Negative temperature coefficient of resistivity in samples of Nb 3Ge irradiated by electrons or heavy ions

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Negative temperature coefficient of resistivity in samples of Nb 3Ge irradiated by electrons or heavy ions

F. Rullier-Albenque, L. Zuppiroli, F. Weiss

To cite this version:

F. Rullier-Albenque, L. Zuppiroli, F. Weiss. Negative temperature coefficient of resistivity in samples of Nb 3Ge irradiated by electrons or heavy ions. Journal de Physique, 1984, 45 (10), pp.1689-1698.

�10.1051/jphys:0198400450100168900�. �jpa-00209910�

1689

Negative temperature coefficient of resistivity ⁱⁿ samples

of Nb3Ge irradiated by ^{electrons or} heavy ^ions

F. Rullier-Albenque, ^L. Zuppiroli

Section d’Etude des Solides Irradiés, ^{Centre d’Etudes} Nucléaires, ^{B.P. n°} 6, 92260 Fontenay-aux-Roses, ^France and F. Weiss

Laboratoire des Matériaux, ENSIEG, ^ER 155, ^B.P. 46, 38402 St Martin d’Hères, ^France (Reçu le 2 décembre 1983, révisé le 4 avril 1984, accepté ^{le 21} juin 1984)

Résumé.

Des échantillons de Nb3Ge ^ont été irradiés à basse température (T~ ²⁰ K) par deux projectiles

aussi violemment différents que des électrons de 2,5 ^{MeV et} des ions lourds d’environ 100 MeV (fragments ^de

fission de l’uranium). ^La résistivité

été mesurée

fonction de la température ^{avant et} après irradiation. Pour les deux types ^de particules des valeurs négatives ^du coefficient de résistivité

la température d03C1/dT _{sont mesurées}

et corrélées à la résistivité résiduelle. Ces propriétés ^de transport ^{anormales sont discutées et} comparées

résultats des récentes théories consacrées à

sujet. Qualitativement, il semble clair _que, s’agissant ^d’un système

où le libre parcours des électrons ^est de l’ordre de la distance interatomique, ^le changement ^de signe ^du coefficient

d03C1/dT ^{est relié à un} processus de localisation des électrons

la participation d’interactions électron-phonon ^et

électron-électron. En d’autres termes, l’ecart ^entre les valeurs expérimentales ^et la théorie de Boltzmann du trans-

port ^{dans un} gaz d’électrons presque libres ^est dû à l’interférence entre les collisions sur les impuretés, les collisions

électron-phonon ^et (ou) les collisions électron-électron. Alors que la théorie de ^Belitz et Schirmacher

l’effet tunnel induit par les défauts dans les métaux très désordonnés semble la plus appropriée pour expliquer les résultats d’irradiation

ions lourds, ^bien qu’elle n’inclue que les interactions électron-phonon, les résultats des irra- diations

électrons révèlent la nécessité d’inclure les interactions coulombiennes

dans le modèle de Alts’huler et Aronov.

Abstract.

Nb3Ge samples have been irradiated at low temperature (T ~ ²⁰ K) by ^two drastically ^different

irradiation projectiles : ^{2.5 MeV} electrons and ~ 100 MeV heavy ^ions (uranium ^fission fragments). ^The resistivity

has been measured

temperature before and after these irradiations. For both projectiles negative temperature coefficients of resistivity d03C1/dT

measured and correlated with the residual resistivity p. These anomalous trans-

port properties

discussed and compared ^to the results of the recent theories

the subject. ^{It is} qualitatively

clear that the change ^of sign ^{of the} temperature coefficient of resistivity, occurring ⁱⁿ

system ^{where the electronic}

mean

free path is of the order of the atomic distance, is indeed related to some localization process, helped by electron-phonon and electron-electron interactions. In other words, the deviation of the resistivity from the Boltz-

mann

nearly free electron behaviour is due to some interference effect between impurity scattering and electron-

phonon and/or electron-electron scattering. ^{While the} theory ^{of defect induced} tunnelling ⁱⁿ strongly ^disordered

metals developed by Belitz and Schirmacher

to be the most appropriate ^{for the} explanation ^{of the} heavy ^ion

irradiation results, although it includes electron-phonon interactions only, electron irradiation results reveal the

necessity ^of including ^the interplay with Coulomb interactions

in the model of Alt’shuler and Aronov.

