ELECTRON-PHONON INTERACTIONSINVESTIGATIONS OF ELECTRON-PHONON INTERACTIONS IN COPPER AND ALUMINIUM

HAL Id: jpa-00215093

https://hal.archives-ouvertes.fr/jpa-00215093

Submitted on 1 Jan 1972

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

ELECTRON-PHONON

INTERACTIONSINVESTIGATIONS OF

ELECTRON-PHONON INTERACTIONS IN COPPER AND ALUMINIUM

A. Long

To cite this version:

A. Long. ELECTRON-PHONON INTERACTIONSINVESTIGATIONS OF ELECTRON-PHONON INTERACTIONS IN COPPER AND ALUMINIUM. Journal de Physique Colloques, 1972, 33 (C4), pp.C4-73-C4-79. �10.1051/jphyscol:1972416�. �jpa-00215093�

EL ECTRON-PHONON INTERACTIONS

INVESTIGATIONS OF ELECTRON-PHONON INTERACTIONS IN COPPER AND ALUMINIUM

A. R. L O N G

Royal Society Mond Laboratory, University of Cambridge, Free School Lane, Cambridge, England

Resume. - On resume des enquCtes recentes sur I'intensite de I'interaction entre les phonons de hautc frequence ( 2 100 GHz) et les electrons de conduction dans I'aluminium et Ie cuivre.

On introduit un systcme Clementaire qui eniploie des jonctions tunnel supraconductrices pour produire ct ddtecter les phonons, et du monoxidc de silicone comme isolant. LC monoxide de silicone a des proprictcs qui affectent critiquement I'interprctation des rksultats. La derniere partie de cet article est un compte rendu preliminaire des enquetes dans lesquelles on fait une distinction entre les interactions de phonons longitudinaux et de phonons transversaux.

Abstract. -- Wc summarise recent investigations of the strength of the interaction between high frequency ( 2 100 GHz) phonons and the conduction electrons in aluminiuni and copper. Thc experiments use a simple arrangement with superconducting tunnel junctions as phonon trans- duccrs. Silicon monoxide, which is used as an insulating material, is found to have certain proper- ties which critically affect thc interpretation of the experiments. The paper concludes with a preli- minary account of experiments in which the interactions of longitudinal and transverse phonons are distinguished.

1.1 INTRODUCTION. - In this paper we report experimental investigations of the interactions between high frequency phonons ( 2 100 GHz) and the conduction electrons in a superconducting metal (alu- minium) and in a normal metal (copper). For our experimental work we have used in the main alumi- nium-aluminium superconducting tunnel junctions as phonon transducers.

The principles of phonon generation and detection using superconducting tunnel junctions are straight- forward. Referring to the junction I-V characteristic sketched in figure 1, in region 11 current flows by a pair breaking mechanism, and quasiparticles extra t o the equilibrium thcrmal population a r e injected

FIG. I . - S-S tunnel junction characteristic (schematic).

into each film composing the junction. These extra quasiparticles may recombine to the ground state, the energy being carried away by phonons at the gap frequency. The detcction process is equally straight- forward. In region I, the current flowing I(,,K N]R, where N is the number density of quasiparticles pre- sent, and R is the high bias (or normal) resistance of the junction. G a p frequency phonons impinging on the junction biased in region 1, will break Cooper pairs, increase N and thereby produce a detectable change in current 61. Now 61 K 6N ^K I, z where 1, is the input phonon current, and z the quasiparticle recombination lifetime, o r mean time a quasiparticle takes t o recombine t o the ground state emitting a phonon which is subsequently lost to the system. If 6N G NT, the thermal quasiparticle density, then recombination is dominated by the thermal quasi- particles already present and

Hence the detector sensitivity 6111, will increase expo- nentially with AlkT.

1 . 2 QUASIPART~CLE LIFETIME. - From the foregoing analysis it is obvious that, if detailed measurements are t o be made with tunnel junction phonon detectors, then it is necessary t o know something about the quasiparticle lifetime in aluminium. Measurements of this quantity for aluminium evaporated on sapphire substrates have recently been made by Gray et al. [I].

