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The interaction between an electron and the polarization modes of a metal-insulator interface
0038-1098/87 $3.00 + .00 © 1987 Pergamon Journals Ltd.
Solid State Communications, Vol. 63, No. 3, pp. 245-249, 1987. Printed in Great Britain.
THE INTERACTION BETWEEN AN ELECTRON AND THE POLARIZATION MODES OF A METAL-INSULATOR I N T E R F A C E F. Sols* and R.H. Ritchie Health and Safety Research Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
(Received 1 February 1987 by A.A. Maradudin) We study the interaction between an electron and the polarization modes of a metal-insulator interface by employing a self-energy approach. The coupling between the metal surface plasmons and the insulator surface excitons is explicitly considered. The surface optical phonons are shown to be screened out by the plasmon field. We derive expressions for the charge-coupling Hamiltonians of the bulk and interface modes and calculate a self-energy profile that includes recoil effects due to the quantum nature of the electron's motion.
THE I N T E R A C T I O N BETWEEN an electron and the polarization modes of a solid-solid surface interface plays an important role in a variety of electron tunnelling and transport processes. The presence of the electron modifies the polarization of the medium and such modification acts back on the electron. In this Communication we address this many-body problem by means of a self-energy approach. First we analyze the polarization modes near an interface, namely, their frequencies and their interaction Hamiltonian with an external electron. The bulk polarization modes of the metal and the insulator are also taken into account. Then we calculate the projected local self-energy following the method introduced by Manson and Ritchie [1]. The real part of the selfenergy gives the contribution of the polarization modes to the effective potential profile which the electron sees near the interface. Details of the calculation, as well as its application to metal and insulatorsurfaces, will be given elsewhere [2]. We consider an interface between two semiinfinite media, 1 and 2, located in the regions z < 0 and z > 0, respectively. We assume that these two media are described by bulk dielectric functions of the type ei(to) =
1 - ~. 092 __ 092
i =
1, 2,
(1)
where index j is summed over all the effective oscillators contributing to the response. The frequencies to~ of the interface modes must satisfy the matching con* Present address: Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. 245
dition [3] e,(to,) + e2(to~) = 0.
(2)
It can be shown [2] that the Hamiltonian describing the interaction between the interface and an external charge e is given by I7" = ~ Fo~ et°'~e-elZl(ao~ + a_+Q~), Q~ ~e2fito~
r~-
AQ
2~ =
2~
(3a)
(3b)
-~'-~-'~to-~ .7 2 2 2 ' (~o~ - too) J
(3c)
where r = (Q, z) is the charge position vector, the index ~ runs over the interface modes (whose frequencies satisfy equation (2)); A is the area of the interface, hQ is the momentum of the interface mode; and a ~ and ao~ are the corresponding creation and anihilation operators. The interface between a metal and an insulator corresponds to the case e,(co)
=
e2(to) =
I -
1
fl2/co 2,
to2
n~ --
(4)
2
(.O e
to2
~)2 --
2,
CO t
(5)
where hf~p is the energy of the bulk metal plasmon; ["I(.De is an effective excitation energy of the electronic transitions in the insulator (which, following Mahan [4], we will call excitons); ~ = 4nn%to~, n being the atomic density and % the static atomic polarizability; co, is the frequency of the transverse optical phonons, ~2 = to~ _ to2, where tot is the longitudinal optical phonon frequency.
