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Microscopic theory of covalent-ionic transition at metal-semiconductor interfaces
SURFACE
SCIENCE 37 (1973) 24-29 0 North-Holland
MICROSCOPIC
THEORY
Publishing Co.
OF COVALENT-IONIC
AT METAL-SEMICONDUCTOR
TRANSITION
INTERFACES
J. C. PHILLIPS Bell Laboratories,
Murray
Hill, New Jersey 07974,
U.S.A.
The microscopic mechanism responsible for the reduction of Schottky barrier heights at interfaces between metals and covalent semiconductors is not effective at similar interfaces between metals and ionic semiconductors. The critical polarizability .sc at which the transition between one regime and the other takes place is calculated theoretically to be 7, in good agreement with experimental values of 5 to 6.
It has been generally recognized for some years that metal-semiconductor (M-S) interfaces can be classified into two broad groups based on the semiconductor S. The height (pa of the barrier to electronic conduction in “ionic” semiconductors or insulators (e.g., ZnO or SiO,) is given by the Schottky relation’) (PB =
(PM
-
VS,
(1)
where (Pi and ‘ps are the work functions of the metal and semiconductor, respectively. In “covalent” semiconductors, on the other hand, barriers are generally much smaller and can be described by (PB
=
a ts)
[%I
-
%I
+
‘PBe ts)
3
(2)
with 0.05 5a(S) 50.2. Here (Pae is a small (-a few tenths of an eV), nearly constant term associated with bulk mismatch at the interface. It has been shown by Kurtin, McGill and Mead2) that the experimental values of a(S) obtained from eq. (2) follow systematic chemical trends as a function of the Pauling electronegative difference AX=X(A)-X(B) of the atoms A and B which make up the semiconductor S. A rather abrupt transition from “covalent” to “ionic” behavior occurs near AX=O.7, corresponding to compounds such as CdS. So far, however, a satisfactory microscopic mechanism for the transition has not been proposed. When the need for small values of a(S) for S one of the covalent semiconductors (such as Si or Ge) first became apparent, Bardeen proposed3) that the reduction in barrier height at the interface could be understood in terms of the filling of surface states. In the intervening decades much effort has been lavished on the properties of surface states, especially at semi24
MICROSCOPIC
25
THEORY
conductor-vacuum interfaces. However, such “clean-surface” states bear no obvious relation to the interfacial surface states hypothesized by Bardeen. Moreover, their connection with the covalent-ionic transition is quite remote and even apparently contradictory. It is only in ionic materials that surface states should be stable at the interface with a metal, as pointed out by P. W. Anderson and Inkson4). From the universality of the transition in a (S(AX)) described by Kurtin, McGill and Mead it would appear that a satisfactory description of the interfacial region should not rely on single-particle basis states (which are
IONIC-COVALENT TRANSITION AT METAL-INSULATOR INTERFACES
/~ 0
_I 10
5
15
e-1
Fig. 1. A plot of the index a(S) of surface states defined in ref. 2 as a function of a semiconductor dielectric polarizability 4~ = EO- 1. The quantity a(S) is a measure of the extent to which surface polarization is able to reduce the barrier height at metal-semiconductor interfaces from the value expected in the limit of no polarization, i.e., the difference in work functions of metal and semiconductor. From this plot it is concluded that the maximum surface polarizabihty is about equal to the bulk dielectric polarizability of ZnSe, i.e., esmax = 5 to 6.