Because emitted phonons have a very small probability of escaping from a n aluminium film into the sapphire

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972416

C4-74 A. R. LONG substrate, before they break further Cooper pairs and are absorbed [2], the population relaxation time does not reflect underlying electron-phonon relaxation rates, but depends instead on the rate of phonon escape from the film. It is straightforward to prove [3]

that the relaxation time

where d is the film thickness, C is a constant which can be derived from statistical arguments and N(0) d/NT is a dimensionless function of the para- meter AlkT, N(0) being the density of electron states at the Fermi level. v , is a typical velocity of sound for the transverse phonons, which as we shall see dominate the recombination process, and a , is a pho- non transmission coefficient from aluminium to sap- phire, averaged over angle. The available data fits this model well for d values in the range l 000 to 3 000

a,

we obtain an a, value of 0.055 (.f 10 %), the low value being an indication of the severe acoustic mismatch between sapphire and aluminium.

1 .3 AN EXPERIMENTAL ARRANGEMENT TO INVESTI- GATE THE PROPERTIES OF PHONON COUPLED TUNNEL JUNCTIONS. - Most of the experiments to be describ- ed in this paper were performed using two junctions of equal area, evaporated one on top of the other on a sapphire substrate, but separated by a thin insulating layer of silicon monoxide, which ensured that the coupling between them was by means of phonons, not electrons (figure 2). For the basic measurement,

AI/ AI Tunnel Junction

R-

A l / A I Tunnel Junction

I

FIG. 2. - TWO junctions coupled by a silicon monoxide film (schematic).

the detector junction was biased at approximately 3412 from a low impedance source, and a current Tg passed through the generator junction. To achieve greater senstivity this input current was chopped

mechanically at approximately 80 Hz, and the signal current 61 synchronously detected. A typical plot of 61 against I, is reproduced in figure 3a : note the linear region I where the thermal quasiparticles domi- nate recombination, and the reduced sensitivity in region 11, where the extra population becomes compa- rable with the thermal. In figure 3b we plot ~ I , I ,

FIG. 3a. - Typical transfer characteristic (taken at 0.61 OK).

Scales : I, = 100 pA/division, 61 = 50 nA/division.

FIG. 36. - Plot of transfer gradient, 61/Ig against AlkT for the same sample.

for the linear region I on a logarithmic scale versus AlkT. At intermediate AlkT values, the variation is approximately exponential as predicted, but it flattens off at low T due to lifetime saturation [4].

1.4 ENERGY SPECTRUM OF THE PHONON FLUX.

- Consider a tunnel junction which is being used to generate phonons. It the extra quasiparticle density due to the generating current is small compared with the thermal density NT, then the excitations will have a thermal distribution characteristic of the tempe- rature of the experiment T. It is then straightforward to prove that the recombination phonons generated will have approximately a Bose energy distribution

N,(w) ^K W' e-Pw[l

+

INVESTIGATIONS OF ELECTRON-PHONON INTERACTIONS IN COPPER A N D ALUMINIUM C4-75

for

with a lower cut-off at o = 2 A . Here o is the phonon energy and P ⁼ l/kT. This may readily be verified experimentally by transmitting from a generator junc- tion with a small gap to a detector with a larger gap

(A, < A,). The proportion of all phonons produced

with energy sufficient to break pairs in the detector junction is then

J W 0 2 e - B u d o

H(AX* A,) = 5 --

02 e-'L d o

2 A ,

the approximation being valid when A,/kT, AJkT ^$- 1.

In figure 4 points a and b, we plot 61/1, values on a logarithmic scale versus A,/kT. The two sets of data correspond to the inversion of generator and detector functions between the junctions of a superimposed pair, evaporated as described in paragraph 1.3. For transmission from large to small gap junction, figure 4a, the expected nearly exponential dependence on A,/kT is observed, but in the other direction, figure 4b, the gradient is rather lower. Figure 4c

(d) For junction parameters. The culrent flowing in region T (Fig. 1) at a bias of 3 A12 is approximately 2A/R.N/N(O)A at large A/kT values (AlkT > 3).

Hence the detected signal varies as I/R.