J. Physique ⁴⁵ (1984) ^{1689-1698 OCTOBRE} 1984,

Classification

Physics ^Abstracts

61.80F201361.80J201372.15E

1. Introduction.

There has been a great deal of interest in the A-15

compounds ^{such as} V3Si, Nb3Sn ^or Nb3Ge ^because

of their various peculiar properties. ^{First of} all, they ^are among the compounds ^{with the} highest

superconducting transition temperatures. Secondly, they exhibit anomalies in various normal state pro-

perties. ^In particular, the fact that the A-15 resistivities do not increase linearly ^with temperature ^{and struc-} tural damage but rather tend to saturate at about 120 to 160 u.cm, corresponding roughly ^{to a mean}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198400450100168900

free path ^I of the order of the interatomic distance a,

has attracted considerable attention [1, 2]. ^The

disordered character of these alloys is also reflected in the breakdown of Matthiessen’s rule, ^the _tempe-

rature coefficient of resistivity (TCR) dp _becoming

dT

even negative ^{at low} temperatures ^for sufficiently

disordered samples [3, 4]. ^{This last} point appears ^to be quite ^{universal as} emphasized by Mooij ^who

found such correlations between dlnp and the resisti- dT

vity p for a large class of materials [5]. The critical value of resistivity beyond which the TCR becomes

negative ^{at room} temperature ^{was found} by Mooij

to be approximatively 150 uQ . cm. ^{It is} striking ^to

notice that this value is precisely ^{the one we have}

mentioned for the A-15 saturation resistivity.

Presently ^{the A-15} superconductor ^{with the} highest

transition temperature, Nb3Ge, ^is prepared by sputter- ing, coevaporation ^{or chemical} vapour deposition.

These elaboration methods allow the A-15 Nb3Ge

metastable phase ^to be stabilized and provides ^films

with transition temperatures ^{which can reach 23 K.}

The transition temperatures, Tc, residual resistivities and temperature coefficients have been shown to be

strongly correlated to each other and to all depend

on the global degree of disorder of the material [3, 4, 6].

But the precise ^nature of this disorder is not at present clear. One _may wonder if the disorder is concentrated

mainly ^at the atomic scale such as disorder in amor-

phous alloys (metallic glasses) ^or if it is related to an

inhomogeneity ^of any kind which could depend ^on

the « metallurgical )) qualities of the films. In the present paper ^we try ^to discuss this problem ⁱⁿ

connection with a set of new radiation damage experiments ⁱⁿ Nb3Ge ^{films. We} have tried to change

the level and the spatial distribution of damage by using ^{two kinds of} very different irradiation projec-

tiles : 2.5 MeV fast electrons which produce mainly

isolated point defects and ~ 100 MeV heavy ^ions (uranium ^fission fragments) ^which deposit high energy densities in the form of electronic excitation and atomic collisions. Studies of the effects of electron, neutron, a-particle ^and heavy-ion irradiation on the electrical resistivity have been reported by ^other investigators [3, ^{4, 6,} 7]. ^{One of the} originalities ^of

the present work lies in irradiating ^{the same material} by projectiles ^{which are} expected ^to produce very different damage distributions.

The outline of this paper is ^as follows. In section 2

we describe the experimental techniques ^and present

our results. In section 3 the different models which have been proposed ^to explain the anomalous beha- viour of resistivity ^are reviewed and experimental

results are discussed in connection with the theoretical

predictions and the defect structure in irradiated

Nb3Ge samples ^is analysed. ^{Section 4 contains our}

conclusion.

2. Experiments.

2.1 SAMPLES. - All the Nb3Ge samples used in this work have been prepared by ^{chemical vapour} depo-

sition. The elaboration techniques ^are described in detail elsewhere [8, 9]. Except ^{for one} (Nb3Ge-9)

the Nb3Ge ^{films (of} thickness from 1.5 to 4.5 J.1m)

are deposited ^on 300 J.1m thick sapphire ^substrate.

The resistivities of these samples ^{were measured} by

a standard four points method. The estimated error

in the absolute values of resistivity ^{is about 20} % primarily ^{due to an} imprecise knowledge ^{of the} sample thickness. For the last sample (Nb3Ge-9) depositions ^{are made on} both sides of a steel foil [9].

In this _case, the sample ^{is about 20} tim thick. This

relatively small thickness is _very helpful for electron irradiation because high electron beam current can

then be used without producing important heating

of the sample. ^{On the} other hand the resistivity

determination is more difficult. At each measurement

point, ^{we have to measure} both the resistance of the

Nb3Ge-9 sample (Nb3Ge ^on steel) and the resis- tance of a steel sample ^{of the same nature. In this} way

we were able to obtain the absolute Nb3Ge resistivity

with an estimated error of 40 %.

The initial sample specifications ^are given ^{in table I.}

The different critical temperature Tc ^and resistivity p

(25 K) ^{values are} the results of different growth ^condi-

tions.

2. 2 IRRADIATIONS.

2. 2 .1 Electrons.

The electron irradiation has been carried out in the Vinkac low temperature facility.

This device, described elsewhere [10], consists of a liquid hydrogen cryogenerator coupled ^{to a Van der Graaff} accelerator, allowing irradiation between 18 and 25 K.