The approximation of a single dispersionless exciton can be fairly realistic for a noble gas solid. It is less accurate for an ionic solid (although some compounds like LiF can be reasonably well described by the dielectric function (5) [5]), and it provides only a very qualitative description of the response of covalent insulators. In spite of their simplicity, the dielectric functions (4) and (5) can provide interesting information on the interplay between plasmons, excitons, and optical phonons in a metal-insulator interface. The frequencies of the metal plasmons and insulator excitons lie in the optical range, while those of the insulator optical phonons belong to the infrared region. The insulator dielectric constant in the region (Ot '¢~ (2) '~ (De is e~ = 1 + f~eZ/(De 2, and the static dielectric constant is ~o = e~ + ~z/(D~. Introducing expressions (4) and (5) in equation (2), and noting that (Dr '~ Op and cot ,~ ~e, we obtain the following solutions: (Dip
=
(,0
(Die =
0 9 2_
=
1 2 + ~(f~p
2o02 +
(Dio =
(D+
(D,,
(6b)
(De2 + f~2/2"
(7)
This limit is not realized in common materials. However, it is useful to consider it for a qualitative discussion, since the mathematical expressions are greatly simplified. In particular, equations (6) become (Dip
e~ + 1
- -
(Die -
2
(De
2 =
(8) which, introduced in equation (3c), yield the coupling parameters --
2 e~ + 1
~'/e
~k(r) (OIl?Jn)(n~bkll?[O~ko)
(6a)
where the subscripts a = ip, ie, and io stand for plasmon, exciton, and optical phonon, respectively. This notation must be taken as illustrative, since the real modes are of a mixed nature. It is interesting to consider the limit of a very fast exciton field
n,~/2 ¢
values are recovered when e~ = 1. (Dieand 2ie are the frequency and coupling parameter of the surface exciton, unaffected by the comparatively slow plasmons and optical phonons. The expression for 2,.0is particularly noteworthy. Both in the bulk and in the surface of an insulator, the optical phonons are affected by the fast but limited ((De ¢ 0) electronic screening from the ionic nuclei. However, in the interface with a metal, the electrons whose coherent collective excitation gives rise to the surface plasmon are able to screen very strongly the slow optical phonons, which thus couple negligibly to an external charge. This implies that the optical phonon contribution to the electron self-energy can be completely neglected. This interpretation is consistent with the fact that the frequency COioessentially differs 2 = (D,(~o 2 from the free surface value COso + 1)/ (e~ + 1)[2,4]. We can now calculate the effective local selfenergy of an electron from the expression [1, 2, 6] (10)
f~)
+ [(f~ + 20)2 + f~)2 - ~'~*'eJR .2~2v/2,
2ip
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MODES OF A M E T A L - I N S U L A T O R I N T E R F A C E
246
--
where 10) and [qJ0) are the eigenvectors corresponding to the initial states of the medium and the particle, respectively, In) and [~Ok) are those of the intermediate states, ~bk(r) = (rlqJk), ek and t0 are the eigenfrequencies of the electron's motion, fi(D.o is the excitation energy of the n-th eigenstate of the medium, I2 is the Hamiltonian given the interaction between the electron and the polarization modes, and t/ -~ 0 + . The equation (10) for the electron self-energy is obtained by unfolding the matrix element (01]/0lVIn~/k) in the expression for the second order energy shift, as was proposed by Manson and Ritchie [1]. It can be viewed as the effective self-energy profile experienced by an electron in the quantum state IqJ0). If a plane-wave basis set is taken for the unperturbed motion of the particle and the Hamiltonian (3) is introduced in equation (10), one obtains the selfenergy [1, 2] e2 e Olzl (2x) 2 ~ 2~Q2= f d2Q f dk Q2 +--~
E(z) -
e ikz
eo~ - l e~ + 1
x
02 + k2 + 2kok + 2KoQ + Qz _ it 1,
(11) ~io
-
( ~ " t - 1 ) 2 (O)tx~ 4 e0-~ \~,/ -
0.
(9)
Clearly, (D~pand 2~p correspond to a surface plasmon modified by the presence of a polarizable medium which, in the limit (7), follows adiabatically the interface plasma oscillations. The free plasmon surface
where Q= = (2m(DJfi) 1/2, m is the mass of the particle, and v = h(K0, ko)/m is the velocity of the particle. In order to calculate the total self-energy of an electron near an interface, one must also consider the contributions from the bulk modes of both media.