sensitive to the details of the energy-band structure E,(k)), but should rather emphasize properties of the overall valence-electron charge distribution. One such property is the bond polarizability. In fig. 1 we see that a(S) is indeed a smooth function of the low-frequency electronic dielectric constant Q(S). The transition from covalent to ionic behavior takes place around a0 = E, z 5 or 6. The task of a microscopic theory is to derive a value for a,. The function of the ubiquitous (and microscopically elusive) surfaces states is to make possible charge redistribution giving rise to a double layer. This charge redistribution is characterized microscopically in terms of alteration
26
J. C. PHILLIPS
of the zero-point energy of surface plasmons. The charge redistribution contributes to the surface energy and the work function, and it changes the surface polarizability. The author has suggested previouslys) that surface plasmons should play a crucial role in the covalent-ionic M-S interfacial transition, but the previous discussion was technically incomplete. Now a conceptually satisfactory discussion is possible based on application and modification of the theory of Schmit and Lucas of surface energies of metals 6). According to Schmit and Lucas, the surface energy of a metal derives from changes in the zero-point energies of longitudinal and transverse plasma oscillations of frequency oL= wP and o,=O. At the metal-vacuum interface wt, = wr = w, = oP/J2, the surface plasmon frequency. The total zero-point collective energy is lessened through the reduction in wL at the surface (corresponding to the reduction in electron-electron repulsion in the electron density tailing into the vacuum), but this is more than compensated by the increase in oT (which is analogous to a reduction in Van der Waals correlation energies of electrons near the surface). Thus the long-range or collective part of the surface energy of a metal is always positive and is given by M
cs
-
+I~.! /joMk2
LR-167T
pc’
where k, is a characteristic wavenumber cutoff. The agreement of eq. (3) with experiment is generally good, indicating that single-particle (short-range) contributions to o are less significant than cLR. At the metal-semiconductor interface the surface energy arR is available for producing a double-layer in the semiconductor through charge-transfer excitation. The average energy required to produce such an excitation is he& where the dielectric constant a0 of the semiconductor is given by &g
=
I +
(w;/w;,‘~
The electric field associated with the difference (1) in work functions of bulk metal and bulk semiconductor can be screened only if the surface energy 0;s of the metal is sufficiently large compared to the charge-transfer excitation energy Ao~ per unit area. The phase-space cutoffs for the oscillators associated with these two quantities are expected to be nearly equal, just as was the case in eq. (3). Thus one can assign oscillator energies per atom to the metal and semiconductor of &0~(J2 - I ) and J&J& respectively. (The factors of + arise in the former case from zero-point energies, in the latter case because a charge-transfer excitation involves two atoms.) The threshold semiconductor polarizability for charge-transfer excitations can be cal-
MICROSCOPIC
culated by equating these energies and neglecting and CD:in the interfacial region. This gives CD;=0$(~~2 EC= l+(J2-
27
THEORY
the difference
between
WY
- l),
(5)
1)_2=7.
(6)
Short-range (dangling-bond) corrections will tend to reduce the value of a,, because residual covalency at the interface means that not all the surface energy of the metal is available for charge-transfer (formation of Bardeen “surface states”). Thus we consider that the result (6) is in excellent agreement with the experimental value c,z5 to 6. The discussion of the development of a double layer in the interfacial region given here, based on charge redistribution through energy supplied by and to collective modes, is quite different from the conventional explanations) based on filling of “surface states” lying near the Fermi energy and in the forbidden energy region of width AEcv between the conduction and valence band edges7). Indeed it appears that the single-particle energy level diagram does not represent a useful approach to describing the dielectric properties of the interfacial region7). One can use the discussion given here to place an upper limit of eS&, gives rise to instability of the interfacial region. This instability is of interest in connection with efforts to enhance superconducting transition temperatures through preparation of metal-semiconductor sandwichess99). There are a number of semiconductors with layer structures, such as GaSelO) (Se-Ga-Ga-Se sandwiches) and SnS,ll) and PbI, (X-M-X sandwiches). If these compounds had cubic structures, they would probably be metals, inasmuch as they are electron-rich (electron/atom ratio>4) and contain heavy elements. However, every cubic metal may be unstable as a bulk, three-dimensional system if its surface energy CJis negative. Another possibility is that its surface energy is positive but too small, and therefore it is unstable against formation of a layer-like structure based on sandwiches. If this is the case, then the entire structure of the crystal has undergone a cubic, metallic-+ layer, semiconductor transition. The dielectric constants of such layer semiconductors should be governed by approximately the same argument as discussed above for metal-semiconductor interfaces. The values
28
J.C.PHILLlYS TABLE
1
Some dielectric constants of semiconductors and semimetals; in the case of semimetals the interband dielectric constant is given Material . SnS2
SnSe2 Cdlz PbIz GaSe ZnSe GeSe PbTe Bi C(graphite) C(diamond)
Structure __-__._....._.”_~
Eo(interband)
Layer Layer Layer Layer Layer Cubic Cubic Cubic Pseudo-cubic Layer Cubic
7.5 7.5 4.8 4.2 4.7 5.9 22 33 100” 4.8 5.7
& Ref. 14.
