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Method for Obtaining Second Derivatives in Electron Tunneling
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J. W. T. Dabbs in:
Oak Ridge National Laboratory 4 Y's a f*
Oak Ridge, Tennessee
In studying the conductance of superconductor-insulator-super-
conductor tunneling Junctions, it has been found most helpful to Observe
the derivative of the conductance as * * of revealing more clearly
the rather small variations which appear.” Thus, the second derivative
of the I-V characteristic is useful in observing effects associated with
peaks in the phonon spectrum of the superconductor, as well as smaller
effects associated with van Hove and other critical points in the
spectrum.”
A method previously unused for obtaining a’I/aw” in tunnel junctions
as a function of applied bias has been developed. Instead of the use Of
an electromechanical rate generator" or of observing second-harmonic
generation’ caused by the non-linearity of the junction, an intermodu-
lation ("beat") frequency signal is observe". If a voltage of the form
v - v. a sin at + B sin at - (l)
is applied to the junction, and if * * f(v) is analytic, then an expansion
into a Taylor series about the point Vo gives, among others, the following
terms in the current through the Junction (and thus through a series
resistor):
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It is seen that Eq. (1) leads immediately to
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d"I lim 2
(#) " A, B-> 0 AB "ata (6)
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Similarly Eq. (5) gives
(£), # = #,G) Io/A * (7)
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It should be noted that the second harmonic term gives
# - £2,5-3"</* (8)
Thus the intermodulation signal for a given (a 1/av’) value is twice as
large (for A - B) as the second harmonic signal at frequency a/2x or
8/2x. The next higher order term is 6 times larger in the inter-
modulation case (with A = B) than in the second harmonic case, but this
is not felt to be a drawback, since the higher order terms would normally
be quite small. The basic approach described here was suggested by the
work of Finston et al. on measurements of secondary electron flux
generated in various media under x-ray irradiation." A number of recent
-5-
developments in transistor circuits have permitted a much greater degree
of reliability and ease as compared with the earlier apparatus.
Figure l shows a block diagram of the circuits used. The two
most unusual features are (a) the use of a locked-phase oscillator
system” which eliminates the effects of relative frequency drift among
the frequencies fi - a/2x, fe - 8/2x and the sum frequency fitte and
(b) the use of a number of identical plug-in operational DC transistor
amplifiers (1) as buffer (B) and summing (S) amplifiers and (2) in a
special form of active bandpass filter with high and adjustable Q and
adjustable center frequency. This filter was developed at ORNL for other
applications.” Other features of interest are the use of very low flicker
noise navistor (6CWh) preamplifiers" and a rather unusual constant current
bias supply circuit. suggested to the author by C. E. Nielsen of Ohio
State University. Details of these elements are shown in Fig. 2. The
feedback loop of Fig. l also includes active filter elements and is
intended to maintain a nearly constant AC voltage across the junction as
required by the analysis given above, while the AC current through the
series resistor R is monitored with the aid of the synchronous demodulator
("chopper").
It should be noted that junction capacity effects have been
omitted in the analysis. The switches are shown in position for a
measurement of dI/dV; changing all switches saves the configuration for
4°1/av” measurements. In the first case, the capacitive currents are
approximately in quadrature with the desired signal and are largely
eliminated by the demodulator; in the second case this is not necessarily
true, but these currents add at most a very small constant background
-li-
at the demodulation frequency, which is made low for this reason. Indeed,
such currents should not be observable for typical junctions at biases
larger than those where the conductance of the junction becomes appreciable.
The constant current supply when connected across the points c-c
of Fig. l provide the bias across the junction. This voltage is fed to
a floating input, high impedence, chopper-stabilized D.C. amplifier and
then to the X coordinate input of an X-Y recorder. The Y input is
obtained from a variable, long time- constant averaging filter” which is
inserted between the chopper demodulator and the X-Y recorder. The
capacitors used in the filter were 200 Volt, polystyrene film units.
Because of the press of other work, the entire arrangement shown
in Fig. l has not been completed; in particular, the performance of the
voltage feedback loop has not been ascertained. The performance of
other elements of the system has appeared excellent, however, and the
system is presented here in the hope that some of the ideas involved will
be useful to others. Obvious modifications in the arrangement shown in
Fig. 1 would permit recording as a function of current rather than
voltage, or the elimination of one demodulator by using additional
active filters, for example.
The author gratefully acknowledges the advice and assistance of
R. A. Dandl in connection with the use of the operational amplifiers and
filters, and the suggestion of the constant current supply by C. E.
Nielsen.
l.
2.
5.
*
Research sponsored by the U. S. Atomic Energy Commission under
-5-
REFERENCES
contract with the Union Carbide Corporation.
J. M. Rowell, P. W. Anderson, and D. E. Thomas, PRL 10, 534 (1965).
D. E. Thomas and J. M. Klein, RSI 31, 920 (1965).
D. J. Scalapino and P. W. Anderson, Phys. Rev. 123, A921 (1964);
A. F. G. Wyatt, PRL 13, 160 (1961) and references cited there.
R. A. Finston, H. H. Hubbell, Jr., W. G. Stone, and R. D. Birkhoff,
ORNL-2752 (Oct. 1959), unpublished.
Furnished by Philamon Laboratories, Inc., Westbury, L. I., New York
(Model TFGX-5125).
F. T. May and R. A. Dandl, RSI 33, 587 (1961). The detailed circuits
shown in the upper half of Fig. 2 include corrections of errors in
the original paper. Considerably improved amplifiers of this type
with higher frequency cape.bilities (10°E.) have also been developed
recently (R. A. Dandl, private communication).
P. C. Caringella and W. L. Eisenman, RSI 32, 654 (1962).
N. T. Larsen, RSI 32, 1200 (1962), Fig. 2.
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S = PLUG IN DC COUPLED OPERATIONAL AMPLIFIER (SUMMING)
C-C = CONSTANT BIAS CURRENT SOURCE INPUT POINTS
D = DYNAMICS INST, CO. 4 (32 DC CHOPPER AMP, (MOD. FOR EXT, OUTPUT )
M=#McINTosh 225 AMPLIFIER
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DATE FIL
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L E G A N. O. T J C E
This report was prepared as an account o' Goverament sponsored work. Neither the United
States, nor the Commission, nor any person acting on behalf of the Commission: *
A. Makes any warranty or representation, expressed or implied, with respect to the accu-
racy, completeness, or usefulness of the information contained in this report, or that the use
of any information, apparatus, method, or process disclosed in this report may not infriage
privately owned rights; or .
B. Assumes any liabilities with respect to the use of, or for damages resulting from the
use of hay information, apparatus, method, or process disclosed in this report.
As used in the above, "person acting on behalf of the Commission” includes any ena-
ployee or contractor of the Commission, or employee of such contractor, to the extent that
such employee or contractor of the Commission, or employee of such contractor prepares,
disseminates, or provides access to, any information pursuant to his employment or contract
with the Commission, or his employment with such contractor.
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