The mean hI/l, value at A/kT = 5 for fifteen sets of data, corrected in the above manner to a junction resistance of I SZ and a gap value of 200 peV, is (2.70

+

shows the small to large gap results, corrected by dividing each 61/1, value, by the value of 17 at the appropriate temperature. The data now varies approxi- mately as exp(A,/kT), confirming our simple ideas.

Measurements similar to these have been perform- ed on a large number of different samples having gap ratios (values of A,/A,) in the range 1.0 to 1.5.

In all cases behaviour similar to the above was obser- ved.

1 . 5 ABSOLUTE SEXSITIVITY MEASUREMENTS. - A large number of sets of transmission data are available for junctions coupled by means of silicon monoxide.

To compare them it is necessary to correct them in the following way :

(a) For the reduced detector sensitivity in the case A , < A , as indicated in paragraph 1 . 4 above.

(b) For the energy dependence of the detector relaxation time A,-' as indicated in paragraph 1.2.

(c) For the detectorjunction thickness. Ourjunctions are generally evaporated to a constant thickness (470 A

per film or 940

A

, 4. - Influence of gap differences (a) A , > Ad. (b) A , <

( c ) Ar i Ad, data corrected b y dividing by 'Lfunction. ^Ad.

1 . 6 ATTENUATION IN SILICON MONOXIDE, ^COMPARI-

SON TECHNIQUE. - The above comparison suggests that silicon monoxide does not appreciably attenuate the phonons. This was confirmed by an independent experiment in which three pairs of identical junctions were evaporated at the same time, but separated by different thicknesses of silicon monoxide. Signals between equivalent junctions in each pair, suitably corrected for variations in detector junction resistance (A values being the same for equivalent junctions), were compared. The results are summarised in table I.

There is some evidence here that thickness dependent attenuation does occur, but it is weak on the scale of silicon monoxide thicknesses used.

6

Absorption data for silicon monoxide

Corrected and Junction Thickness normalised Identification of SiO Signal

- - -

57 -+ 68 3 550 A I

1 3 - + 24 2 200 A 1.02

+

911 -+ 1012 1 350 A 1.09

+

There is another important point to note here, namely that, by comparing results taken with different junction pairs evaporated at the same time, we can eliminate much of the large random error observed in comparing results of different experiments (4 1.5).

This Comparison Technique is the basis of all our more accurate work.

1 .7 ATTENUATION AT ALUMINIUM-SILICON MONO- XIDE BOUNDARIES. - If, on the basis of paragraph 1.6, one neglects the silicon monoxide completely, and predicts the signal expected in transmission between two junctions from the magnitude of the lifetime of free junctions (§ 1.2), then one finds that the observed signal is only 116 of that expected. One is therefore forced to conclude that the silicon monoxide-aluminium boundaries cause some attenuation of the high frequency plzonon flux. We have assembled some evidence in favour of this concIusion by measuring the quasi- particle lifetime in junctions overlaid with silicon monoxide. The lifetime appears to be independent of the overlay (375 or 1 125 A of SiO, or I 000 SiO plus 8 500

A

The fact that the quasiparticle lifetime is indepen- dant of the thickness of the silicon monoxide film in proximity with the junction, and appears to be dominated by the surfaces is an important point in the analysis of the absorption measurements, to be described in section 2. It means that a direct compa- rison of the signal obtained with and without an absorbing layer can give us a reasonably accurate estimate of the attenuation.

2. Investigation of electron phonon interaction rates in aluminium and copper. - 2.1 METHODS AVAILABLE.

- As has already been stated (4 1 .2), we cannot obtain any direct information about electron-phonon relaxa- tion rates in aluminium by looking at the quasi- particle relaxation time. In order to make progress in this direction, we performed a different type of expe- riment, looking at the attenuation length for high frequency phonons injected into clean aluminium films. This measurement is described briefly in para- graph 2.3. In order to prepare the way, the rather simpler equivalent experiment, performed to look at the attenuation of high frequency phonons in copper,

is touched on briefly in paragraph 2.2. Full accounts of these experiments will be published in due course [3].

Such measurements as these do not give us any information as to how longitudinal and transverse phonons interact independently with the conduction electrons. Such information can only be obtained by separation of the different phonon modes, which move with different velocities, and looking at their behaviour independently. Some preliminary experi- ments of this type are described in section 3.