247
MODES OF A METAL-INSULATOR I N T E R F A C E
Vol. 63', No. 3
These modes are not affected by the interface. If the semi-infinite medium lying in z > 0 is described by a dielectric function of the type (1) (after removing the index i), the interaction between the bulk modes and an electron is given by the Hamiltonian
E,(z ~ -
e2QbP
=
~)
2
(1 )e2Q,o
=-~
~)
X , ( z --, +
(16a)
'
l? = O(z) ~ ~ Aqpd°'esin(pz)(bqp + b+q~), (12a)
--(1-
4xe2hco#
~-1
Vq2
=
2a,
(12b)
2 2 f~) cop 2 ' 2----2
(12c)
• (~o~ -
~;)
where the index fl is summed over the different bulk modes satisfying e(eg#) = 0, V is the volume of the medium, q = (Q, p) and bq# and b,~ are the ladder operators of the mode (q, fl). In a metal described by (4), the bulk plasmon frequency is tnbp = lap and thus 2bp 1. Correspondingly, in an insulator described by (5), the frequencies are ~0b2e= eooO92, for the bulk exciton, and W2o = (e0/ eo~)co,2 for the optical phonon. The corresponding coupling parameters are 2bo = ( l / e ® - l/e0) and 2b~ = (1 -- 1/e~). When the Hamiltonian (12) is introduced into equation (10) for the self-energy, an expression is obtained which is identical to (11) except for the change ~ ~ fl and (e-Qlzle ikz) --+ (1 - e Qlzleikz). For brevity, we only give here the resulting expression for the total self-energy of an electron with velocity v --* 0: =
e2 ~
X,(z)
=
The third term of equation (16b) is the well known classical image potential for a charge embedded in a medium of dielectric constant e0 and located close to the interface with a metal. The equivalent term in an insulator surface only differs in a factor (e0 - 1)/ (e0 + 1). Now we can see that, in the classical result, the fact is implicit that the surface optical phonons play no role in the self-energy near a metal-insulator interface. Our previous analysis provides a deeper understanding of the origin of this classical image potential. We have applied our results to the interface between AI and LiF. These two materials are fairly well described by dielectric functions (4) and (5). For AI we have taken fi~p -- 15.8eV, which corresponds to a jellium density of rs -- 2.07. For LiF, the following parameters have been used [5]: haJt = 38meV, hcoe = 16.8eV, e~ = 1.92, and e0 = 8.9. In Fig. 1 we show the contribution of the interface modes to the self-energy of an electron moving to the right. The separate contributions of the exciton- and 5
l
i
l
l
e2
~- ~ 2~Q~ + 41z[
o
_
~ -_ . - - ~ . . : ,
. . , - - . - ~ ~
'[
,3, where the indexes ~ and fl are summed over the interface and bulk modes (those of a metal in z < 0 and those of an insulator in z > 0), and
S(x) = 1 -- 2E3(x),
(14a)
E,(x) being the exponential integral [7]
~
-10
-is --20
i -8
e- x
E. (x) =
dx --~--.
(14b)
1
It is easy to show that [2] ~.,(0)
=
e2
(16b)
4Zeo "
Q,p p>0
A2q#-
e2
~--~) e2Qbe2
- ~ - ~ 2,Q~.
(15)
From equation (13), and assuming that the limit (7) holds, it is easy to obtain the following asymptotic expressions:
I -6
,
I -4
i
I -2
i
I
I
I
I
0
2
4
6
8
Z (angstroms)
Fig. l. Real part of the self-energy due to the metalinsulator interface modes for an electron moving to the right with velocity v = 0.93 a.u. The parameters corresponding to the system AI/LiF have been taken. The dashed-dotted line shows the contribution from the surface plasmon-like modes, and the dotted line shows that of the surface exciton-like modes. The frequencies and couplings have been taken from equations (3) and (6) of the text.
