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SUPERCONDUCTING TUNNELLING AND PHONONSSUPERCONDUCTING TUNNEL JUNCTIONS AS PHONON SOURCES AND
DETECTORS
A. Dayem
To cite this version:
A. Dayem. SUPERCONDUCTING TUNNELLING AND PHONONSSUPERCONDUCTING TUN-
NEL JUNCTIONS AS PHONON SOURCES AND DETECTORS. Journal de Physique Colloques,
1972, 33 (C4), pp.C4-15-C4-20. �10.1051/jphyscol:1972404�. �jpa-00215081�
SUPERCONDUCTING TUNNELLIIVG A N D PHONONS
SUPERCONDUCTING TUNNEL JUNCTIONS AS PHONON SOURCES AND DETECTORS
A. H. DAYEM
Bell Telephone Laboratories, Incorporated Holmdel, New Jersey 07733
Abstract.
- Experimental results pertaining to the behavior of superconducting diodes when used in phonon generation and detection are discussed in some detail for a generator voltage
V $ 6 A .
We show that many aspects of the behavior can be fully explained by an average recom-
bination lifetime and the quasiparticle density of states of a superconductor. We also show how this average recombination time can be derived from experimental dependence of the signal on the detector bias which, in addition, clearly shows the influence of the ac Josephson effect. Further analysis shows that this experimental behavior is consistent with the Riedel Singularity in the ac Josephson current at a bias voltage
= 2 A .Finally, we discuss one aspect which can only be inter- preted in terms of the energy dependence of the recombination and relaxation rates and the detailed spectral distribution of the emitted phonons. This aspect is the deviation of the signal from its linear dependence on the generator current for V
< 4 A .This deviation has the form of a Lorentzian with a peak at voltage
Vm.By simple arguments, we show that the spectral width of the recombination phonon peak is given by
Vm - 2 A.By calculating the spectrum of the generated phonons and hence the detector signal we show that the above predictions are essentially correct.
1.
Introduction. -It is well known, both theore- tically and experimentally, that the decay of excita- tions in a superconductor occurs via electron-phonon interaction. Thus, in a superconductor of energy gap 2 A, we encounter two distinct decay processes.
The first is a relaxation process in which an excited quasiparticle emits or absorbs a phonon of energy w in a transition to a level (E + w)
2 A, where Eis the particle energy measured from the Fermi level.
The second is a recombination process in which two excited quasiparticles of energies El and E2 form a Cooper pair at the Fermi level, emitting a phonon of energy (El + E2) > 2 A. Pair breaking is the reverse process to recombination. The relaxation process is relatively slow for small values of w because both the phonon density of states and the coherence factor in the superconductor are small for small
w ,especially near the top of the gap where the coherence factor tends to zero. In fact, a finite energy range exists in the vicinity of A, where recombination is faster than relaxation. This range is obviously responsible for the finite width of the phonon peak at 2 A in the emitted phonon spectrum.
It is obvious, therefore, that no matter what method is employed to produce excited quasiparticles in a superconductor, the decay processes will result in the emission of phonons with a unique spectrum characterized by a well defined peak at energy
E2 A.
The quasiparticle excitations can be produced by electron tunneling or by the absorption of incident photons or phonons. In electron tunneling the density of injected particles at a given energy is known to a great degree of accuracy, and the range over which
excitations are produced is simply controlled by the applied voltage. Similarly, a tunable laser or a micro- wave source, or a tunable phonon source will provide easy control over the energy range where the excita- tions are produced. On the other hand, if one only requires a phonon beam with the unique spectrum described above, any radiation source which is neither coherent nor monochromatic will suffice provided that the incident radiation contains quanta of energy
> 2 4 .
The use of superconducting tunnel diodes as pho- non generators and detectors was first reported by Eisenmenger et al. [I], [2] and was later used by Kinder et al. [3], by Dayem et al.
[4],and by Dynes et al. [5]. The use of an incoherent phonon source (a heater) was recently reported by Narayanamurti et al. [6]. In all those experiments the superconductor used was in thin film form evaporated on an insulat- ing single crystal, typically sapphire or germanium, which provides a medium for transmitting the gene- rated phonons to the detector.