2 . 2 STEADY ^STATE ABSORPTION MEASUREMENTS IN

COPPER. - The film configuration for phonon absorp- tion measurements in copper is shown in figure 5.

C l e s AlJunfion-

- -

-

SiG

t

Copper f

I

SiO

- Clean ALJungon- -

-

FIG. 5. ^- Film configuration for absorption experiments in copper (schematic).

The experimental copper film is separated from the tunnel junction transducers by silicon monoxide films.

At the same time as the experimental sample is eva- porated, a control sample without a copper film is also laid down. This enables us to allow for random variations of film and junction parameters by the comparison technique (9 1.6). Values of the transfer gradient 81/Ig ; suitably corrected to unity collector resistance, are normalised by dividing by the transfer gradient observed in the control sample. In figure 6 ,

FIG. 6. - Normalised signal ^S versus AlkT for copper absorp- tion experiments. Parameter is d x (Ag/200), i. e. thickness of copper corrected for variations of input phonon energy. (A, is

in peV.)

INVESTIGATIONS OF ELECTRON-PHONON INTERACTIONS IN COPPER AND ALUMINIUM (24-77

this normalised transfer gradient, denoted S, is plotted against AlkT for a number of different samples (note the logarithmic ordinate scale). S decreases monotonically with copper film thickness, the para- meter in figure 6 , indicating a significant bulk atte- nuation effect.

We expect that the attenuation will be constant at low temperatures, as all electron-phonon events result in a loss of phonons from the flux, but that at higher temperatures, an increasing proportion of the events will be ineffective [3]. This behaviour is evident in figure 6. Unfortunately we cannot in general follow the behaviour of S to a low enough temperature to find the limiting value, because of lifetime saturation effects which render the behaviour of our detectors unreliable [4]. We are therefore forced back on a semi-empirical technique to fit the high temperature data [3]. We use a I-dimensional analogue for the attenuation, writing S in the low temperature limit in the form e-dlA' where d is the copper thickness and A' is related to the attenuation length. If one assumes that phonons are emitted into the copper isotropically, it is a straightforward matter to relate A' to the true attenuation length A, which is also assumed to be isotropic. One further correction is made to the data, that for input phonon energy. A should scale as UA,.

The mean free paths obtained on this basis, corrected to a A, value of 200 peV, are consistent, for film thick- nesses varying from 4 700 to 12 100 A, with a mean value 9 900 k 400 A. For comparison we may calcu- late the mean free path for the attenuation of longi- tudinal phonons using a free electron model for copper [5]. At 400 peV this is 5.1 pm, indicating an important contribution to the electron-phonon scat- tering due to the non free-electron nature of the metal.

2.3 STEADY SIATE ABSORPTION MEASUREMENTS I N A L U M I N I U M . - The measurements of phonon attenua- tion in aluminium use a very similar film system to that for copper, except that the junction electrodes are dirty, high gap value films, and the centre film is of clean aluminium (1 2 500

a)

However the mode of analysis we adopt is rather different, because, even in the case of complete absorption by the centre film we expect to see a finite signal, namely the thermal proportion I7 of the flux re-emitted from the centre film (energy gap A,), which would be collected by the detector, gap A,. We there- fore write the expected signal, normalised to that developed in the control sample, in the empirical form :

FIG. 7. - Normalised signal S versus A d k T for~alurninium absorption experiment. Full line, 17 function. Hatched lines, empirical form for S developed in text, for different values of

d!A'.

with centre films of two different thicknesses. Again we plot signals S, normalised to a control sample, on a logarithmic scale against A,/kT. We also enter on the same curve the relevant Z l value [3], calculated from experimental parameters. The hatched lines correspond to relationships of the form (3) with the values of d/A' indicated. Note that the thicker film attenuates more strongly, and that in the limit of a thick centre film the signal tends to I7.

