J. Phys. Chem. 1986, 90, 1755-1759 be so much influenced by the nature of the solvent. It is impossible to reliably distinguish overall molecular motion from internal rotation from the present N M R data, so we assumed u (= R 1 / R 2 )to be 1-2. The following finding may be considered to be strong evidence in support of this assumption. Imanari et al.l9 measured I3C spin-lattice relaxation times for the biphenyl derivative with four methyl groups at ortho positions. In this molecule, since the internal rotation is hindered, the ratio of the relaxation times of meta and para carbons gives the estimation of u. According to their data, the u value becomes about 1 . 1 . Therefore, the present assumption of u = 1-2 seems to be quite reasonable. The potential function obtained gives the torsional force constant of 0.1 10 m d y d as the second derivative of the potential, and this force constant, in turn, results in the torsional frequency of 104 cm-]. In the Raman and the inelastic neutron scattering spectrum,20there exist several weak bands in the region around (19) Imanari, M.; Ohuchi, M.; Ishizu, K. J . Magn. Reson. 1974,14, 374. (20) Krebs, K.; Sandroni, S.; Zerbi, G. J . Chem. Phys. 1964, 40, 3502.
1755
100 cm-I, and it seems reasonable to assign one of them to the torsional mode of biphenyl. Recently, Almenningen et al.* examined the gas-phase electron diffraction of biphenyl and obtained the potential function where V, = 0.5, V, = -6.2 kJ/mol, and '6 = 44O. This result does not agree with the corresponding values of 9.93, -9.00 kJ/mol, and 37O in this work. The reason for the difference probably lies partly in the difference of the phase of biphenyl, but it is not clear at the present time. On the other hand, the potential curves calculated by Casalone et a].' and by Almlof8 are quite close to the present one. However, since these theoretical calculations were carried out for an isolated molecule, further work is necessary to fully confirm these good agreements. Acknowledgment. We express our gratitude to Mr. K. Kushida of Varian Instruments Ltd. for the use of the Varian XL-300 N M R spectrometer for ZH measurements. We also thank Professor M. Hirota of Yokohama University for his helpful suggestions. Registry No. Biphenyl, 92-52-4.
Dielectric Constant Dependence of Intensities in Inelastic Electron Tunneling Spectroscopy K. W. Hipps,* Ursula Mazur, and Richard Knochenmusst Department of Chemistry and Chemical Physics Program, Washington State University, Pullman, Washington 99164-4630 (Received: August 9, 1985; In Final Form: December 27, 1985)
The role of the dielectric properties of the insulating barrier in theories of inelastic electron tunneling spectroscopy is considered. Attention is drawn to the fact that the dielectric constant is frequency dependent and that many existing theories predict unphysical behavior as the dielectric constant undergoes dispersion at frequencies corresponding to absorption by the insulator. Attempts to improve agreement between experiment and theory through use of the complex form of the interaction potential were partially successful, but the results are still not satisfactory. It appears that assuming a frequency-independentdielectric constant, equivalent in this study to the use of the free space dielectric constant, in the theoretical expression for intensities gives the best results of the three methods tested.
Introduction The detailed interpretation of molecular vibrational spectra as observed by inelastic electron tunneling spectroscopy (IETS) has been mostly limited to the analysis of peak positions, intensities being insufficiently understood from a theoretical viewpoint to allow more than general inferences. However, there is motivation to seek a better theoretical understanding since IETS intensities should, in principle, carry substantial information about the orientations of the species in tunnel junctions. The theoretical treatments which have been most used to interpret spectra are due to Scalapino and Marcus (SM),' and to Kirtley, Scalapino, and Hansma.2 Both of these assume a linear intensity/coverage relationship. Cunningham, Weinberg, and Hardy (CWH)3 have also proposed a useful theory of tunneling intensities which predicts a nonlinear intensity/surface coverage relation. All three treatments give qualitatively correct values for the intensity of a representative transition, but none of them give good relative intensities within a particular spectrum. Because of its conceptual and computational tractability, the theory of intensities proposed by Kirtley, Scalapino, and Hansma* (KSH) has received somewhat more attention than others. It predicts the correct order of magnitude for vibrational inelastic intensities but poorly reproduces experimental intensity patterns.& This theory incorporates a number of approximations, and it is 'Present address: Department of Chemistry, University of Bern, Freiestrasse 3, 3000 Bern 9, Switzerland.