The phonon detector is typically a superconducting diode of energy gap d the generator gap and is bias- ed a t a voltage V, smaller than its own gap. Incident phonons of energy larger than the detector gap will, through pair-breaking, sustain an increase in the steady state quasiparticle population by an amount which depends on (Jz), where J is the number of phonons absorbed per sec and
zis the recombination lifetime.
The increase in population results in an increase in the tunneling current by an amount denoted by I,, the signal current. In practice, we do not measure I, directly but rather the signal voltage S produced at
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972404
C4- 16 A. H. DAYEM
a fixed bias current (constant current bias source).
For small signals the following relations are valid
where R, is the dynamic resistance of the detector characteristic at the bias point. Most of the experi- mental results can be fully explained, at least qualita- tively, by ignoring the spectral distribution of J(m) and energy dependence of z(E). The next three sec- tions will deal with such experimental behavior. The last section, however, will describe experimental observations that require for their interpretation a full knowledge of J(m) and z(E).
The experimental results presented here will be limited to O < V ,<
6A, where V is the generator voltage and were obtained with the generating and detecting diodes both made out of the same material, viz Pb or Sn.
2. Dependence of S on Generator Current
L - Since both the relaxation and recombination processes can result in emission of phonons of energy
22 A, the S
-I characteristic should possess one quantum point at I(V)
=1(4 A) at which a sudden change in slope should occur. For V > 4 A,
Sshould possess a specific behavior which reflects the effect of the superconductors density of states,p(E)
=E(E2 -A2)-%, on the distribution of injected population and hence on the emitted phonons. Such behavior can easily be calculated by representing the generator by a three level system [4]
:the Fermi level at E
= 0which contains all Cooper pairs and two excited levels one at E
=A the other at E
=3 A, interacting with phonons of energy
=2 A. The detailed balance [7]
equations for this system yield the obvious result that the number of 2 A-phonons escaping from the diode is simply ( I + 2 12)/e, where e is the electronic charge, I is the total generator current and I, is that portion of Icoming from injection in the range 3 A < E < V - A . I, remains zero for V < 4 A and will reflect the effect of the density of state p(E) on the number of emitted phonons for V > 4 A. The detector is adequately represented by a two level system and the detailed balance yields the increase in the steady state particle population
where No and Mo are the thermal equilibrium values of particle and phonon populations respectively, and
yis the phonon escape rate. Note that J l y is the increase in phonon population above its thermal equilibrium value. It is obvious from (2) that in one extreme (Jly
6M,) n and hense
Sdepend linearly on J and in the opposite extreme the signal will vary like the square root of J. Since J is proportional to ( I
-I-2 12), the dependence of the signal on the gene- rator current may easily be calculated from the well known expression for the tunneling current at T
=0.
The results are shown in figure 1 where the solid curve represents the linear limit, and the dashed square-root limit
;both I and S have been normalized to their respective values at V
=6 A. Note that S(6 A)/S(4 A) will vary from 3.1 in the linear limit to 1.76 in the square root limit.
FIG. 1. - Theoretical behavior of the signal in the linear (solid) and square-root (dashed) limits.
Any experimental plot of S vs. I, such as that shown in figure 2, should represent an intermediate case between the linear and square root extremes, and should possess a kink at V
=4 A immediately follow- ed by a well-defined peak in the slope, which replaces the theoretical singularity clearly obvious in figure 1.
Trapped flux in the generating diode is, in our expe- rience, the only factor that drastically changes the outcome of a measurement. When flux is trapped
I (ARBITRARY U N I T S )
FIG. 2. - Experimental curves showing generator voltage and detected signal vs. generator current. These results we obtained for Pb diodes and correspond to a case intermediate between
the linear and the square-root limits.
SUPERCONDUCTING TUNNEL JUNCTIONS AS PHONON SOURCES AND DETECTORS C4- 17
in the generator, its order parameter will spatially vary from
0to A . This results in smearing the sharp rise at V
=4 A , and in decreasing, or even completely wiping out, the peak in the derivative.