248
MODES OF A M E T A L - I N S U L A T O R I N T E R F A C E
plasmon-like modes are also shown for comparison. The limit (7) has not been taken. However, it can be shown [2] that, at least for the system AI/LiF, the use of limit (7) gives reasonably good results, due to some compensation between the corrections that the realistic coupling introduces in the plasmon- and excitontype contributions. The self-energy profiles which we have calculated are induced by the interaction of the electron with the plasmons, optical phonons, and excitons. Since there are other physical effects which also contribute to the total effective potential (exchange, charge layers, etc.), simple rules to take into account these other physical terms are desirable. To describe the motion of an electron between the conduction bands of a metal and an insulator, we suggest the simple rule that a step function be added such that the difference between the bulk saturation terms equals the difference between the bottom of the conduction bands, as obtained from experimental data or from more refined calculations. The resulting potential profiles will look more like the actual potential experienced by an electron. Due to the band bending that takes place in metal-semiconductor junctions, the relative position of the conduction bands will differ from that which corresponds to the separated surfaces. Since this band bending takes place in a region of about 100/~ or more, we can assume that the actual saturation values correspond to the bands position at the junction. For the system A1/LiF, our calculations yield a relative height of 6.2eV. A reasonable value for the actual difference could be 13.8eV (it should lie between 11.6 and 17.3 eV) [8]. Thus the height of the added step potential is 7.6 eV. We have introduced the same step for the self-energy at finite velocities, which is consistent with the initial assumption that the unperturbed motion of the electron is described by plane waves. In Fig. 2 we plot the effective potential (real part of the calculated self-energy plus the step function) seen by an electron crossing perpendicularly the interface with velocities v = 0, + 0.93, and + 1.5 a.u. near an AI/LiF interface. It must be noted that the saturation values Eb(+ ~ ) do not have a monotonic dependence on the charge velocity. In conclusion, we have studied the interaction between an electron and the polarization modes near an interface by means of a self-energy approach that explicitly takes into account quantum recoil effects. For the case of a metal and an insulator we have derived expressions for the frequencies and the electron-coupling Hamiltonians of the interface modes. The basic interplay between the metal plasmons and the insulator excitons and optical phonons has been discussed. Since the energies of the excitons and the
0
.
.
.
.
i
.
.
.
.
Vol. 63, No. 3 i
. . . .
i
. . . .
...._. -5
-10
-15
...J
--20
, -10
,
,
,
i
. . . . -5
I
. . . .
0 z
l = , , , 5
10
(angstroms)
Fig. 2. Real part of the total self-energy near an A1/LiF interface for an electron moving to the right with velocities v = 0 (solid line), v = VF = 0.93 (dashed-dotted line), and v = 1.5 a.u. (dashed line). The metal is in z < 0 and the insulator is in z > 0. See the text for comments on the step potential introduced a t z = 0. plasmons are comparable, we have considered the resulting coupled interface modes. The surface optical phonons are strongly screened by the metal surface plasmons and play no role in the interface. Further improvements of this model should include a more realistic bulk response for both the metal and the insulator (with a more detailed treatment of the electrodynamical problem at the interface), a more suitable basis set describing the electron's unperturbed motion (such as the solutions of a step potential with different effective masses on each side), and a proper consideration of the Pauli exclusion principle in the calculation of the expression (10) for the self-energy. Acknowledgements - - We thank J.R. Manson and F. Flores for helpful discussions. This research was sponsored jointly by the U.S.-Spain Joint Committee for the Scientific and Technological Corporation, under IAG 40-1516-84, and the Office of Health and Environmental Research and the Division of Electric Energy Systems, U.S. Department of Energy, under contract DE-AC05-840R21400 with the Martin Marietta Energy Systems, Inc.
REFERENCES 1. 2. 3. 4.
J.R. Manson & R.H. Ritchie, Phys. Rev. B24, 4867 (1981). F. Sols & R.H. Ritchie, to be published. E.A. Stern & R.A. Ferrell, Phys. Rev. 120, 130 (1960). G.D. Mahan, Elementary Excitations in Solids,
Vol. 6'3, No. 3
. .
MODES OF A METAL-INSULATOR INTERFACE
Molecules, and Atoms, Part B, p. 93, (Edited by J.T. Devreese, A.B. Kunz, and T.C. Collins), Plenum Press, New York, (1974). E.D. Palik & W.R. Hunter, Handbook of Optical Constants of Solids, (Edited by E.D. Palik), Academic Press, New York, (1985). J. Mahanty, K.N. Pathak & V.V. Paranjape, Phys. Rev. B33, 2333 (1986).
7.
8.