0022-3654/86/2090-1755$01.50/0
natural to try to test and improve upon them. A first step in this direction was made by Hipps and Knochenmuss' who added a new term to the electron-molecule interaction potential. The theoretical approaches of C W H and of Kirtley have a common feature. They predict that the intensity of a given transition should be inversely proportional to the square of the dielectric constant of the insulating barrier at the frequency of the observed transition. The Scalapino and Marcus treatment, however, ignores the dielectric properties of the tunneling insulator. Thus, there are two opposing views which must be resolved. Either the dielectric properties of the insulator are important and should be properly included, or they are not and can be ignored. It is this aspect of the theories that we consider in the present paper. While one might argue that electrostatic models are inappropriate for the very thin layers studied by tunneling spectroscopy, especially at frequencies near the absorption of the barrier, the fact remains that theories are being used to predict intensities which (1) Scalapino, D. J.; Marcus, S. M. Phys. Reu. Lett. 1967, 18, 459. ( 2 ) Kirtley, J.; Scalapino, D. J.; Hansma, P. K. Phys. Rev. B 1976, 14, 3177. (3) Cunningham, S. L.; Weinberg, W. H.; Hardy, J. R. Appl. Surf.Sci. 1979, 2, 640. (4) Kirtley, J.; Hall, J. Phys. Rev. B 1980, 22, 848. (5) Godwin, L. M.; White, H. W.; Ellialtioglu, R. Phys. Rev. B 1981, 23, 5688. (6) Yang, W.-J.; White, H. W. Surf.Sci. 1982, 118, 303. (7) Hipps, K. W.; Knochenmuss, R. J . Phys. Chern. 1982, 86, 4477.
0 1986 American Chemical Society
1756 The Journal of Physical Chemistry, Vol. 90, No. 9, 1986
Hipps et al.
assume that it is appropriate to use a dielectric constant treatment. The question we wish to address here is, “Do these treatments work?” The KSH model assumes that the molecule which scatters the tunneling electron can be represented as a set of point charges located on the nuclei, the magnitudes of which reflect in some way the charge distribution in the molecule. Alternatively, one can use symmetry coordinates and the dipole derivatives associated with the symmetry coordinate^.^ The electron and molecule then interact via an electrostatic potential (showing only the component at the vibrational frequency, Q):
where j indexes the atoms or symmetry coordinates in the molecule, Z j is the partial charge on atom j or the dipole derivative associated with symmetry coordinate j , rj is the distance from atom j to the electron, and t is the dielectric constant of the insulator in which the molecule is embedded. The distance dependence, F(rj),is not simply l/lrjl because of interactions with the induced images in the nearby metals.2 This potential results in an expression for the inelastic intensity which is proportional to l/c2. The intensity formula given by Cunningham et aL3 is proportional to the square of the dipole derivative for the normal mode of vibration and also proportional to 1 /e2. In calculations of intensity utilizing the Kirtley or the Weinberg method, the value of E has normally been taken as three for alumina insulators. This is approximately correct for frequencies above 2000 cm-I, but is not valid at lower frequencies. The alumina barrier has an absorption band in the 500-1000-~m-~ region. Because of this absorption, the dielectric “constant” becomes frequency dependent and complex, rather than constant and purely real. Eriksson et aL8 have measured the dielectric properties of alumina films (300 to 3500 nm thick) in the frequency range of interest and they find that the real part of the dielectric constant passes through zero and becomes negative throughout the region from about 920 to 700 cm-I. The real part of t then rises to a value of near 6 at 200 cm-’. Thus, the above potential functions cannot be correct, since use of the real part of t results in division by zero at two frequencies, implying an infinite energy of interaction and also infinite inelastic tunneling intensity. This result is not acceptable theoretically or experimentally. Infinite interaction energy makes no physical sense, and experimental intensities are neither infinite nor even unusually large in this region. A simple way to modify the theory to account for absorption by the insulator is to replace the real dielectric constant with the complex dielectric constant. Then, in the barrier V(r,t) = - ( e 2 / t ) C Z j / ( R j- rl J
where carets are used to denote complex quantities. Both Z and R are complex due to their time dependence. If harmonic motion is assumed this dependence is exp(iQt). This form of the potential energy is essentially the same as that given by Z i m a ~ ~ Using first-order time-dependent perturbation theory and including the image charges we find that the final expression for the tunneling intensity is exactly as before except that the real dielectric constant is replaced by the magnitude of the complex dielectric constant. Obviously this converges to the old theory as the imaginary part o f t vanishes. Figure 1 shows the relationship of the alumina absorption band to the magnitude of the complex dielectric constant. The magnitude of t curve is computed from the data of Eriksson et al.,8 and the alumina absorption is as observed by IETS. To test the E dependence of either this result or the old theory would ideally require measurement of the intensities of vibrations which differed only in frequency. Then any changes in intensity (8) Eriksson, T. S.; Hjortsberg, A.; Niklasson, G. A.; Granqvist, C. G. Appl. Opt. 1981, 20, 2742. (9) Ziman, J. M. Principles of the Theory of Solids; Cambridge University Press: London, 1964; 1st ed, Chapter 5.1.