In the above discussion, we have seen one aspect in the behavior of the signal which can be analyzed in terms of an average recombination time
2in the detector. In the linear limit, e
=constant-and is solely determined by the excited particles present in the detector in the absence of any incident phonons. In the square-root limit, e is determined mainly by the excitation produced in the detector by the incident phonons. The experimental curve shown in figure 2 represents a behavior intermediate between the two extremes. To study the dependence of this average
zon the bias point one keeps the phonon flux constant and measures the signal vs. the detector bias VB as shown in the next section.
3. Variation
of z withV,.
-The dc characteristic of an Sn detector is shown in figure 3 for VB
,<2 A . This diode has a normal state dc resistance
FIG. 3. - Experimental curves of the detector current and signal vs. detector voltage.
At the operating temperature, the dc current expected from the thermally excited particles should be - .81 mA.
This value is at least one order of magnitude smaller than the current shown in figure 3. Also shown in the same figure is the signal voltage S obtained with a fixed generator voltage V
=6 A . One clearly sees how the highly nonlinear nature of the I,
-VB cha- racteristic has produced a complicated dependence of S on VB. AS mentioned in the introduction, it is the signal current Is rather than the signal voltage
Sthat contains the required information. To extract Is from the data shown in figure 3 we use numerical calculation and the relation
The implicit assumption in this relation is that I, is a slowly varying function of VB, and is everywhere small compared with I,. Usingeq. (3), however, requires
an accurate calibration of both the dc channel and the pulsed signal channel so that the difference (VB
-S(VB)) can be evaluated with an accuracy better than + 2 %. This calibration was carefully carried out using a digital voltmeter and 10 sec inte- gration time.
The signal current thus obtained is depicted in figure 4. One finds that I, decreases with increasing
FIG. 4. - Signal current deduced numerically from experi- mental data depicted in the preceding figure.
VB in a manner similar to the inverse behavior of I,.
One suspects, therefore, that the tunnel current IB must sustain an increase in the quasiparticle popula- tion in the detector, thus altering e and hence I,.
By making the phonon intensity J sufficiently small, the incident phonons can be used as a probe to sample the excited state of the diode at a given voltage. The response signal of the diode determines the degree of excitation through its dependence on the effective recombination lifetime.
Represent the detector by a two level system in which excitations are produced by IB and J. The par- ticle balance is then given by
and the phonon balance is
where fi and r are constants defining the respective transition probabilities. If N, is the particle popu- lation in the absence of J, one gets from (4) and (5)
where we have used the thermal equilibrium relations
r ~
=; PM, and defined the experimental lifetime
by e,
=(1 + ,812 y)/(2 rN,). Now the incident pho-
nons
Jwill increment the particle population by an
amount n ,
=N
-N , given from (4)-(6) by
C4-18 A. H. DAYEM
for n,
@N , . Since I, an,, both n , and I, are inversely
proportional to N, and one may put N,/No
=Z,,/I, in ( 6 ) to get
where I,
=No e/4
7,.From the experimental results depicted in figures 3 and 4, and from the linear depen- dence of I,-' on ZB, we can numerically obtain the constants I, and I,,. The computation was carried out by producing a least-square straight line fit for the interval VB < A and yielded Zo
=.314 mA and I,,
=.452 mA. Using this value of I,, and taking T
=1 OK and a combined film thickness of
3000 A,
one obtains zo
=2.5 x s, which is in good agreement with Eisenmenger's measurements [2].
It is evident that using this procedure to determine z0 as a function of temperature will ensure that all measured values do not include, inadvertently, effects coming from extraneous sources, especially those which would increment the quasiparticle population.
4. Josephson effect and the Riedel singularity.
- Thus far, we have considered the effects of the dc tunnel current IB which lead to the determination of Zo and I,, in eq. (8). The results of this analysis are depicted in figure 5 where the right hand side of
FIG. 5. - Right-hand side of eq. (8) vs. detector current. Shown also is the straight line fit in the range Vg < A. The calculated contribution of the ac Josephson effect is depicted by the dashed
line.
eq. (8) is plotted vs. I,. We notice the excellent fit between the straight line and the experimental data for VB < A. Even the sharp drop in I, at VB
=A has been totally included in this straight portion.