In addition, ^{it is} possible ^to isolate the irradiation cryostat ^{from the} cryogenerator ^{and to warm} up the

sample by ^a linear increase of the temperature ^between 20 to 300 K with heating ^{rates from 0.1 to 10 Kelvin}

per minute. The sample Nb3Ge-9 ^{and a steel sam-} ple ^were irradiated together with 2.5 MeV electrons up ^{to a} dose of 4 ^x 102° e/CM2 corresponding ^{to an}

electron flux of about 5 x 1014 e/cm2/s. ^{The irradia-}

tion temperature ^never increased above 22 K. The

sample resistivity ^{was measured} continuously during

irradiation. After _given irradiation times, the irradia- tions were interrupted ^{and the} resistivity ^{versus tem-}

perature ^curves p(T) ^were determined between 18 and 60 K after annealings ^{at 300 or 60 K.}

Electron irradiations are known to produce ^isolat-

ed point defects. In a diatomic compound ^{such as} Nb3Ge ^{it is} possible ^to determine their number by using ^a calculation due to Lesueur [12]. Knowing

the two respective displacement ^threshold energies [14] this calculation gives the fraction of atoms displac-

ed (displacements ^{per atom} (dpa)) for each atomic

species. The outline of this calculation and its appli-

cation to Nb3Ge ^are indicated in the appendix. ^{It is}

1691

Table I.

Sample specifications.

shown that an electron flux of 5 ^x 1014 e/cm2 . s

corresponds ^{to a} damage production ^{rate of about}

5 ^x 10- 8 dpa/s.

2. 2. 2 Ions.

In order to irradiate the films with

heavy ^{ions we have used sources} of fission fragments containing ⁹⁰ % ^{of 2 3 5 uJ. The} samples together ^with

uranium or uranium dioxyde sources were irradiated with thermal neutrons producing the fission of 2 3 5 U nuclei. The neutron irradiations of these ^« sandwiches))

were performed in the Vinka low temperature facility

of the Triton reactor of Fontenay-aux-Roses. ^This

device allows neutron irradiation at liquid hydrogen temperature [11]. The thermal neutron flux was

about 7 x 1011 n/cm2/s corresponding ^{to a fission} fragment ^flux of order of 4 x 1010 FF/CM2/S ^for

a bulk uranium source. During irradiation the tem-

perature always remained below 35 K.

Uranium fission fragments ^are expected ^to deposit high densities of _energy both in electronic excitation and atomic collisions. One can calculate the latter contribution and evaluate the number of atomic dis-

placements (see appendix). ^For the thermal neutron flux used in our experiment (7 ^{x 1011} th.n./cm2 . s)

the damage production ^{rate is found to be of the}

order of 7 ^x 10-6 dpa/s.

2. 3 EXPERIMENTAL RESULTS.

2. 3.1 Electrons.

Figure ¹ shows the evolution of

Fig. ^1.

Evolution of resistivity

temperature

of the Nb3Ge-9 sample under electron irradiation. See table I for the sample description. ^{The arrows} indicate the

resistivity ^minima.

the resistivity ^versus temperature ^curves p(T) ^for

different electron doses. Although ^{it is not} very visible in this figure, ^a resistivity minimum is observed after

a dose of 2.3 ^x 102° e/cm2 (corresponding ^to

~

0.02 dpa). It appears ^at about 40 K and is weakly

shifted towards higher temperature with irradiation.

On figure ^{2 the} negative TCR is shown with a more

appropriate scale; ^this particular ^{curve obtained}

at the end of irradiation after annealing ^{at 300 K is} reproduced together ^{with the curve before} irradiation.

Fig. ^2.

Comparison between the shapes ^of p(T)

before and after irradiation for the Nb3Ge sample ^irradiated

with 2.5 MeV electrons.

During this electron irradiation the critical tem-

perature Tc

has decreased at a rate Ap (Ap Ap ^{AP is the} corresponding resistivity increase) equal ^to

-

0.1 K/pf2. cm, which is the value that we have

already reported ^for previous electron and neutron irradiation [13].

2.3.2 Fission fragments.

Eight ^different Nb3Ge samples ( 1 ^to 8) ^were irradiated with fission fragments.

Figure ^{3 shows some} typical p(T) ^{curves obtained}

before and after irradiation. These curves are simi-

Fig. ^3.

^{Some of} typical p(T)

for as-grown ^and fission fragment irradiated Nb3Ge samples. ^{See table I}

for the sample descriptions.

lar to those reported ^after a-particle [3] ^or heavy-ion [4]

irradiations. The most damaged samples ^{exhibit a} negative TCR in the whole temperature range investi-

gated (20-300 K). ^{In these cases the} p(T) ^{curves are}

very similar to that observed for amorphous Nb3Ge [14].

2. 3 . 3 Comparison between electron and fission frag-

ment irradiations.

The most striking ^{resdlt of the}

present study ^{is the} appearance of a negative ^TCR

after both electron and fission fragment irradiations.

But whereas the negative ^{TCR is} extending up ^{to room}

temperature after fission fragment irradiation, ^it

appears only ^{at low} temperatures (T ⁵⁰ K) ^after

electron irradiation.