Figure 1. The magnitude of the complex dielectric constant of y-alumina as calculated from the data of ref 8 (solid line), and the alumina absorption band as observed by IETS (dashed line).
should be due to variations in c with frequency (barring any frequency-dependent effects arising from the computation of the transition matrix element, vide infra). This is not possible, but comparison of vibrations of isotopically substituted molecules can, with some additional information, provide a similar test. The specific vibrations considered here are the C H modes of formic acid. These were chosen because C H modes are generally fairly localized due to the small relative mass of the hydrogen isotopes. Also, the formate ion is small enough so that the size distinction between normal modes and symmetry modes is small. Thus, the dependence on dipole derivative in Kirtley’s formula4 becomes quite similar to the equations of Cunningham et aL3 and SM. Furthermore, the tunneling spectrum of formic acid is w e l l - k n o ~ nand ~ ~ some ~ ~ ’ of ~ the frequency shifts on deuteration occur in a range where the dielectric constant of alumina changes significantly.
Experimental Section Reagent grade formic acid was used without further purification. Fully deuterated formic acid was obtained from Chemalog Corp., 99.5% isotopic purity with 5% DzO,and used as received. Tunnel junctions were made by depositing A1 metal in vacuo torr and on a glass substrate at a base pressure of 5 X oxidizing the surface in a 75-mtorr oxygen glow discharge (450 V ac). 500 mtorr of previously degassed acid vapor was introduced to the vacuum chamber from a valved side arm for 20 min. The system was then evacuated to 9 X lo-’ torr, and the top metal layer (Pb) was deposited. IETS spectra were measured on a modified version of a previously described instrumentI6 which measures d2V/df-. A second lock-in amplifier was used to record the dV/dI signal at the modulation frequency, and an additional digital voltmeter was used to measure the modulation voltage across a series resistor of value at least 100 times larger than the junction resistance. This information was mathematically combined with the standard second harmonic signal to give d * I / d V spectra which were then stored on disk. The conductance curve was also stored on disk for later use. dZ2/dV data are more rigorously comparable to theory than the more common dZV/dP results. All measurements were made at 4.2 K. The modulation voltage used was 1.3 mV rms. Each spectrum recorded was the sum of 6 to 20 scans taken (10) decheveigne, S.; Gauthier, S.; Klein, J.; Leger, A,; Guinet, C.; Berlin, M.; Deforneau, D. Surf. Sci. 1981, 105, 377. (11) Brown, N. M. D.; Floyd, R. B.; Walmsley, D. G. J . Chem. Soc., Faraday Trans. 2 1979, 75, 17. (12) Magno, R.; Adler, J. G.J . Appl. Phys. 1978, 49, 4465. (13) Liehr, M.; Thiry, P. A,; Pireaux, J. J.; Caudano, R. Phys. Reo. B 1985, 31, 42. (14) Adler, J . G. Tunneling Spectroscopy, Hansma, P. K., Ed.; Plenum Press: New York, 1982; Chapter 14. (15) Gauthier, S.;Klein, J.; Leger, A,; De Cheveigne, S.; Guinet, C.; Belin, M.; De Fourneau, D. Phys. Rev. B 1982, 26, 3622. (16) Mazur, U.; Hipps, K. W J . Phys Chem. 1979, 83, 2773
The Journal of Physical Chemistry, Vol. 90, No. 9, 1986 1757
Dielectric Constants in IETS
TABLE I: Positions (cm-I) and Assignments for Certain HCOL and DCOf Bands NaHC02"
IETS~
Cu(HCOJ2'
PED'
€1
1368 1015
1394 1035
1398 1028
In-Plane C H Bend 1.91 0.86
1071 919
1061 902
1072 912
Out-of-Plane C H Bend 1.oo -0.12
100% PCH 100% a-CD
1595 1580
1622 1614
1580 1575
Asymmetric C O Stretch 2.10 2.10
112% C O 123% C O
2830 2131
2890d 2193d
2914 2213
C H Stretch 2.81e 2.42e
101% C H 99% CD
78% H C O 93% DCO
Q(H)lW)d
+ 13% C O
1.35
1.18
+ 27% H C O + 12% DCO
1.oo
1.32
"Reference 17. bThis work: urn and kh data. 'Reference 18. dMost intense component. 'Extrapolation of Eriksson's data.