of q, for layer-like materials are collected in table I. They appear to conform to this reasoning. Some dielectric constants of chemically similar cubic semiconductors are also entered in this table. The values of E@can go as high as 100 for cubic semiconductors. Another interesting case is that of diamond~graphite. Here we would argue that it is the small value of c0 in diamond that makes this transition possible, i.e., cubic semiconductors are unstable against layer formation only if their dielectric constants are small enough already so that the interband ~o~arizabi~ity does not change appreciably because of the transition. This argument is made more plausible by the small changes that are known to occur in e, when a cubic semiconductor becomes amorphous l2). Direct experimental evidence for the smallness of Ed in the interfacial region has been obtained for the specific case of cesiated Si surfaces13). From the measured interfacial plasmon energy of 7 eV one infers +=5.2, in good agreement with the experimental values of E, at Schottky interfaces. References 1) W. Schottky, Z. Physik 118 (1942) 539. 2) S. Kurtin, T. C. McGill and C. A. Mead, Phys. Rev. Letters 22 (1969) 1433. 3) J. Bardeen, Phys. Rev. 71 (1947) 717. 4) J. C. Inkson, J. Phys. C 4 (1971) 591. 5) J. C. Phillips, Phys. Rev. B 1 (1970) 593. 6) J. Schmit and A. A. Lucas, Solid State Commun 11 (1972) 415,419. 7) Note that in most semiconductors fiwg % Al& (e.g., in Ge one has fiwg = 4.3 eV and AhEcv= 0.8 eV). See also ref. 5. Recently J. C. Inkson, J. Phys. C 5 (1972) 2599, has
MICROSCOPICTHEORY
8) 9) 10) 11) 12) 13) 14)
29
attempted to apply the ideas of ref. 5 to discuss the interfacial covalent-ionic transition, but his discussion is rendered irrelevant by his failure to distinguish between Eg (the average bond energy) and AE,, (of order the smallest direct energy gap). The plasmons he discusses are perturbations on the (rigid) single-particle band structure of the semiconductor, and make no allowance for the massive redistribution of charge and polarizability which arises because of the collective effects discussed here and in ref. 6. D. Allender, J. Bray and J. Bardeen, to be published. J. C. Phillips, Phys. Rev. Letters, in press. J. C. Phillips, Phys. Rev. 188 (1969) 1225. C. Y. Fong and M. C. Cohen, Phys. Rev. B 5 (1972) 3095. J. Stuke, J. Non-Crystalline Solids 4 (1970) 1. J. D. Levine, Surface Sci. 34 (1973) 90. W. S. Boyle and A. D. Brailsford, Phys. Rev. 120 (1960) 1943.
Discussion Question (by G. HEILAND): I would like to draw your attention to the fact that the contact between clean cleaved Si surfaces andevaporatedmetalsmay still be more complicated than is assumed in the models of Schottky and of Bardeen. Two different metals have been evaporated simultaneously on pairs of Si surfaces created by the same cleavage process. If the differential capacity is plotted as l/C2 versus bias voltage there appears a small but reproducible difference of slope that cannot be attributed to a difference in donor concentration or dielectric constant of the Si. The model of an interfacial layer and of surface states may need some refinement to explam the experimental data [H. Harreis and G. Heiland, Surface Sci. 24 (1971) 6431. Speaker’s reply (by J. C.
PHILLIPS):
Yes.
Question (by S. RABII) Did you find any trends in plotting compounds?
l/[&(0)-1]
as a function of (Az)a for the IV-VI
Speaker’s reply (by J. C. PHILLIPS): Because of the presence of nonbonding s electrons in AnBlo-” compounds, the analysis of e(0) is more complicated than for AnB8-n compounds. However, some chemical trends can be identified. Question (by J. D. Dow): In discussing the surface energy, you seem to treat surface plasmons and dipole layers independently. Would you please discuss the modifications of this treatment necessary for an ionic solid, in which a significant fraction of the oscillator strength is exhausted by the excitons, the surface modes are longitudinal excitons, and the bulk excitons are relatively tightly bound? Speaker’s reply (by J. C. PHILLIPS): The transition fromcovalent to ionic behavior at the metal-semiconductor interface takes place in compounds such as CdSzSel_z which are covalent in the bulk (small fraction of oscillator strength in excitions, spectroscopic ionicity about 0.6). For x values for which the compound has become so ionic that a large fraction of the oscillator strength is associated with excitons (alkali halides, ionicity 0.9 or more) the transition at the surface has been completed and the materials are in the Schottky limit (low polarizability and no double layer). In this limit the question therefore becomes irrelevant.