Figure ^4, where the temperature coefficient of

resistivity ^{values are} plotted ^{versus the} resistivity

values at room temperature for all the samples, ^shows clearly that there is a strong correlation between these two quantities : ^the larger ^the resistivity p, the lower the temperature coefficient dp ^which

dT becomes negative ^{at a} resistivity ^{of the order of}

150 uQ2. cm.

In order to compare irradiation effects at low temperature ^{we have} represented ^{the same} plot ^at

25 K on figure ^{5. It can been seen on this} figure ^that

Fig. ^4.

Correlation between temperature coefficient of resistivity (TCR) ^and resistivity ^{values at} 280 K for all the samples (as-grown, ^electron

^fission fragment ^irra- diated).

^{and o} symbols ^refer respectively ^{to electron-}

and fission fragment-irradiated samples.

1693

Fig. ^5.

Correlation between temperature coefficient of

resistivity (TCR) ^and resistivity ^{values at} 25 K for all the

samples (as-grown, ^electron

^fission fragment irradiated).

*

and . symbols ^refer respectively ^to electron- and fission

fragment-irradiated samples.

a correlation between dp ^{(25 K) and p (25 K) is}

again ^visible but the values for ion and electron irradiations are not situated on the same correlation

curves. At 25 K the change ^{from a} positive ^{to a} nega- tive TCR occurs for 60 _p 80 uQ. cm ^{for elec-}

tron irradiation and for 120 _p 160 uQ. ^{cm for}

ion irradiation.

3. Discussion.

3.1 THE DIFFERENT MODELS EXPLAINING NEGATIVE TEMPERATURE COEFFICIENTS OF RESISTIVITY.

The observation of a negative coefficient of resistivity

and its correlation with the resistivity has.stimulated

a lot of theoretical work. The numerous theories can

be classified in the five following categories.

1) The extended Ziman theories [15-18] ^{have been} remarkably successful to explain ^the resistivity ^versus temperature ^{curve of metal} glasses [19-21]. Like in the

theory ^{of the} resistivity ^of liquid ^metals [15], ^{the con-} ducting ^{electrons are} considered to be quasi-free

and weakly scattered due to the random position ^of

atoms (pseudopotentials). ^The resistivity ^{can be}

related to the pair distribution function (or, ^{in the} k-space, ^{to the} X-ray interference function). ^The posi-

tion of the Fermi wave vector kF ^with respect ^{to the}

peak position ^{of the} X-ray interference function deter- mines the sign ^{of the} temperature coefficient. The main

success of this model was to explain ^the change ⁱⁿ

the temperature coefficient of resistivity ^{when the}

Fermi wave vector is shifted by changing ^{the compo-}

sition of glassy ^metallic alloys [19, 20].

This diffraction model was used by Bieger et ^al. [22]

to explain ^the shape ^{of the} resistivity ^{curves of some}

A-15 compounds. ^{In our case,} it is worth mentioning

that the p(T) ^{curves for the most} heavily ^irradiated Nb3Ge samples (Fig. 3) ^are very similar to those of many other metallic glasses [14, 19-21].

There are two serious criticisms, ⁱⁿ principle, concerning ^the application ^{of the extended Ziman}

theories to transition metal glassy alloys ^{and to A-15} compounds. ^First these models are all based on a

nearly free electron approach. ^One may wonder if

they ^can properly describe electronic systems ^for which a tight binding approach should be much more

appropriate. Secondly, ^most of the Ziman extended theories assume weak coherent scattering of electrons

despite the fact that the mean free paths ^{are of the}

order of one atomic distance and that the resistivities

are approaching ^{the «} maximum metallic resistivity ^»,

which indicates that these alloys ^{are not} far from the

regime of localization. These remarks have inspired

some of the following alternative models.

2) Imry evoked the possible ^{role of} incipient

Anderson localization in describing the resistivities of highly damaged ^A-15 alloys ^{and more} generally

in explaining ^the universality ^{of the} Mooij correlation in highly disordered metals [24]. ^His interesting specu- lations are based on the recent ideas of Thouless [25]

and the scaling theory of localization of Abrahams

et al. [26] ^which predict ^{that the zero} temperature resistance of a disordered electronic system depends

on its length ^{scale in a universal manner. Extensions}

of these theories to higher temperatures ^{are somewhat} controversial [27, 50]. They ^{are based on the} compari-

son of the different lengths characteristic of the elec- tron propagation (critical localization length ç, ^elastic

mean free path le, size of the sample ^{L, inelastic mean}

free path li ^the temperature dependence of which plays

the main role in the resistivity variation). ^Moreover,

localization theories _are, basically, independent ^elec-

tron theories and they ^{cannot be} applied crudely

to a system ^{in which a} particularly strong ^electron-

phonon interaction and a substantial electron-elec- tron interaction are expected ^{to occur.}

It is worth mentioning here that localization toge- ther with Coulomb interactions have been envisaged by Anderson et al. [31] ^{to be} responsible ^{for the « uni-}

versal » degradation of the transition temperature

T, ⁱⁿ high-T, superconductors ^{like A-15} compounds.