with a 1-s time constant at a rate of 0.4 mV/s. Integrated intensities were calculated by simultaneously fitting (nonlinear least squares) a quadratic polynomial background and three or four Gaussian peaks to the data in the regions 650-1 130, 1220-1 750, 1950-2450, and 2525-3225 cm-'. Two Gaussians were used to represent the contribution of the AI-0 band in the lowest energy region. The peak positions reported for all but the C-H/D stretching bands are the peak values of the Gaussian bands obtained from the fitting procedure and corrected for Pb superconductivity. The C-H/D stretching region required several Gaussians to produce a satisfactory fit and only the corrected position of the most intense component is given. To obtain experimental values for comparison to theory, two methods were used. In the first case, the asymmetric carboxylate stretch served as an internal standard. This vibration shows essentially no frequency shift on deuteration and is not strongly dependent on the C H modes. The intrinsic intensity of this vibration was assumed to be independent of H / D isotopic substitution. The ratio of the C H (or CD) peak to the asymmetric carboxylate stretching band in each spectrum was taken to obtain a normalized intensity. In the second case, the measured values of the intensity were averaged and used directly. This latter method would be appropriate if one assumed saturation coverage on the oxide surface, an assumption often made for small carboxylic acids."
Results and Discussion Many tunneling spectra of formic a ~ i d ~ have . ' ~ 'been ~ reported. The species in the junction is always the carboxylate rather than the protonated acid. The peak positions as measured in this study (corrected for superconductivity of the top metal) for the energy region of interest are shown, with assignments, in Table I. Typical spectra for the 650-1 800-cm-I region are shown in Figures 2 and 3 and are most similar to those of Adler and of Gauthier. The relevant formate vibrations having large hydrogenic contributions are the C H out-of-plane bend at 1060 cm-' which shifts down to 902 cm-I on deuteration, and the in-plane C H bend at 1394 cm-l which shifts to 1035 cm-I, the C-H stretching band near 2890 cm-I which shifts to 2194 cm-'. Also of interest is the asymmetric carboxylate stretch at 1622 cm-l. This latter band shifts by only 8 cm-I on deuteration. These frequencies are in '~ reasonable agreement with previous work on f ~ r m a t e . ~ * ' *Also given in Table I are the observed positions of the bands of interest in sodium formate" and copper formate.I8 It should be noted that the in-plane bend of H C 0 2 - partially overlaps with several weak bands, resulting in less reliability in the integrated area for this peak. However, we estimate the amount of overlap to be less than 10% and this is not significant to the points made below. For a purely hydrogenic motion, the expected isotope frequency ratio on deuteration (Q(H)/O(D)) is about 1.4. The in-plane bend and stretching frequency ratios for formate are close to this at (17) Kidd, K. G.; Mantsch, H. H. J . Mol. Srrucr. 1981, 85, 375. (18) Heyns, A . M . J . Mol. Strucr. 1985, 127, 9 .
z
N
(u
>
P
(u
U
50
1225
ENERGY cm - 1
Figure 2. IETS spectrum of the protonated form of the formate ion on alumina. TABLE 11: Integrated d21/d V 2 Intensities Divided by Peak Conductance for Various Formate and Formate-d2 Bands" file
1394
1061
1622
rk26 rk28 khO 1 urn20 um19 av std dev
0.900 1.00 2.04 2.14 2.43 2.206
1.65 1.87 3.65 3.61 4.19 3.81b
0.307 0.454 0.883 0.924 0.941 0.916b
um16 um15 um13 um14 av std dev
10561 1622
13941 1061
2.93 2.20 2.31 2.32 2.58 2.46 0.29
5.37 4.12 4.13 3.91 4.45 4.40 0.58
0.55 0.53 0.56 0.59 0.58
Equivalent Deuterated Species Bands 0.565 0.875 0.520 1.09 1.68 0.568 0.886 0.519 1.09 1.71 0.585 0.905 0.574 1.02 1.58 0.616 0.948 0.572 1.08 1.66 0.584 0.904 0.546 1.07 1.63 0.03 0.06 C H strC
um16 umO 1 kh02 kh03 urn19 khO I
13941 1622
12.40 13.66 13.27
0.65 0.64 0.65 0.65
C D strc
1622
ratio
6.60 8.34 10.30
0.520 0.676 1.10 0.680 0.941 0.883
12.7 12.3 9.4 18.2 14.5 15.0
"Tabulated intensities were multiplied by 10'. only. 'Integrated area of entire complex band.