249
Handbook of Mathematical Functions, (Edited by M. Abramowitz and I.A. Stegun) p. 232, National Bureau of Standards, Washington, D.C., (1964). F. Guinea, J. Sanchez-Dehesa & F. Flores, J. Phys. C16, 6499 (1983); a detailed explanation is given in [2].
Solid State Communications, Vol. 63, No. 3, pp. 245-249, 1987. Printed in Great Britain.
THE INTERACTION BETWEEN AN ELECTRON AND THE POLARIZATION MODES OF A METAL-INSULATOR I N T E R F A C E F. Sols* and R.H. Ritchie Health and Safety Research Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
(Received 1 February 1987 by A.A. Maradudin) We study the interaction between an electron and the polarization modes of a metal-insulator interface by employing a self-energy approach. The coupling between the metal surface plasmons and the insulator surface excitons is explicitly considered. The surface optical phonons are shown to be screened out by the plasmon field. We derive expressions for the charge-coupling Hamiltonians of the bulk and interface modes and calculate a self-energy profile that includes recoil effects due to the quantum nature of the electron's motion.
THE I N T E R A C T I O N BETWEEN an electron and the polarization modes of a solid-solid surface interface plays an important role in a variety of electron tunnelling and transport processes. The presence of the electron modifies the polarization of the medium and such modification acts back on the electron. In this Communication we address this many-body problem by means of a self-energy approach. First we analyze the polarization modes near an interface, namely, their frequencies and their interaction Hamiltonian with an external electron. The bulk polarization modes of the metal and the insulator are also taken into account. Then we calculate the projected local self-energy following the method introduced by Manson and Ritchie [1]. The real part of the selfenergy gives the contribution of the polarization modes to the effective potential profile which the electron sees near the interface. Details of the calculation, as well as its application to metal and insulatorsurfaces, will be given elsewhere [2]. We consider an interface between two semiinfinite media, 1 and 2, located in the regions z < 0 and z > 0, respectively. We assume that these two media are described by bulk dielectric functions of the type ei(to) =
1 - ~. 092 __ 092
i =
1, 2,
(1)
where index j is summed over all the effective oscillators contributing to the response. The frequencies to~ of the interface modes must satisfy the matching con* Present address: Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. 245
dition [3] e,(to,) + e2(to~) = 0.
(2)
It can be shown [2] that the Hamiltonian describing the interaction between the interface and an external charge e is given by I7" = ~ Fo~ et°'~e-elZl(ao~ + a_+Q~), Q~ ~e2fito~
r~-
AQ
2~ =
2~
(3a)
(3b)
-~'-~-'~to-~ .7 2 2 2 ' (~o~ - too) J
(3c)
where r = (Q, z) is the charge position vector, the index ~ runs over the interface modes (whose frequencies satisfy equation (2)); A is the area of the interface, hQ is the momentum of the interface mode; and a ~ and ao~ are the corresponding creation and anihilation operators. The interface between a metal and an insulator corresponds to the case e,(co)
=
e2(to) =
I -
1
fl2/co 2,
to2
n~ --
(4)
2
(.O e
to2
~)2 --
2,
CO t
(5)
where hf~p is the energy of the bulk metal plasmon; ["I(.De is an effective excitation energy of the electronic transitions in the insulator (which, following Mahan [4], we will call excitons); ~ = 4nn%to~, n being the atomic density and % the static atomic polarizability; co, is the frequency of the transverse optical phonons, ~2 = to~ _ to2, where tot is the longitudinal optical phonon frequency.