SCIENCE 37 (1973) 24-29 0 North-Holland
MICROSCOPIC
THEORY
Publishing Co.
OF COVALENT-IONIC
AT METAL-SEMICONDUCTOR
TRANSITION
INTERFACES
J. C. PHILLIPS Bell Laboratories,
Murray
Hill, New Jersey 07974,
U.S.A.
The microscopic mechanism responsible for the reduction of Schottky barrier heights at interfaces between metals and covalent semiconductors is not effective at similar interfaces between metals and ionic semiconductors. The critical polarizability .sc at which the transition between one regime and the other takes place is calculated theoretically to be 7, in good agreement with experimental values of 5 to 6.
It has been generally recognized for some years that metal-semiconductor (M-S) interfaces can be classified into two broad groups based on the semiconductor S. The height (pa of the barrier to electronic conduction in “ionic” semiconductors or insulators (e.g., ZnO or SiO,) is given by the Schottky relation’) (PB =
(PM
-
VS,
(1)
where (Pi and ‘ps are the work functions of the metal and semiconductor, respectively. In “covalent” semiconductors, on the other hand, barriers are generally much smaller and can be described by (PB
=
a ts)
[%I
-
%I
+
‘PBe ts)
3
(2)
with 0.05 5a(S) 50.2. Here (Pae is a small (-a few tenths of an eV), nearly constant term associated with bulk mismatch at the interface. It has been shown by Kurtin, McGill and Mead2) that the experimental values of a(S) obtained from eq. (2) follow systematic chemical trends as a function of the Pauling electronegative difference AX=X(A)-X(B) of the atoms A and B which make up the semiconductor S. A rather abrupt transition from “covalent” to “ionic” behavior occurs near AX=O.7, corresponding to compounds such as CdS. So far, however, a satisfactory microscopic mechanism for the transition has not been proposed. When the need for small values of a(S) for S one of the covalent semiconductors (such as Si or Ge) first became apparent, Bardeen proposed3) that the reduction in barrier height at the interface could be understood in terms of the filling of surface states. In the intervening decades much effort has been lavished on the properties of surface states, especially at semi24
MICROSCOPIC
25
THEORY
conductor-vacuum interfaces. However, such “clean-surface” states bear no obvious relation to the interfacial surface states hypothesized by Bardeen. Moreover, their connection with the covalent-ionic transition is quite remote and even apparently contradictory. It is only in ionic materials that surface states should be stable at the interface with a metal, as pointed out by P. W. Anderson and Inkson4). From the universality of the transition in a (S(AX)) described by Kurtin, McGill and Mead it would appear that a satisfactory description of the interfacial region should not rely on single-particle basis states (which are
IONIC-COVALENT TRANSITION AT METAL-INSULATOR INTERFACES
/~ 0
_I 10
5
15
e-1
Fig. 1. A plot of the index a(S) of surface states defined in ref. 2 as a function of a semiconductor dielectric polarizability 4~ = EO- 1. The quantity a(S) is a measure of the extent to which surface polarization is able to reduce the barrier height at metal-semiconductor interfaces from the value expected in the limit of no polarization, i.e., the difference in work functions of metal and semiconductor. From this plot it is concluded that the maximum surface polarizabihty is about equal to the bulk dielectric polarizability of ZnSe, i.e., esmax = 5 to 6.
