Transport properties in dilute alloys

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Transport properties in dilute alloys

F.J. Blatt

To cite this version:

F.J. Blatt. Transport properties in dilute alloys. J. Phys. Radium, 1962, 23 (10), pp.597-601.

�10.1051/jphysrad:019620023010059700�. �jpa-00236644�

597

TRANSPORT PROPERTIES IN DILUTE ALLOYS (*) By ^{F. J.} BLATT,

Michigan ^State University, Department ^of Physics, ^East Lansing, Michigan, ^{U. S. A.}

Résumé. 2014 On passe ^en revue, aussi bien dans le domaine théorique qu’expérimental, ^les progrès

réalisés dans l’étude des propriétés ^de transport ^des alliages dilués ; ^on porte ^{une attention} particu-

lière aux valeurs obtenues dans les mesures classiques ^{de ces} propriétés ^{en vue d’en tirer des infor-}

mations sur la structure de bande des alliages.

Abstract.

²⁰¹⁴

Recent theoretical and experimental progress relating ^to transport properties ⁱⁿ

dilute alloys ^is reviewed, ^with particular ^attention paid ^to the value of conventional transport

measurements in providing information on the electronic band structure of alloys.

LE JOURNAL DE PHYSIQUE 23, OCTOBRE 1962,

The purpose of this _pope is primarily ^{to review}

some of the recent progress in our understanding

of transport processes ⁱⁿ dilute alloys ; ^{in the}

course ^of this review, particularly toward the end,

1 shall mention some as yet unpublished ^results pertinent ^{to this} topics

’

There are of course diverse avenues of experi-

mental and theoretical progress, all aimed ^at the

same goal : ^better understanding ^{of the electronic}

structure of alloys, ^{and of} thèse, ^the study ^of transport phenoinena ^{is but one of many. Others,}

for example, magnetic [1], ^thermal [2], ^and optical properties [3], ^diffusion [4], ^{and rosonance} pheno-

mena [5], ^{have revealed a wealth of} information that will be treated in considérable détail in the

course of this conférence. ^It is the relationship

between various physical properties ^{and whatever}

model of an alloy ^{we construct which seems to me}

the most significant ^and interesting aspect, ^and consequently ^{also more crucial than the result of}

any ^one individual measurement, ^{no matter how}

ingeniously ^{devised or how} carefully performed.

I shall therefore allow mygelf occasional reference to topics ^that appear elsewhere ^on the program.

Today experimental techniques ^{and their theore-}

tical interprétation ^with respect ^to pure ^metals have far outdistanced conventional transport ^mea- surements. ^{To be} sure, the latter played ^a very

important ^{and valuable} role, ^even only ^{a few}

years ago ; for example, ^from thé relative magni-

tudes ^{of the} electrical and thermal conductivities

’

of metals, ^Klemens [6] deduced that the Fermi sur-

faces of copper,-silver ^and gold touched the ^zone

;

boundary, ^{and that the same was} probably ^also

true for the heavy ^alkali metals, cesium and rubi- dium. Convincing ^{data on the Fermi surface of}

copper, however, ^{is to be found not in conven-} tional transport measurements, ^{but in the measu-} rcments of Pipp4rd ^oh the anomalous rosistance [7], (*) Supported by U. S. Air Force Oiîîce of Aerospace Research, and the National Science Poundatioii.

the de Haas-van Alphen ^{work of} Shoenberg [8],

the cyclotron ^résonance measurements of Langén- berg ^{and Moore} [9], ^{and the} magnéto-tésistànce

results of Klauder and Kûnùer [10], ^{ànd Ales-}

seévskii and Gaidukov [11J. The value ôf these

experiments ^{W88 of course} gfeàtly ^{eilhànèed land to}

some extent they ^were inspired by ^{recent thoore-}

tical work, notably ^of Pippard (7J, ^Hârtisôn [12],

and Lifshitz and coworkers [13]. Durmg ^the past

few years, de Haa§-VâH Alphen, tnagftet6-l’esig-

tance, Hall, magnéto-thermal, magnotô-acôustie,

and cyclotron çemnancé effects have been studied and have beén observe on a hogt ouf _pure met- als [14], including ^{an alkAli métal} [15] ^{and somme of}

the transition mutais [16]. *’ ^PÓBVerful though ^these techniques ^may be, they unfortunatély ^canm4 help

very ^{much in our séarch} fot a better und-erstanding

of alloys. ^{Oné of the} principal tèquîreihents ^for

thèse èxpérimènts ^{ig. à} long electronio mean free

pàth-thàt 19 ^to say, high püèity. Côn-seq-uêntly,

if we wish ^to focus ^out attention on âlloys, ^conven-

tional transport measurements continue to play ^a

very useful ^role.