However, for VB > A contributions coming from IB are too small to account for the experimental result.
The character of the deviation suggests the onset of losses produced by the ac Josephson effect.
At a bias voltage VB, there is an electromagnetic field in the strip line forming the diode [8], [9]. The field is driven by the ac Josephson current whose
frequency is 2eVB/h [lo], and whose amplitude at T
= 0 OKis j,
=(A/a2 R,) K(VB/2 A) [I 11, where a2 is area of the diode and K is the complete elliptic integral of the first kind. The amplitude of
j,has a singularity at VB
=2 A, which was pointed out by Riedel [12].
The ac losses in the strip line remain small as long as VB < A. However, at VB > A, the photon energy becomes larger than the gap, and the ac losses rapidly increase due to absorption of photons by Cooper pairs. This leads to a further increase in the steady particle population and hence a further reduction in z.
We have made an approximate estimate of these losses using the transmission line equations derived by Eck et al. [8], after incorporating a loss term in the form of a distributed series resistance given by the product of the skin resistance and the ratio asla, [13], the photon absorption coefficient in the superconduct- ing state relative to its normal state value. We thus obtained an expression defining
n,the number of photons absorbed per sec in the superconducting films (on either side of the barrier),
with
and
where A is a constant depending on the properties of the diode and the applied magnetic field, E and K are complete elliptic integrals, and
R,is the skin resistance in the normal state a t a frequency 2 Alh.
These expressions give the dashed line shown in figure 5. The agreement is reasonably good, thus providing an experimental observation of the Riedel singularity in the ac Josephson current. The value of R, thus obtained is about 4.9 x the room tempe- rature resistance of a thin film
1 500A thick, 1 mm wide and 2 mm long.
We now turn to the effects of the energy dependence of the relaxation and recombination rates on the emit- ted phonon spectrum and the detected signal.
5.
Spectral width of recombination peak in Sn. -Since the phonon density of states
-t0 as the energy
w -t0, particle relaxation involving a low energy
phonon is relatively slow, particularly in the vicinity
of the gap edge where, in addition, the coherence
factor
+0. Hence, there is a finite energy range near A,
where the recombination rate is comparable or even
faster than the relaxation rate resulting in the finite
width of the recombination phonon peak. When par-
ticle injection is limited to this energy range, marked
effects are produced in the detected signal which,
SUPERCONDUCTING TUNNEL JUNCTIONS AS PHONON SOURCES AND DETECTORS C4- 19
when properly analyzed, lead to the experimental
determination of the spectral width. These effects can be easily understood if we consider the injection of N particles in each of two discrete levels one at A, the other at E'. If one neglects particle relaxation from E' to A, the recombination gives N/2 phonons of energy 2 A and an equal number at energy 2 E'.
In the detector, these phonons produce particle exci- tations which in the steady state have a population Nz at
Aand Nz' at E'. The detected signal is propor- tional to N(z + z') > 2 Nz, since the recombination time z' at E' is larger than z at A . Thus, if the incident shower contains phonons of energy > 2 A, the detect- ed signal will have a positive deviation from linearity in its dependence on the generator current for V <
4 A .The deviation of the signal from linearity will continue to increase when the number of levels receiving injection is increased as long as the relaxa- tion rate remains relatively small. When the injection width is made sufficiently large to include levels with appreciable relaxation rate, excitations injected into these latter levels will decay via the two step process and the signal deviation starts dropping. Since relaxa- tion will transfer injected particles not only to the top of the gap but also to a finite energy range adjacent to it, one may deduce that the emitted phonon spec- trum at larger currents will settle to a width equal to (V, - 2 A) where V,,, is the generator voltage at which the maximum signal deviation occurs.