The approximation of a single dispersionless exciton can be fairly realistic for a noble gas solid. It is less accurate for an ionic solid (although some compounds like LiF can be reasonably well described by the dielectric function (5) [5]), and it provides only a very qualitative description of the response of covalent insulators. In spite of their simplicity, the dielectric functions (4) and (5) can provide interesting information on the interplay between plasmons, excitons, and optical phonons in a metal-insulator interface. The frequencies of the metal plasmons and insulator excitons lie in the optical range, while those of the insulator optical phonons belong to the infrared region. The insulator dielectric constant in the region (Ot '¢~ (2) '~ (De is e~ = 1 + f~eZ/(De 2, and the static dielectric constant is ~o = e~ + ~z/(D~. Introducing expressions (4) and (5) in equation (2), and noting that (Dr '~ Op and cot ,~ ~e, we obtain the following solutions: (Dip
=
(,0
(Die =
0 9 2_
=
1 2 + ~(f~p
2o02 +
(Dio =
(D+
(D,,
(6b)
(De2 + f~2/2"
(7)
This limit is not realized in common materials. However, it is useful to consider it for a qualitative discussion, since the mathematical expressions are greatly simplified. In particular, equations (6) become (Dip
e~ + 1
- -
(Die -
2
(De
2 =
(8) which, introduced in equation (3c), yield the coupling parameters --
2 e~ + 1
~'/e
~k(r) (OIl?Jn)(n~bkll?[O~ko)
(6a)
where the subscripts a = ip, ie, and io stand for plasmon, exciton, and optical phonon, respectively. This notation must be taken as illustrative, since the real modes are of a mixed nature. It is interesting to consider the limit of a very fast exciton field
n,~/2 ¢
values are recovered when e~ = 1. (Dieand 2ie are the frequency and coupling parameter of the surface exciton, unaffected by the comparatively slow plasmons and optical phonons. The expression for 2,.0is particularly noteworthy. Both in the bulk and in the surface of an insulator, the optical phonons are affected by the fast but limited ((De ¢ 0) electronic screening from the ionic nuclei. However, in the interface with a metal, the electrons whose coherent collective excitation gives rise to the surface plasmon are able to screen very strongly the slow optical phonons, which thus couple negligibly to an external charge. This implies that the optical phonon contribution to the electron self-energy can be completely neglected. This interpretation is consistent with the fact that the frequency COioessentially differs 2 = (D,(~o 2 from the free surface value COso + 1)/ (e~ + 1)[2,4]. We can now calculate the effective local selfenergy of an electron from the expression [1, 2, 6] (10)
f~)
+ [(f~ + 20)2 + f~)2 - ~'~*'eJR .2~2v/2,
2ip
Vol. 63; No. 3
MODES OF A M E T A L - I N S U L A T O R I N T E R F A C E
246
--
where 10) and [qJ0) are the eigenvectors corresponding to the initial states of the medium and the particle, respectively, In) and [~Ok) are those of the intermediate states, ~bk(r) = (rlqJk), ek and t0 are the eigenfrequencies of the electron's motion, fi(D.o is the excitation energy of the n-th eigenstate of the medium, I2 is the Hamiltonian given the interaction between the electron and the polarization modes, and t/ -~ 0 + . The equation (10) for the electron self-energy is obtained by unfolding the matrix element (01]/0lVIn~/k) in the expression for the second order energy shift, as was proposed by Manson and Ritchie [1]. It can be viewed as the effective self-energy profile experienced by an electron in the quantum state IqJ0). If a plane-wave basis set is taken for the unperturbed motion of the particle and the Hamiltonian (3) is introduced in equation (10), one obtains the selfenergy [1, 2] e2 e Olzl (2x) 2 ~ 2~Q2= f d2Q f dk Q2 +--~
E(z) -
e ikz
eo~ - l e~ + 1
x
02 + k2 + 2kok + 2KoQ + Qz _ it 1,
(11) ~io
-
( ~ " t - 1 ) 2 (O)tx~ 4 e0-~ \~,/ -
0.
(9)
Clearly, (D~pand 2~p correspond to a surface plasmon modified by the presence of a polarizable medium which, in the limit (7), follows adiabatically the interface plasma oscillations. The free plasmon surface
where Q= = (2m(DJfi) 1/2, m is the mass of the particle, and v = h(K0, ko)/m is the velocity of the particle. In order to calculate the total self-energy of an electron near an interface, one must also consider the contributions from the bulk modes of both media.