They argue that the anomalous behaviour of the electron diffusion constant associated with localiza- tion leads to an increase of the repulsive interaction between Cooper pair electrons, which results in a

decrease of Tc. ^{Once the} resistivity ^{exceeds some}

critical value p,, Tc ^{is found to decrease as a universal}

function of p/pc.

3) ^The electron-phonon interaction is the main

concern of Girvin and Jonson [23] and Belitz and Schirmacher [49] ^{in their} study ^{of the} resistivity ^of

disordered metals. In normal metals the additional disorder associated with phonons ^{causes the} mobility

to decrease with temperature. ^{The new} feature which appears in the high resistivity regime ^is the fact that the phonons ^can increase rather than decrease the electron mobility. ^{A new} contribution called defect induced tunnelling appears in the conductivity ^when

the Boltzmann hypothesis ^of independent scattering

events fails. Jonson and Girvin developed ^a theory

in the case where the usual adiabatic approximation

breaks down. Although their numerical calculations

give ^a qualitative agreement ^{with the} Mooij ^correla- tion, they ^{do not} give ^{the correct} numbers : this theory

would be more appropriate ^{for the} description ^of

metals with resistivities of the order of 1 000 pQ . ^cm

than for metals with resistivities of the order of 100 gfl . cm. Belitz and Schirmacher have developed

very similar ideas with a different formalism. We will show that their results can account rather accu-

rately ^{for the} heavy ion irradiation results shown in the present paper.

4) Whereas the theories of the previous category

concentrate on the electron-phonon interaction, ^Alt’-

shuler and Aronov [32, 33] proposed ^a description ^of

disordered materials which deals with interacting

electrons in the _presence of weak impurity scattering.

In three dimensional systems, the interaction model

predicts ^that 1 /p ^{varies as} 1/ T ^{at low} temperatures [32].

Such a dependence ^{has been} observed, for the low temperature electrical resistivity of semi-metallic bis- muth in form of evaporated ^{films and even of} polyme-

ric sulfur nitride (SN)x [34].

5) Several other theories have been proposed ^to explain ^{the anomalous} transport properties of disorder- ed metals. In the two level tunnelling ^model [28, 29, 51], ^the negative TCR arises from a Kondo type exchange between the conduction electrons and two

equivalent ^atomic positions separated by ^{a low} energy barrier. This model has been applied ^to amorphous Nb3Ge ^{by Tsuei [14].}

In a general way, ^a negative ^{TCR can} be observed each time the electrons have to cross a distribution of barriers even if the distances between these barriers

are larger ^than atomic distances and are more related to the macroscopic granularity ^{of the} samples. ^This

introduces a new class of models which, ^{to our know-} ledge, ^{have not} yet ^been applied ^to Nb3Ge. ^{Let us} point ^{out that a} theory of disordered materials _gene-

rally characterized by large conducting regions sepa- rated by ^small insulating barriers has been proposed recently by Sheng [30].

3.2 CORRELATIONS BETWEEN THE TEMPERATURE COEF- FICIENT OF RESISTIVITY dpld T AND THE RESISTIVITY.

Figures ^{4 and 5} clearly show that there is a strong ^corre- lation between the temperature coefficient of resis-

tivity -72013 ^{and the} resistivity ^{p, not} only ^{at room tem-}

perature (Mooij plot) ^{but also at low} temperatures.

This correlation is better in the former case than in the latter where two different curves can be distinguished depending ^on the irradiation type. ^{Thus the kind} of defects created does influence the behaviour of the

resistivity. ^{It is} remarkable _anyway that the change

of sign of the TCR occurs around 100 u. cm ^{for both}

cases, that this value is close to the saturation value of the resistivity ^at high temperature ^{and is not so far} from values obtained at the same temperature ⁱⁿ other systems ^with quite different electronic _proper- ties [48].

The systems of interest are dirty metals with mean

free paths of the order of one atomic distance. This indicates that they ^{are not} far from the regime ^{of loca-}

lization. A priori, the models of category ^{2 in sec-} tion 3.1 are the most attractive for explaining ^the properties of interest but we have already ^mentioned

that they ^{are not able} to account for the shapes ^of

the p(T) ^curves that have been observed in heavily

irradiated samples (Fig. 3). The theories able to

explain ^the strong deviation from the Boltzmann behaviour leading ^{to a} negative ^TCR up ^{to room}

temperature ^are the models of categories 1, ^{3 and 4} in section 3.1. All of them explain ^the negative ^TCR by ^some interference effect between two scattering

events.