sensitive to the details of the energy-band structure E,(k)), but should rather emphasize properties of the overall valence-electron charge distribution. One such property is the bond polarizability. In fig. 1 we see that a(S) is indeed a smooth function of the low-frequency electronic dielectric constant Q(S). The transition from covalent to ionic behavior takes place around a0 = E, z 5 or 6. The task of a microscopic theory is to derive a value for a,. The function of the ubiquitous (and microscopically elusive) surfaces states is to make possible charge redistribution giving rise to a double layer. This charge redistribution is characterized microscopically in terms of alteration
26
J. C. PHILLIPS
of the zero-point energy of surface plasmons. The charge redistribution contributes to the surface energy and the work function, and it changes the surface polarizability. The author has suggested previouslys) that surface plasmons should play a crucial role in the covalent-ionic M-S interfacial transition, but the previous discussion was technically incomplete. Now a conceptually satisfactory discussion is possible based on application and modification of the theory of Schmit and Lucas of surface energies of metals 6). According to Schmit and Lucas, the surface energy of a metal derives from changes in the zero-point energies of longitudinal and transverse plasma oscillations of frequency oL= wP and o,=O. At the metal-vacuum interface wt, = wr = w, = oP/J2, the surface plasmon frequency. The total zero-point collective energy is lessened through the reduction in wL at the surface (corresponding to the reduction in electron-electron repulsion in the electron density tailing into the vacuum), but this is more than compensated by the increase in oT (which is analogous to a reduction in Van der Waals correlation energies of electrons near the surface). Thus the long-range or collective part of the surface energy of a metal is always positive and is given by M
cs
-
+I~.! /joMk2
LR-167T
pc’
where k, is a characteristic wavenumber cutoff. The agreement of eq. (3) with experiment is generally good, indicating that single-particle (short-range) contributions to o are less significant than cLR. At the metal-semiconductor interface the surface energy arR is available for producing a double-layer in the semiconductor through charge-transfer excitation. The average energy required to produce such an excitation is he& where the dielectric constant a0 of the semiconductor is given by &g
=
I +
(w;/w;,‘~
The electric field associated with the difference (1) in work functions of bulk metal and bulk semiconductor can be screened only if the surface energy 0;s of the metal is sufficiently large compared to the charge-transfer excitation energy Ao~ per unit area. The phase-space cutoffs for the oscillators associated with these two quantities are expected to be nearly equal, just as was the case in eq. (3). Thus one can assign oscillator energies per atom to the metal and semiconductor of &0~(J2 - I ) and J&J& respectively. (The factors of + arise in the former case from zero-point energies, in the latter case because a charge-transfer excitation involves two atoms.) The threshold semiconductor polarizability for charge-transfer excitations can be cal-
MICROSCOPIC
culated by equating these energies and neglecting and CD:in the interfacial region. This gives CD;=0$(~~2 EC= l+(J2-
27
THEORY
the difference
between
WY
- l),
(5)
1)_2=7.
(6)
Short-range (dangling-bond) corrections will tend to reduce the value of a,, because residual covalency at the interface means that not all the surface energy of the metal is available for charge-transfer (formation of Bardeen “surface states”). Thus we consider that the result (6) is in excellent agreement with the experimental value c,z5 to 6. The discussion of the development of a double layer in the interfacial region given here, based on charge redistribution through energy supplied by and to collective modes, is quite different from the conventional explanations) based on filling of “surface states” lying near the Fermi energy and in the forbidden energy region of width AEcv between the conduction and valence band edges7). Indeed it appears that the single-particle energy level diagram does not represent a useful approach to describing the dielectric properties of the interfacial region7). One can use the discussion given here to place an upper limit of eS
28
J.C.PHILLlYS TABLE
1
Some dielectric constants of semiconductors and semimetals; in the case of semimetals the interband dielectric constant is given Material . SnS2
SnSe2 Cdlz PbIz GaSe ZnSe GeSe PbTe Bi C(graphite) C(diamond)
Structure __-__._....._.”_~
Eo(interband)
Layer Layer Layer Layer Layer Cubic Cubic Cubic Pseudo-cubic Layer Cubic
7.5 7.5 4.8 4.2 4.7 5.9 22 33 100” 4.8 5.7
& Ref. 14.