In order to convert the experimental d/A' values into consistent attenuation length A, one needs, a s before to make a number of corrections :

(a) for the input phonon energy (A scale as 1/43 ; (b) for the non-infinitc electron mean free path (see for examplc [I]) ;

(c) for the enhanced attenuation occurring in superconducting metals versus the attenuation a t equivalent energy in the normal state [6] ;

( d ) from the I-dimensional model to the true mean free path A , assumed isotropic (see $ 2.2).

The attenuation length in clean, normal (i. e. non- superconducting) aluminium for an input phonon energy of 400 peV is found to be 5 400

+

superconducting aluminium (A = 200 peV) at the gap frequency, this would be equivalent to 3 400 A 400 A.

The calculated attenuation length for longitudinal phonons on the free electron model for superconduct- ing aluminium is 6 500 A.

where d is the thickness of the centre film, and A' is, 2.4 A DISCUSSION OF THE STEADY STATE ABSORPTION

as before, related to the absorption length. MEASUREMENTS. - A full description and discussion of In figure 7 we show two sets of data, taken with these absorption techniques will be published in a pairs of junctions evaporated at the same time, but future paper [3]. However we should emphasise that

C4-78 A . K. LONG the values obtained depend to a considerable extent on the models used in the analysis, which are difficult to justify rigorously. We feel confident however that the values we have quoted are approximately correct, because (a) the values we obtain are consistent with one anothcr and ( h ) there is no strong dependence of the values on absorbing film thickness, indicating that we havc a true bulk attenuation.

3. Mode separation experiments. - 3.1 IN.I ^RO-

DUCTION TO MODt S t P A R A T l O N CXPEKIMFNTS. - AS the next stage in the breakdown of electron-phonon interaction rates in simplc metals, we make a vcry common approximation for the phonon spectrum.

We divide the phonons into two groups, pure longi- tudinal and pure transverse, with two transverse and one longitudinal mode for each q vector. The phonons are allotted velocities of sound o, and v, equal to tlic appropriate bulk values, and are assumed to couple independelitly with the conduction electrons, with mean free paths A , and A , .

We can make an independent investigation of the different phonon typcs by evaporating transducers diametrically opposite one another on either side of a parallel sided single crystal of insulating material, in our casc a C-cut artificial sapphire. The C-axis in sapphirc is a pure mode axis, with degenerate trans- verse modcs, and so, as we arc selectilig phonons moving perpendicular to the sapphire boundary, we can make the approximation that longitudinal phonons within the transducer couple with longitudinal pho- nons in the sapphire, and si~nilarly for transverse.

There are a number of difficulties associated with this technique. Notably one needs to know information about reflection and refraction at the boundaries (though for perpendicular incidence both these problems are tractable), and about the phonon focus- sing within the body of the sapphire 17). However thcse difficulties can be eliminated by suitable normalisation experiments.

3 . 2 P R E L I M I N A K Y E X P E R I M E N T S WITH A L U M l N l L M TRANSDUCERS. - The following arguments assume that the transition probability for a plionon incident normally on the aluminium/sapphire boundary is close to unity, as is expected. For a thin phonon generator (2 dlA,, 2 dlA,+ 1) the ratio of longitudinal to transverse phonons emitted (LIT ratio) accurately reflects their relative rates of production. However for a thick phonon generator (2 dlA,, 2 d/A, $ I), where the majority of phonons produced are reabsorbed, all information about the intrinsic relative rates of pro- duction is lost. The LIT ratio becomes lncrely a

1 L

function of the relevant velocities of sound.

_/_

01 . , 0, .

Similarly, for a thick detector, all phonons impinging on the junction are detected whatever their type, whereas for a thin junction proportions 2 d/A, of the

longitudinal and 2 dlA, of the transverse are detected.

Hence, in order to investigate differences between longitudinal and transverse phonons, one needs !/]in transducers. This is not as restrictive a criterion as in the lifetime cxperiments, where reabsorption dominates even for thc thinnest transducers, for, in this type of experiment, one is working at perpendicular inci- dence on thc boundary and thc rclevant a values (calculated from the acoustic inipedcinces) are close to unity.