247
MODES OF A METAL-INSULATOR I N T E R F A C E
Vol. 63', No. 3
These modes are not affected by the interface. If the semi-infinite medium lying in z > 0 is described by a dielectric function of the type (1) (after removing the index i), the interaction between the bulk modes and an electron is given by the Hamiltonian
E,(z ~ -
e2QbP
=
~)
2
(1 )e2Q,o
=-~
~)
X , ( z --, +
(16a)
'
l? = O(z) ~ ~ Aqpd°'esin(pz)(bqp + b+q~), (12a)
--(1-
4xe2hco#
~-1
Vq2
=
2a,
(12b)
2 2 f~) cop 2 ' 2----2
(12c)
• (~o~ -
~;)
where the index fl is summed over the different bulk modes satisfying e(eg#) = 0, V is the volume of the medium, q = (Q, p) and bq# and b,~ are the ladder operators of the mode (q, fl). In a metal described by (4), the bulk plasmon frequency is tnbp = lap and thus 2bp 1. Correspondingly, in an insulator described by (5), the frequencies are ~0b2e= eooO92, for the bulk exciton, and W2o = (e0/ eo~)co,2 for the optical phonon. The corresponding coupling parameters are 2bo = ( l / e ® - l/e0) and 2b~ = (1 -- 1/e~). When the Hamiltonian (12) is introduced into equation (10) for the self-energy, an expression is obtained which is identical to (11) except for the change ~ ~ fl and (e-Qlzle ikz) --+ (1 - e Qlzleikz). For brevity, we only give here the resulting expression for the total self-energy of an electron with velocity v --* 0: =
e2 ~
X,(z)
=
The third term of equation (16b) is the well known classical image potential for a charge embedded in a medium of dielectric constant e0 and located close to the interface with a metal. The equivalent term in an insulator surface only differs in a factor (e0 - 1)/ (e0 + 1). Now we can see that, in the classical result, the fact is implicit that the surface optical phonons play no role in the self-energy near a metal-insulator interface. Our previous analysis provides a deeper understanding of the origin of this classical image potential. We have applied our results to the interface between AI and LiF. These two materials are fairly well described by dielectric functions (4) and (5). For AI we have taken fi~p -- 15.8eV, which corresponds to a jellium density of rs -- 2.07. For LiF, the following parameters have been used [5]: haJt = 38meV, hcoe = 16.8eV, e~ = 1.92, and e0 = 8.9. In Fig. 1 we show the contribution of the interface modes to the self-energy of an electron moving to the right. The separate contributions of the exciton- and 5
l
i
l
l
e2
~- ~ 2~Q~ + 41z[
o
_
~ -_ . - - ~ . . : ,
. . , - - . - ~ ~
'[
,3, where the indexes ~ and fl are summed over the interface and bulk modes (those of a metal in z < 0 and those of an insulator in z > 0), and
S(x) = 1 -- 2E3(x),
(14a)
E,(x) being the exponential integral [7]
~
-10
-is --20
i -8
e- x
E. (x) =
dx --~--.
(14b)
1
It is easy to show that [2] ~.,(0)
=
e2
(16b)
4Zeo "
Q,p p>0
A2q#-
e2
~--~) e2Qbe2
- ~ - ~ 2,Q~.
(15)
From equation (13), and assuming that the limit (7) holds, it is easy to obtain the following asymptotic expressions:
I -6
,
I -4
i
I -2
i
I
I
I
I
0
2
4
6
8
Z (angstroms)
Fig. l. Real part of the self-energy due to the metalinsulator interface modes for an electron moving to the right with velocity v = 0.93 a.u. The parameters corresponding to the system AI/LiF have been taken. The dashed-dotted line shows the contribution from the surface plasmon-like modes, and the dotted line shows that of the surface exciton-like modes. The frequencies and couplings have been taken from equations (3) and (6) of the text.