of q, for layer-like materials are collected in table I. They appear to conform to this reasoning. Some dielectric constants of chemically similar cubic semiconductors are also entered in this table. The values of E@can go as high as 100 for cubic semiconductors. Another interesting case is that of diamond~graphite. Here we would argue that it is the small value of c0 in diamond that makes this transition possible, i.e., cubic semiconductors are unstable against layer formation only if their dielectric constants are small enough already so that the interband ~o~arizabi~ity does not change appreciably because of the transition. This argument is made more plausible by the small changes that are known to occur in e, when a cubic semiconductor becomes amorphous l2). Direct experimental evidence for the smallness of Ed in the interfacial region has been obtained for the specific case of cesiated Si surfaces13). From the measured interfacial plasmon energy of 7 eV one infers +=5.2, in good agreement with the experimental values of E, at Schottky interfaces. References 1) W. Schottky, Z. Physik 118 (1942) 539. 2) S. Kurtin, T. C. McGill and C. A. Mead, Phys. Rev. Letters 22 (1969) 1433. 3) J. Bardeen, Phys. Rev. 71 (1947) 717. 4) J. C. Inkson, J. Phys. C 4 (1971) 591. 5) J. C. Phillips, Phys. Rev. B 1 (1970) 593. 6) J. Schmit and A. A. Lucas, Solid State Commun 11 (1972) 415,419. 7) Note that in most semiconductors fiwg % Al& (e.g., in Ge one has fiwg = 4.3 eV and AhEcv= 0.8 eV). See also ref. 5. Recently J. C. Inkson, J. Phys. C 5 (1972) 2599, has
MICROSCOPICTHEORY
8) 9) 10) 11) 12) 13) 14)
29
attempted to apply the ideas of ref. 5 to discuss the interfacial covalent-ionic transition, but his discussion is rendered irrelevant by his failure to distinguish between Eg (the average bond energy) and AE,, (of order the smallest direct energy gap). The plasmons he discusses are perturbations on the (rigid) single-particle band structure of the semiconductor, and make no allowance for the massive redistribution of charge and polarizability which arises because of the collective effects discussed here and in ref. 6. D. Allender, J. Bray and J. Bardeen, to be published. J. C. Phillips, Phys. Rev. Letters, in press. J. C. Phillips, Phys. Rev. 188 (1969) 1225. C. Y. Fong and M. C. Cohen, Phys. Rev. B 5 (1972) 3095. J. Stuke, J. Non-Crystalline Solids 4 (1970) 1. J. D. Levine, Surface Sci. 34 (1973) 90. W. S. Boyle and A. D. Brailsford, Phys. Rev. 120 (1960) 1943.
Discussion Question (by G. HEILAND): I would like to draw your attention to the fact that the contact between clean cleaved Si surfaces andevaporatedmetalsmay still be more complicated than is assumed in the models of Schottky and of Bardeen. Two different metals have been evaporated simultaneously on pairs of Si surfaces created by the same cleavage process. If the differential capacity is plotted as l/C2 versus bias voltage there appears a small but reproducible difference of slope that cannot be attributed to a difference in donor concentration or dielectric constant of the Si. The model of an interfacial layer and of surface states may need some refinement to explam the experimental data [H. Harreis and G. Heiland, Surface Sci. 24 (1971) 6431. Speaker’s reply (by J. C.
PHILLIPS):
Yes.
Question (by S. RABII) Did you find any trends in plotting compounds?
l/[&(0)-1]
as a function of (Az)a for the IV-VI
Speaker’s reply (by J. C. PHILLIPS): Because of the presence of nonbonding s electrons in AnBlo-” compounds, the analysis of e(0) is more complicated than for AnB8-n compounds. However, some chemical trends can be identified. Question (by J. D. Dow): In discussing the surface energy, you seem to treat surface plasmons and dipole layers independently. Would you please discuss the modifications of this treatment necessary for an ionic solid, in which a significant fraction of the oscillator strength is exhausted by the excitons, the surface modes are longitudinal excitons, and the bulk excitons are relatively tightly bound? Speaker’s reply (by J. C. PHILLIPS): The transition fromcovalent to ionic behavior at the metal-semiconductor interface takes place in compounds such as CdSzSel_z which are covalent in the bulk (small fraction of oscillator strength in excitions, spectroscopic ionicity about 0.6). For x values for which the compound has become so ionic that a large fraction of the oscillator strength is associated with excitons (alkali halides, ionicity 0.9 or more) the transition at the surface has been completed and the materials are in the Schottky limit (low polarizability and no double layer). In this limit the question therefore becomes irrelevant.