One of the most important eârly contributions to the theory ^of transport ⁱⁿ alloys. ^{is the Weil}

known paper ^of Mott [17], ^why pr6vided ^{a theote-}

tical explanation ^{of the} empiriéàl ^{fuie formulated} by ^Linde and ^{now known as} Lifide’s ^rule [18].

Another Very important gtëp ^{fotwed Wàs takeh}

somé fifteon yeorg lator by ^Friedél [19]î ^{who con#1-}

dered in detail the perturbation ^caüèèd- by ^a single impurity ^{in a} gans of frée eleéirono, Friedel de- monstï’ated first, ^{that the} screening chargé ^about

the impurity ^{bears a} simple ^{relation to the} phase

shifts of a partial ^wave àfiàly8is, and, second, ^he

found that the screening charge ^{dues not decrease} monotonically ^{as would be ôbtaenëd} by ^the simple

Fermi-Thomas model [20], ^but oàcillates, ^{the Wave} Length ^{of thé} oscillations being ^{related to thé Fermi}

momentum and the amplitude docroaging ^{with the} cube ^of the distance ^{froln thé} impurity ^conter.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:019620023010059700

598

It is difficult to overestimate the importance ^of

this work. In the area of electronic transport, ^it

has provided ^the basis for our current under-

standing ^{of the} resistivities of alloys ^{and their}

thermoelectric properties [21]. Moreover, ^{it has}

been extremely useful in discussions of diffusion of

impurities ^{in metals} [22], ^of positron ^annihi-

lation [23], and the oscillations in the electronic charge density appear ^to be crucial to the expla-

nation of the Knight ^{shift and of} quadrupolar

eff ects in solid solutions [24].

I shall consider recent transport measurements

shortly. ^{Let me first make a} few remarks con-

cerning ^the development ^{of the} theory, particu- larly ^{in recent} years, which has witnessed various modifications and improvements ^{of the} original

work of Friedel. These concern themselves pri- marily with the removal of certain simplifying assumptions and restrictions in Friedel’s theory- namely, 1) ^{the use} of free electron wave function

(plain waves) ; 2) ^the assumption ^of spherical

energy surfaces in k-space ; 3) the absence of elastic distortion due to alloying ; 4) ^infinite dilution of

solutes ; 5) ^the validity ^{of the one electron} approxi-

n1ation.

(1) ^The restriction to plain ^{wave functions has}

been considered by ^Roth [25] and Harrison [26]

who have indicated the correct treatment of elastic

scattering ^{of Bloch waves. Blandin} [27] ^{has con-}

cerned himself primarily ^{with the} oscillatory charge fluctuations and has shown that these appear in much ^{the same} way when Bloch func- tions are used for the unperturbed eigenfunctions

in place ^of plain ^waves.

(2) ^Gautier [28] ^hais suecessfully removed the restriction to spherical energy surfaces. He finds that in the case of copper, distortions ^{in the Fermi}

surface bring ^forth non-periodic oscillations in the

screening charge density along ^certain crystal- lographic directions.

(3) ^The effects of elastic distortion about impur- ity ions have been considered by ^Blatt [29] ^follow- ing ^the suggestion of Harrison. The calculated residual resistivities of the noble metal alloys ^were

found to be in excellent agreement ^with experi- ment, ^{and these same ideas have now been} applied

with considerable success to calculations of the

Knight shift in solids [24] ^{and of the} resistivity ^of liquid ^solutions [30].

(4) ^Caroli [31] ^and Flynn [32] ^have investigated

the effects of finite impurity concentration.

Caroli’s conclusions concerning the oscillations of the charge density ^about impurity centers may be summarized by saying ^{that the most} important

term in the charge density fluctuations due to two

neighboring impurities ^{is the direct sum of the}

individual oscillatory terms, and that interférence effects appear ^to play only ^a minor role. Flynn

ari ived at similar conclusions as regards ^{the resi-}

dual resistivity. Finally, ^{we corne to the last} point---namely, ^the question concerningthevalidity

of the one-electron approximation. This very

complicated problem, ^the reformulation of the

impurity resistivity ^{for an} interacting electron gas, has been tackled by Langer [33]. His f ormal solu- tions are too complex ^for practical evaluation ; however, Langer ^and Ambegaokar [34] ^{have shown}

that in the appropriate perturbation approxi- matiûn, the Friedel sum rule holds even for a

systém ^of interacting electrons. The density ^fluc-

tuations have also been re-examined using many-

body techniques, ^{and once} again ^the expressions

reduce to Friedel’s in the appropriate ^limits [35].