In the Ziman extended model, ^the origin ^{of the} negative ^{TCR has to} be found in the interference bet-

ween waves diffracted by ^two adjacent ^{centres of the} amorphous material. But because A-15 alloys ^are

narrow d-band metals with bandwidths of approxi- matively ^{0.3 eV} it is reasonable to exclude the extended Ziman theories ^to explain ^the temperature ^variations of the resistivity.

In the Alt’shuler and Aronov interaction model the interference between impurity scattering ^and

Coulomb scattering ^is responsible for the anomalous behaviour of the resistivity : the electron impurity scattering process is influenced by ^the presence of the

surrounding interacting electrons and in turn Cou- lomb interactions between electrons are enhanced when these particules ^are substantially slowed down

by frequent collisions with impurities. ^{We have tried}

to fit the low temperature parts ^of the 1 curves ^{to the}

p

T1/2 law of the interaction model. In the case of the electron irradiation (curves ^{similar to this of} Fig. 2)

the signal ^to noise ratio is approximately ^{5. Thus}

numerous temperature increasing ^{functions can be}

fitted to 1/p including ^{a T 1/2} variation. In the case of ion irradiations the parts ^{of the curves with} negative

TCR are far more extended but the fit is less good

1695

below 40 K. It is better above 60 K in a region ^where

it has less sense.

Finally, the interference between impurity ^and phonon scattering ^{is also} very important ^{for under-} standing the deviations from the Boltzmann theory

as mentioned by Girvin and Jonson who have em-

phasized ^{that in the} dirty ^{limit the electrons are} diffusing ^so slowly ^that they ^{are sensitive to the time} dependence ^{of the} phonon amplitudes. ^Unfortuna- tely, their numerical calculations imply ^{that when}

the TCR changes sign, ^the resistivity ^is by ^{one order}

of magnitude larger ^{than the} experimental ^value.

A theory ^of phonon controlled conductivity ^{in the} high resistivity metals has been developed ^more recently by Belitz and Schirmacher [49] ^{on the basis}

of similar concepts. They obtained for the temperature dependent conductivity

-

Mo ^and Lo ^are static contributions due to disorder.

--

MT ^{is the} generalized phonon scattering ^rate

which reduces to the nearly free electron expression

when the mean free path ^is large compared ^{to the}

Fermi wavelength ÀF.

-

LT ^{is the new term due to} the interference _pro-

cess that they ^call phonon controlled tunnelling ^rate.

The random tunnelling process increases with temperature ^{as does the} scattering ^rate but the former is proportional ^{to the} conductivity ^thus allowing ^for negative TCR. When the time scale for the electronic motion is much smaller than that of the phonons (adiabatic approximation) ^the previous formula is

a generalisation of the Ziman theory ^for amorphous

metals. Thus it is not surprising ^{that such a formalism}

can account for the conductivity ^{of the} heavy-ion

irradiated samples ^of figure ³ (negative coefficients from 20 to 300 K). Figure ⁶ shows that this model

can indeed account for the resistivity ^{curves obtained}

after heavy ion irradiation with Mott’s maximum metallic resistivity value of the order of 700 gf2.cm.

Even if the initial concept of maximum metallic

resistivity ^{is in} contradiction with the scaling theory [26] in which the conductivity ^vanishes continuously,

it has been shown [49], ^{and this} experimental study

also confirms, ^that pm is the relevant resistivity ^scale

near the Anderson transition. It is worth mentioning

that this theory is also able to explain ^the resistivity

saturation at high temperature ^{at a} value of about 150 gfl. cm.

However the small negative dp _{parts of the curves}

dT

for the electron irradiated samples ^can probably ^not

be explained ^{in the same} way and some Coulomb interactions halve probably ^to be included as men-

tioned in the semi-quantitative analysis ^{of Kaveh}

and Mott [50].

Fig. ^6.

Comparison ^{of two} p(T ) ^curves measured after

heavy ion irradiations with the model of Belitz and Schir- macher [49]. ^{Dots are} experimental ^data presented ^on figure ³ (see table I for sample description). Symbols (*)

are

the fits using curves P PM = f T 0 ^{of the figure I (a)} in reference [49]. Here, pm is the value of Mott’s

maximal

resistivity

pm

3 n’hle’kf and 0 is the Debye tempe-

rature equal ^{to 302 K for} N b3Ge [52]. Best fits for p(T)

curves

6 and 9 are obtained using respectively ^{the fourth}

and fifth (from bottom) 1!.... ^{curves of Belitz et al. with}

PM

value of pM of the order of 700 gf2.

The following section is an attempt ^{to determine} in a more microscopic way which are the defects

responsible ^{of these} macroscopic ^{effects on the resis-} tivity ^{and to} compare electron and fission fragment

irradiation.

3.3 DEFECT STRUCTURE IN IRRADIATED Nb3Ge ^SAM-

PLES.