We take as thc scale for A in aluminium the 3 400 A

mcasured in the absorption experiment. A preliminary scrics of measurements has been performed [8] using a thick cc fluorescent

),

Allowance for the phonon focussing may be made by considering c( thick )) transducers, where the LIT ratio is very nearly independent of all parameters of the transducers [3] and may be predicted. We draw on the published data of Eisenmenger [lo] who used Pb/Pb and Sn/Sn tunnel junctions (which with their strong electron-phonon interactions rnay be assumed to bc thick) and obtaincd LIT ratios

-

These preliminary estimates indicate therefore that longitudinal phonons are detected by our thin detector somewhat more readily than transverse i. e. that A , z 3 A,. This result may be inverted statistically [3]

to generate the intrinsic LIT production ratio, which comes out to approximately 114 ; rather more trans- verse phonons are generated by a thin aluminium transducer than longitudinal ones. Hence to a first approximation we may write

A more complcte account of these experiments will be published in due course (31, [8].

4. Future work. - The two types of investigation described in sections 2 and 3 are complementary.

Further absorption experiments are planned to inves- tigate the overall scale of the attenuation of high fre- quency phonons in materials where the interaction is too weak to be investigated by other techniques (e. g.

superconducting tunnelling). The mode separation experiments are still in an early stage ; further work will obviously be necessary to verify the models discussed in section 3.

We should like to thank Dr. C . J. Adkins for the many helpful comments and suggestions he made during the course of this work.

INVESTIGATIONS OF ELECTRON-PHONON INTERACTIONS I N COPPER AND ALUMINIUM C4-79

References

[ I ] GRAY (K. E.), LONG (A. R.), ADKINS (C. J.), Phil. [6] BOBETIC (V. M.), Phys. Rev., 1964, 136A, 1535.

Mag., 1969, 20, 273. [7] TAYLOR (B.), MARIS (H. J.), ELBAUM (C.), Phys. Rev., [2] ROTHWARF (A.), TAYLOR (B. N.), Phys. Rev. Lett., 1971, B 3 , 1462.

1967, 19, 27. ^[S] SZCZEPURA (R. T.), to be published.

[3] LONG (A. R.), to be published. [9] NARAYANAMURTI (V.), DYNES (R. C.), Phys. Rev.

Lett., 1971, 27, 410.

[4] GRAY (K. E.), J. Phys. F., 1971, I , 290. [lo] EISENMENGER (W.), in ⁽⁽ Tunneling Phenomena in [5] See for example KITTEL (C.), ⁽⁽ Quantum Theory of Solids B, Ed. Burstein (E.) and Lundqvist (S.),

Solids )>, Wiley, New York, pp. 326 et Seq. Plenum Press, New York, pp. 371-384.

DISCUSSION V. NARAYANAMURTI. - 1. We have tried an expe-

riment to check the modal dependence of the LIT scattering in Cu with 2 fluorescent Sn generators and a tunnel detector. For -- 3 000A Cu films we did not see any effect. This is not inconsistent with

A N lp.

2. What is the temperature sensitivity of your A1 junctions in the range of thermal quasi-particle tunneling ?

3. Have you seen the 6 A rise in double A1 junction experiments change with film thickness ?

A. LONG. - 1. The D C measurements we have performed do not distinguish the relative amounts of longitudinal and transverse phonon scattering and are not inconsistent with A, and A, being approxima- tely the same.

2. The ratio of within gap to without gap current is in general 1 : 100 at the lowest available tempera- tures. However we do find that the quasi-particle

lifetime in our junctions tends to saturate at the lowest temperatures.

3. Yes. It is of approximately the same magnitude as that observed by Kinder, Lassman and Eisen- menger in tin.

H. J. MARIS. - If the transmission probability at the silicon monoxide/copper interface were very small, would the phonons not bounce back and forth in the copper and be absorbed more readily than hey would in one traverse of the copper film i. e. would not the mean free path estimates presented be too small ?

A. R. LONG. - The boundary scattering a t normal incidence, to which out experiments are most sensitive, is not expected to be very large. Any entranced reflec- tion at large angles of incidence followed by absorption will mean that our mean free path. Calculated assu- ming an isotropic distribution of phonons entering the copper, and attenuated over an isotropic mean free path, is too large rather than too small.

A full analysis of this point will be presented in due course, in a detailed discussion of our experiments.