Thus, ^{the work of the recent} past ^{has further} strengthened ^the ideas first advanced by ^{Mott and} Friedel, ^{and has} greatly enhanced the usefulness of these concepts.

I have so far referred primarily ^{to work which}

relates most closely ^{to the} properties ^{of noble} metals, alloyed ^with non-transition elements.

When the solvents or solutes are transition metals,

new problems ^{arise. Here} too, considerable _pro- gress has been made in recent years, particularly by ^Friedel [36], de Gennes and Friedel [37],

Yosida [38], Kasuya [39] and Rocher [40]. ^Pro- bably ^{the most fruitful} development ^{relates to} spin

disorder scattering. Friedel and de Gennes [37],

Mannari [41], ^and Kasuya [39] have shown that this mechanism should lead to the 1’2 dependent resistivity ^{at low} temperatures obser ved in ferro-

magnetic metals. The results of de Gennes and Friedel have been extended by ^{Weiss and}

Marotta [42] ^{to account for the} resistivity ^{of most} magnetic ^{metals at low} temperatures. ^Béal [43],

and also Rocher and Friedel [44], ^{have made use of}

these same concepts ^to provide ^an .elegant ^des- cription ^{of the} resistivity ^and thermoelectric power of metals which undergo ^an order-disorder trans- formation.

One of the most remarkable and gratifying ^fruits

of this work is the resolution of the puzzle ^{of the}

" resistivity minimum" a nd of thé associated thermoelectric power anomaly, ^{which is observed in}

certain dilute copper alloys [45] ^{as well as in} alloys

of silver, gold, ^zinc [461, ^and magnesium [47].

This phenomenon ^was discovered by ^de Haas, ^de

Boer and van den Berg ^as nearly ^as 1933, ^{in measu-}

rements on what was bel ieved to be pure gold [48].

y

Work ^was continued at Leiden, ^at Oxford, ^and by

°

MacDonald and coworknrs at Ottawa. The results for _many years ^were extremely puzzling ^because

there seemed to be lit11e rhyme ^{or reason for the}

appearance of the phenomenon ^{in such a} profusion

of alloys. Although ^as early ^{as 1956} the work ^of Owen, Brown, Knight and Kittel [49], ^{Jacobs and}

Schmitt [50], ^{and of} Hodgcock [51] strongly ^suggest-

ted that the presence of transition metal imbu-

rities might ^have somef hing to do with the appear-

599 ance of a resistance minimum, ^{it was} only ^as

recently as 1960 that careful work by ^{the Ottawa}

group [52] provided proof ^that indeed the culprits

are transition metal impurities. Already prior ^to

the work of Gold and coworkers, de Vroomen [53],

Brailsford and Overhauser [54] and Dekker [55]

had suggested ^{that the} resistance anomalies might

be due to transition metal impurities alone, ^ad

that _year Kasuya [56] published ^a paper which showed that spin-disorder scattering ^{could account}

not only ^{for the} resistance minimum but also for the associated anomalously large thermoelectric power. Independently, de Vroomen and ^Potters

[57] ^{and also} Bailyn [58] had carried out cal- culations of the resistivity and the associated thermoelectric power due ^to spin ^{disorder scat-} tering, ^{and it now seems} quite ^well established ^that

the resistance minimum is’in fact closely ^{related to}

a cooperative spin dependent interaction between transition metal impurities [59].

As regards ^the thermoelectric power of copper at temperatures well above that of the resistance

minimum, ^{it is now} clear from the work of Blatt and Kropschot [60] ^{that most of the} thermoelectric power of copper between 20 and about 120 OK is the result of phonon drap ^rather than ^{of the}

Sommerfeld-Franz diffusion ^term. The presence of this contribution to the thermoelectric power of

a metal was first suggested by ^Gurevich [61] ^but

was observed and studied initially ^{in semicon-}

ductors [62]. Subsequently, ^Klemens [63], ^Ter

Haar and Neaves [64], Hanna and Sondheimer [65]

and more recently Bailyn [66] ^{and Ziman} [67] ^have

studied the theory of this eff ect in metals, ^{and the}

latter in particular ^have emphasized ^the sensitivity

of the phonon drag thermoelectric power ^to details of the electron-phonon interaction, particularly ^the

relative importance ^of Umklapp and normal scat-

tering. The existence of phonon drag ^{has now}

been demonstrated not only in copper but also in

-

the alkali metals [68] ^{and, in aluminum} [69]. ^The

method employed by de Vroomen et al. for alu- minum is analogous ^{to the one used to separate}

the electronic and lattice contributions to the spe- cific heat at low temperatures. ^{It has a clear} advantage ^over that of Blatt and Kropschot ^{in that}

the two thermoelectric eff ects can be studied in the pure metal without ^{recourse to} alloying. ^{On the}

other hand, ^{it does not allow the} phonon-drag phe-

nomenon to be studied at elevated temperatures.