In an ordered diatomic compound ^AB,

irradiation produces ^point defects either isolated

(electrons) ^or in cascades (neutron ^or heavy ions)

and antisite defects, ^{i.e. A atoms} occupying ^{B sites}

and vice versa. This atomic interchange results in a

decrease of the long range order parameter ^{S. The} existence of antisite defects has been demonstrated in neutron irradiated A-15 compounds using ^neutron

and X-ray diffraction techniques [35, 36]. ^{Due to the}

structural instability ^of high-T, ^A-15 alloys, ^the

presence of defects can result in local atomic rearran-

gements. ^{Testardi et al.} [37, 38] ^have suggested ^that

the major effect of point ^or antisite defects is to

generate microscopic ^strains occurring ^{over dimen-}

sions of the order of some unit cells.

In the case of fission fragment irradiation, ^the large

energy deposited by electronic excitation can be

thought ^to strongly influence the damage production.

Moreover, the observation that p(T) ^{curves for sam-} ples irradiated at the highest ^{doses are} very similar to that for amorphous Nb3Ge ^{could be} interpreted

in terms of local amorphization. ^{Let us} point ^{out that}

a mechanism by ^thermal spikes ^{has been proposed} by ^Lesueur [39] ^to explain ^the Pd80S’20 amorphiza-

tion under fission fragment irradiation. In this assump-

tion, ^fission fragment irradiated Nb3Ge samples

are considered as binary ^mixtures composed ^{of small} amorphous regions ^{in an} undamaged matrix, ^and electrical resistivity ^{can be} expressed within the framework of the effective medium theory [40].

Attempts ^{to fit} experimental resistivity production

curves with this interpretation ^{have been unsuccess-}

ful showing ^{that a’ more} homogeneous ^{model is} required ^to explain ^these resistivity ^results.

It is possible ^to relate the long range order _para- meter S to the resistivity by using ^the relationship

of Muto [47]

where PD and po ^are the electrical resistivities for the disordered and the ordered states, respectively. ^More-

over, ^a functional dependence of S with irradiation based on probability arguments has been derived by

Aronin as

where A is the replacement effectiveness _per atomic

displacement ^c (dpa).

In the case of fission fragment irradiated Nb3Ge samples, ^{a linear} dependence ^{between S} (deduced ^from resistivity measurements) ^{and the number of atomic} displacements, ^{is observed once the dose exceeds}

about 0.04 dpa. Experimental ^{value of A is found to}

be N 6. The same value of A has been reported by

Brown et al. [42] upon measuring the r sistivity ^of

a Nb3Ge sample irradiated by ^fission

f gment.

Then we reach the conclusion th t amorphous

behaviour observed in the most radiation-damaged samples ^{is not due to a local} amorphization ^induced by irradiation but results from a very large ^accumula-

tion of antisite defects. This interpretation is consis- tent with the experimental X-ray results of Cox

et al. [36] ^{in neutron} irradiated Nb3Ge samples;

they ^{found a} relation between the transition tem-

perature ^{and the} long-range ^order parameter up ^to

a dose of - 0.1 dpa ^{and observed a loss of} crystalli- nity ^{after a} dose of N 0.2 dpa.

In the ^case of electron irradiation, antisite defects

probably play ^{also a} role in the damage mechanisms,

but it is not possible ^to determine their contribution from resistivity measurements.

Antisite defects can be created during ^electron

irradiation by replacement ^collision sequences. In A-15 compounds production ^of large ^{amounts of}

antisite defects in such _sequences is anyhow unlikely

because the 102 > ^{atomic rows} along ^which repla-

cements produce disorder, ^are « broken » between the ABA sequences (see Fig. 7), whereas the very

probable ^{collision sequences} along 100 ) ^{do not} produce antisites. A detailed study of electron irra- diation effects in V3Si samples [41] suggested ^that

antisite defects are produced ^{at about the same con-}

centration levels as are point ^defects in electron irradiated A-15 compounds.

Fig. ^7.

Schematic sketch illustrating in the A-15 struc- ture the 102 > directions along ^which replacement ^col-

lisions

produce ^disorder.

Thus the main difference between fission fragment-

and electron-irradiated Nb3Ge samples ^{at a} given

number of dpa appears ^to be a much larger ^concen-

tration of antisite defects in the former (1 ) than in the latter. One can see here the indication that the effect of antisite defects is mainly ^{to affect} the electron-

phonon interaction while point defects tend to inter- fere with the electron-electron interactions.

4. Conclusion.

We have studied the influence of two very different types ^of irradiation (electrons ^{and fission} fragments)

on the transport properties ^of Nb3Ge samples. ^In

both cases a change ^of sign ^{of TCR} dP _{was observed}

dT

and a correlation between dp and the residual resis- dT

tivity ^was established either at high ^{or at low} tempera-

tures. The problem ^{is to} decide if these correlations

are universal in the sense of Mooij. ^In the detail of the

values, they ^{are not} universal and the type ^{of disorder} has some influence ; but in all cases the sign ^{of the}

TCR changes ^at values of the order of 100 uQ. cm

which is a fraction of about 0.15 of the maximum metallic resistivity pm initially suggested by ^Mott (in three dimensions _{Pm ~} hale 2 = ⁶⁰⁰ u cm ^when

a

3 A). ^The scaling ^theories [26] have shown that there is no maximum metallic resistivity ^{in 3 D.}

It seems clear anyway that pm is the relevant resis-

tivity ^{scale near} the Anderson transition [49] though

the conductivity ^vanishes continuously. ^{In the same}

way the Mooij ^criterium give ^{the correct numbers} for

most of the changes ^of sign ^{of the} TCR, ^{even if there}

is no real reason for this correlation to be valid in

detail, ^at any temperature.