The experimental results, upon which the above arguments were based, are illustrated in figure 6 where
GENERATOR CURRENT IN AMP
FIG. 6. -Typical experimental curves for Sn diodes showing the generator voltage, the detected signal and its derivative vs.
generator current.
the generator voltage, the detected signal, and its derivative are shown as a function of the generator current. In figure 7 we show the detailed behavior of the signal and its deviation from linearity. One finds that the deviation resembles a Lorentzian with
I ' 9 ' 1 1 ' "
0 . 0 0.20 040 0.60 0.80 1.00 1.20 1.40 1.60 1.80 GENERATOR CURRENT IN AMP
FIG. 7. - Experimental signal and its deviation from linearity for Sn diodes operating in the linear limit.
a peak at V,
=2.45 A, leading to spectral width of 0.45 A for the phonon recombination peak.
To check the validity of the above argument, we have calculated [14] the emitted phonon spectrum at different injection widths and hence the detected signal as a function of I. We used a simplified model of 40 discrete excited-particle levels in the range A < E
,< 3 A . Only electron interaction with longi-tudinal phonons in normal processes has been taken into account. The phonons interacting with the par- ticles fall into two groups, the relaxation phonons of energy
win 0 <
w < 2 A, and the recombinationphonons of energy
SZ,in 2 A
,< Q< 4 A. Expressions, given in the Bardeen-Cooper-Schrieffer, (BCS), theory of superconductivity [13], defining quasiparticle tran- sitions in which a quantum of radiation is involved, have been used after modification into a form suitable for numerical calculation. A Debye density of states was used for the phonons, while the particle density of states corresponded either to the BCS density, or
NORMALIZED ENERGY u/A
FIG. 8. - Calculated phonon spectra for a Sn generator at an applied voltage = 4 A ; the solid curve for the BCS density, while the dashed for the experimental density of particle states.
C4-20 A. H. DAYEM
to an experimental density derived from I-V tunnel
characteristics. The calculated behavior of the signal shows the predicted deviation in a manner quite similar to the experimental deviation shown in figure 7. The deviation calculated using the experi- mental density of states had a peak at V,
=2.25 A while the width deduced from the calculated spectra is .265 A . This verifies the predictions discussed at the beginning of this section.
In figure 8, we show the calculated phonon spectrum with injection in all 40 levels. Note that the number of relaxation phonons
+0 as
o -+0 as discussed above. The relaxation phonon distribution clearly shows the competition between relaxation and recom- bination processes as described above.
With this, we conclude our present discussion of phonon generation and detection using superconduct- ing diodes.
References
EISENMENGER (W.) and DAYEM (A.),
Phys. Rev.Lett., 1967, 18, 125.
EISENMENGER
(W.),in
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D,Burstein
(E.)editor, Plenum Press, New York,
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(H.),LASZMANN
(K.)and EISENMENGER
(W.), Phys. Lett., 1970, 31A, 475.DAYEM (A.), MILLER (B.) and WIEGAND (J.),
Phys.Rev. B, 1971, 2, 2949.
DYNES (B.), NARAYANAMURTI
(V.),and CHIN
(M.), Phys. Rev. Lett., 1971, 26, 181.NARAYANAMURTI
(V.)and DYNES (R.),
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Phys. Rev. B, 1972,5.DISCUSSION C. J. ADKINS, A. LONG.
-The recombination rate
for a quasiparticle must increase as the energy of that quasiparticle increases above the gap. This is because the phonon phase space increases with the energy of the phonon to be emitted, and so does the matrix element for the electron-phonon interaction. The density of states at the higher energy (the initial energy of the quasiparticle) does not enter the expres- sion. This is because we are stipulating that this par- ticle is present
;we are not asking the likelihood of its being present. Thus only one electron density enters into the calculation, that of the thermal quasi- particle with which the given excitation recombines.
Integration over the thermal population gives a life- time for our quasiparticle
where E is the excitation energy, N, the number density of thermal quasiparticles and A the energy gap in the superconductor.
A. DAYEM. -Your statement that the density of states of the higher energy does not enter, does not seem correct. Summation over all k, and
k jof the
transition rate given in BCS shows that the rate is proportional to
where fi, J;. are the occupation probabilities of the particle levels
iand j. Our knowledge that these states are occupied is expressed by
:f. = f. =
1
L J
and not by
:p i f i ~ j f j = l .
