for jellium and found the force on the electron to be poorly given by the image charge resultisfor distances of less 1°That thecharge-jelliurn response of the charge near thethan test 2A. charge is found to saturate at small distances. This problem is avoided in the present case on two accounts. The first is that insulating systems have finite static dielectric constants, and secondly by, the central cell (that atom or group of atoms nearest to the test charge) is explicitly excluded from the pertinent formulae in this paper, equations (8) and (9). Thus the expression given for the correlation energy is too small for the portion below the surface plane, but is correct outside the surface. Therefore, the conclusion reached about correlation inhibiting the formation of surface states is enhanced by this effect. The short range correlation may be included simply using the models of Kunz, Mickish and Collins.6 Recent unpublished calculations of Euwema and Kunz indicate that the response of an atom to an electron remains nearly linear to a separation of about 5 a.u. Thus the size of the excluded central cell need not be too large.
REFERENCES 1.
MOTT N.F. and LITTLETON M.J., Trans. Faraday Soc. 34,485 (1938).
2.
FOWLERW.B.,Phys. Rev. 151,657 (1966).
3.
TOYOZAWA Y., Frog. Theoret. Phys. (Kyoto) 12, 421(1954).
4.
KUNZ A.B.,Phys. Rev. B6, 606 (1972).
544
ELECTRON CORRELATION OF NONMETALLIC SOLIDS
Vol. 16, No. 5
5. 6.
HEDIN L.,Phys. Rev. 139, A796 (1965);ArkivFysik 30, 19 (1965). BRINKMAN W.F. and GOODMAN’B., Phys. Rev. 149, 597 (1966); INOUE M., MAHUTTE C.K. and WANG S., Phys. Rev. B2, 539 (1970); LIPARI N.O. and FOWLER W.B., Solid State Commun. 8, 1395 (1970); KUNZ A.B. and MICKISH D.J.,Phys. Rev. B8, 779 (1973); MICKISH D.J., KUNZ A.B. and COLLINS T.C., Phys. Rev. B9,4461 (1974).
7.
BENNETT AJ., MCCARROLL B. and MESSMER R.P.,Surf Sci., 24, 191 (l971);Phys. Rev. B3, 1397 (1971); KUNZ A.B., MICIUSH D.J. and DEUTSCH P.W., Solid State Commun. 13,35 (1973).
8.
APPLEBAUM J.A. and HAMANN D.R.,Phys. Rev. B6, 2166 (1972); Phys. Rev. Lett. 31, 106 (1973); VAN DOREN V.E. and KUNZ A.B., Phys. Rev. (to be published); KASOWSKI R.V., Phys. Rev. Lett. 33,83 (1974);
9.
JACKSON J.D., Classical Electrodynamics, John Wiley, New York (1962).
10.
APPLEBAUM J.A. and HAMANN D.R.,Phys. Rev. B6, 1122 (1972).