248
MODES OF A M E T A L - I N S U L A T O R I N T E R F A C E
plasmon-like modes are also shown for comparison. The limit (7) has not been taken. However, it can be shown [2] that, at least for the system AI/LiF, the use of limit (7) gives reasonably good results, due to some compensation between the corrections that the realistic coupling introduces in the plasmon- and excitontype contributions. The self-energy profiles which we have calculated are induced by the interaction of the electron with the plasmons, optical phonons, and excitons. Since there are other physical effects which also contribute to the total effective potential (exchange, charge layers, etc.), simple rules to take into account these other physical terms are desirable. To describe the motion of an electron between the conduction bands of a metal and an insulator, we suggest the simple rule that a step function be added such that the difference between the bulk saturation terms equals the difference between the bottom of the conduction bands, as obtained from experimental data or from more refined calculations. The resulting potential profiles will look more like the actual potential experienced by an electron. Due to the band bending that takes place in metal-semiconductor junctions, the relative position of the conduction bands will differ from that which corresponds to the separated surfaces. Since this band bending takes place in a region of about 100/~ or more, we can assume that the actual saturation values correspond to the bands position at the junction. For the system A1/LiF, our calculations yield a relative height of 6.2eV. A reasonable value for the actual difference could be 13.8eV (it should lie between 11.6 and 17.3 eV) [8]. Thus the height of the added step potential is 7.6 eV. We have introduced the same step for the self-energy at finite velocities, which is consistent with the initial assumption that the unperturbed motion of the electron is described by plane waves. In Fig. 2 we plot the effective potential (real part of the calculated self-energy plus the step function) seen by an electron crossing perpendicularly the interface with velocities v = 0, + 0.93, and + 1.5 a.u. near an AI/LiF interface. It must be noted that the saturation values Eb(+ ~ ) do not have a monotonic dependence on the charge velocity. In conclusion, we have studied the interaction between an electron and the polarization modes near an interface by means of a self-energy approach that explicitly takes into account quantum recoil effects. For the case of a metal and an insulator we have derived expressions for the frequencies and the electron-coupling Hamiltonians of the interface modes. The basic interplay between the metal plasmons and the insulator excitons and optical phonons has been discussed. Since the energies of the excitons and the
0
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.
.
.
i
.
.
.
.
Vol. 63, No. 3 i
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Fig. 2. Real part of the total self-energy near an A1/LiF interface for an electron moving to the right with velocities v = 0 (solid line), v = VF = 0.93 (dashed-dotted line), and v = 1.5 a.u. (dashed line). The metal is in z < 0 and the insulator is in z > 0. See the text for comments on the step potential introduced a t z = 0. plasmons are comparable, we have considered the resulting coupled interface modes. The surface optical phonons are strongly screened by the metal surface plasmons and play no role in the interface. Further improvements of this model should include a more realistic bulk response for both the metal and the insulator (with a more detailed treatment of the electrodynamical problem at the interface), a more suitable basis set describing the electron's unperturbed motion (such as the solutions of a step potential with different effective masses on each side), and a proper consideration of the Pauli exclusion principle in the calculation of the expression (10) for the self-energy. Acknowledgements - - We thank J.R. Manson and F. Flores for helpful discussions. This research was sponsored jointly by the U.S.-Spain Joint Committee for the Scientific and Technological Corporation, under IAG 40-1516-84, and the Office of Health and Environmental Research and the Division of Electric Energy Systems, U.S. Department of Energy, under contract DE-AC05-840R21400 with the Martin Marietta Energy Systems, Inc.
REFERENCES 1. 2. 3. 4.
J.R. Manson & R.H. Ritchie, Phys. Rev. B24, 4867 (1981). F. Sols & R.H. Ritchie, to be published. E.A. Stern & R.A. Ferrell, Phys. Rev. 120, 130 (1960). G.D. Mahan, Elementary Excitations in Solids,
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MODES OF A METAL-INSULATOR INTERFACE
Molecules, and Atoms, Part B, p. 93, (Edited by J.T. Devreese, A.B. Kunz, and T.C. Collins), Plenum Press, New York, (1974). E.D. Palik & W.R. Hunter, Handbook of Optical Constants of Solids, (Edited by E.D. Palik), Academic Press, New York, (1985). J. Mahanty, K.N. Pathak & V.V. Paranjape, Phys. Rev. B33, 2333 (1986).
7.
8.
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Handbook of Mathematical Functions, (Edited by M. Abramowitz and I.A. Stegun) p. 232, National Bureau of Standards, Washington, D.C., (1964). F. Guinea, J. Sanchez-Dehesa & F. Flores, J. Phys. C16, 6499 (1983); a detailed explanation is given in [2].