b u m and kh data
1.35 and 1.32, respectively. The out-of-plane bend ratio is 1.18, indicating that the assumption of localized proton motion is not well met for this mode. Kidd and MantschI7 have performed an exceptionally detailed isotopic study of sodium formate. The principal terms in the potential energy distribution for the vi-
1758
The Journal of Physical Chemistry, Vol. 90, No. 9, 1986
Hipps et al.
TABLE 111: Calculated and Experimental Proton to Deuteron Isotope Intensity Ratios for Several Tunneling Bands" _ I
cm"
Id
5
k[H/Dl
kw/Dl
(14
(e,)
~c[" ( e = 1)
*
IE PI/ Dl-_.___.____._.I_.__ IE [ / Dl
_ _ _ I _ _
C-(H/D) Obt-of-Plane Bend
1061 902
I .oo -0.12
1.01 0.96
0.02
1394 1035
1.91 0.86
1.91 0.89
0.27
1622 1614
2.10 2.10
2.10 2.10
1 .oo
2890d 2193d
2.23' 2.22'
2.8Ie 2.42'
1.31
1.07
1.18
2.1 (0.4)
4.2
.35
2.3 (0.3)
3.8
.oo
1 .o
I .7
32
1.4 (0.3)
1.6 (0.7)
C-(H/D) in Plane Bend 0.29
Asymmetric CO Stretch I .00
C--(€I/D) Stretch 1.31
"The real part of e , its magnitude, or the free space value was used in the calculation as indicated by the parenthetical symbol under the calculated intensity columns. Intensities were scaled by the asymmetric CO stretch intensity. Values in parentheses are standard deviations. 'Absolute intensity ratios were used. dThe most intense component position is given but total band areas were used. 'Estimated from an extrapolation of Eriksson's data are also given in Table I, and Table I1 gives both the real part and magnitude of t. In order to evaluate the calculated isotopic intensity ratio, several assumptions were made. First, all three C-H/D motions were assumed to be pure symmetry coordinates. As discussed above, this is a very good assumption for the out-of-plane bend and for the stretch, but less satisfactory for the in-plane bend. Thus, by taking 6R to be the vector of the appropriate symmetry coordinate displacement, we have
650
1225 ENERGY cm - 1
I abo
Figure 3. IETS spectrum of the deuterated form of the formate ion on
alumina. brations of interest are taken from their work and reported in Table I. The out-of-plane bend, despite its small isotopic ratio, and the C-H/D stretch are essentially pure symmetry coordinates. The in-plane bend is about 80% pure symmetry coordinate. Table I1 displays the results of Gaussian fitting a number of formate bands for a number of different tunnel diodes prepared by different people. Intensities are given in the unitless form recommended by Kirtley and Hall," the integral of d21/dp divided by dI/dVat .ae band maximum. Note that a factor of 3 variation in the intensities of the protonated species bands in the region below 18C J cm-' is observed. The band intensities scaled by the 1622-cm-' band intensity, however, are all quite similar. This result suggests that the use of the 1622-cm-I band as an internal standard is a sensible procedure. This is further reinforced by the concentration-dependent data of Langan and H a n ~ m a . ' ~They found that relatively large variations in the surface coverage of benzoic acid left the relative intensities of all but one CH stretching band unchanged. Table I11 gives the experimental proton to deuteron integrated intensities (divided by conductance) ratios. The next to last column i r based on using the asymmetric carboxylate as an internal standard. The last column is based on the raw (unscaledj values. For reasons given previously, we feel that the scaled data is the preferred set. The necessary dielectric data were taken from the work of Eriksson et a1.* To our knowledge, his are the most coinplet data for thin-film alumina. For reference, values of the rea' part of the dielectric constant at the transition frequencies (19) Langan, J. D.; Hansma, P. K. SurJ. Sci. 1975, 52, 211
where the factor C is weakly dependent on frequency because of k vector differences in the transition moment integral and due to the change in bias voltage required to open inelastic channels of differing energy. Cvaries by