Among all the models proposed ^to explain ^the

anomalous transport properties, ^{the most} appro-

priate ^{seem to us} those which are related to some inter-

(1 ) ^{In this case} disordering by ^thermal spikes ^is probably

the most efficient mechanism.

1697

ference effects between impurity scattering ^{and elec-} tron-phonon ^or electron-electron scattering. ^{In order}

to distinguish between these two models, ^further experimental ^{work is needed. In} particular magne- toresistance measurements should be _very helpful.

Acknowledgments.

It is a pleasure ^{to thank Dr. Y.} Quere ^for many valua- ble suggestions ^{and comments. We wish also to thank}

S. Paidassi who provided ^some samples ^used hire

and J. Dural and R. Blot for their technical assis- tance during ^{the neutron} and electron irradiations.

Several useful comments from the referees are grate- fully acknowledged.

Appendix

CALCULATION OF THE NUMBER OF ATOMIC DISPLACE- MENTS.

1) Electrons.

Lesueur has recently proposed

an analytic calculation of the number of atomic

displacements ⁱⁿ polyatomic compounds [12]. ^The principal limitation of his calculation is that the electronic stopping ^is neglected but, concerning ^elec-

tron irradiation, ^this approximation ^is quite ^valid.

Different assumptions of this calculation (see ^Ref.

[12]) ^{result in very} simple expressions; the number of j ^atoms displaced by ^a primary ^of type k moving ^with

an energy Ek ^{can be} expressed ^{at low} energy in the form

where Ckj ^{is a constant only depending on the k - j}

collision and Ej is the threshold _energy to displace a j ^{atom. Let us} notice here that a j ^{atom can} displace another j ^atom provided ^{that its energy is at least} equal ^{to 2} Ed ; but, ^to displace ^a k-atom, ^its energy has to be at least equal to - E§ ^where _£kj

^{4 Mk Mj}

has to be at least equal ^{to -} _Ajk ^where Åkj

( 4MkM. J )2 . Mk + Mj

In the case of Nb3Ge ^{one finds}

The total cross-sections to displace ^{Nb atoms can}

be then written with respect ^to atomic concentrations

as :

and a similar expression ^for dd e.

Emb (resp. Eme ) represents the maximal _energy transferred to a Nb (resp. Ge) primary ^{atom and}

d 6 (resp.(-rp) ) the differential cross section

dE IVb p ^V/Ge/

for collisions between electrons and Nb (resp. Ge)

atoms.

In a previous experiment ^we have determined the

respective ^threshold energies ⁱⁿ Nb3Ge. By using

numerical results tabulated by ^Oen [45] ^{it is} straight-

forward to calculate these integrals ^{for 2.5 MeV elec-}

trons. The results are :

resulting ^{in a total} dpa production ^{rate of 5 x 10-8} dpa/s ^{for an} electron flux of 5 x 1014 e/cm2/s.

2) ^uranium fission fragments.

Energetic heavy

2) ^Uranium fission fragments.

Energetic heavy

ions like uranium fission fragments ^{lose most of}

their _energy in the form of electronic excitation. In this case an analytic determination of the _energy transferred to lattice, and then of the number of atomic displacements, ^{is not} possible. ^{In a mono-}

atomic compound, ^{one can} calculate the damage

created by ^fission fragment ^sources [44] ^{but such a}

calculation is much ^more complicated for diatomic

compound. ^On the other hand a diatomic compound

like Nb3Ge ^{can be shown to} be assimilated with ^a monoatomic compound ^for calculations of _energy losses. In this _way it is possible ^to determine the energy transferred to lattice by ^{a fission} fragment

source. A typical «damage profile » ⁱⁿ Nb3Ge ^is

shown in figure ^8. Knowing ^{the total} energy ^trans-

Fig. ^8.

Damage profile per average fission fragment ⁱⁿ

Nb3Ge target ^for

^{thick uranium source.}

ferred to

lattice E E = dEn ^{it is possible to}

estimate the number of atomic displacements by

Ed represents ^{here the} average threshold displace-

ment energy that we have determined from the atomic percentage of Nb and Ge and found to be equal ^to

30 eV. With a thermal neutron flux of 7 x 1011 n/cm2/s,

one finds a damage production ^{rate of about 7 x}

10-6 dpa/s ^{for a thick uranium source.}

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