We have recently ^{extended the} measurements to dilute ternary alloys ^and preliminary results indi- cate that, provided phonon drag ^is taken into account properly, ^{one can} predict the thermo- electric behavior of a ternary alloy ^once the pro-

perties ^{of the} binary alloys ^{are known} [70], ⁱⁿ complete analogy ^{to the} prediction of the residual

resistivity ^{of a} ternary alloy ^{from the known scat-} tering ^cross sections of the constituent solutes.

At the ^same time, ^{we have also made use of the} technique of de Vroomen to study ^the phonon drag

contribution ^{to the} thermopower ⁱⁿ metals, ^such

as lead, ^{where the} alloying technique ^{would not be}

suitable [71]. ^{Here we} have observed ^a phono drag contribution at iow temperatures ^which

appears ^to reach its maximum near twice the critical temperature [72].

Finally, Schroeder and Henry [73] ^have recently

measured the thermopower ^{of oc brasses between}

helium and room temperature using alloys ^with

zinc concentrations ranging ^{from a few to about}

40 per cent. ^One plausible explanation ^{of their} results, suggested by ^Gold [74], is that with increa-

sing ^zinc concentration, ^{contact at the} [100] ^faces

is also established. This interpretation suggests

that the thermopower, particularly ^the phonon drag contribution, ^{because of its} sensitivity ^to

certain details of the Fermi surface, ^{will be useful}

in the study of the band structure of some alloys.

Investigations of the thermal conductivity ^of

metallic alloys ^have been motivated primarily by

interest in the lattice contribution to the thermal

conductivity ^{which can} be determined if the solute content is sufficiently high ^so that the electronic

component ^no longer dominates. The electronic component ^can then be calculated from the elec- trical conductivity. ^{The method of} separation ^of

the electronic and lattice contributions has been discussed in detail by Kemp, Klemens and co-

workers [75]. ^At liquid ^helium temperatures, ^the

lattice thermal conductivity ^is generally propor- tional to T2. There are two resistive mechanisms which could lead to this temperature dependence, scattering ^of phonons by electrons and by ^dislo-

cations. Work of Kemp and Klemens [76] sug-

gested ^{that the} dislocation density in annealed dilute alloys might ^be quite large, ^{since the} 1 /T2

contribution to the thermal resistivity ^{was rather} high compared ^to that which would be expected

from the usual concentrations of dislocations in well-annealed pure specimens. ^An alternative pos-

sibility ^{is that} alloying might significantly ^alter

the electronic band structure [77]. Pippard [78]

however, ^has suggested that when the electronic mean-free path ^becomes comparable ^{to the wave-} length ^of the dominant phonons, ^{its further reduc-}

tion can influence the electron-phonon interaction,

and thereby increase in the lattice thermal ^resis- tivity. ^{Recent work} by Lindenfeld and Penne- baker [79] ^on the lattice conductivity of copper

, alloys ^{has shown this} effect very clearly. ^The

observed changes in lattice conductivity ^between

various specimens ^is evidently ^the result of a

change ⁱⁿ electron-phonon interaction as predicted by Pippard, obviating the need of large dislocation densities.

Another-problem, ^{which has} perhaps ^{not received}

as much attention as it deserves, ^{relates to the}

600

thermal motion of the impurities. ^Koshino [80]

ha9 considered scattering ^of conduction, electrons, not by stationary impurity ^{ions but} by ^{ions which} participate in thermal vibrations. He finds that at high temperature ^the vibrations of impurity

ions give ^{rise to a} resistivity contribution linear in

température ^{as is the idéal} resistivity. ^{At low} temperatures, ^Koshino calculates a resistivity ^con-

tribution proportional ^{to the density of impurity}

centers and to the square of ^the température.

There does not appear ^to be much expérimental

information available on this particular point, ^but

as yet unpublished ^{data of Schroeder} [81] suggests

that such a contribution exists. Schroeder finds in alpha phase copper-zinc alloys ^a contribution ^to the resistivity which goes roughly ^as Tg, ^and

which increases monotonically ^{with zinc content.}

In summary, the theoretical work ^{of recent} years, particularly by Friedel and coworkers, by

Ziman, Bailyn, Kasuya, Langer ^and others, ^hàs established a sound framework for thé discussion of transport properties in dilute alloys. Yet, though ^{much is now} known, ^much still remains to be done ; ^for example, although ^{the residual resis-}

tivities of the monovalent metals are now well

understood, ^{the residual} resistivity ^of polyvalent

metals - gallium, ^for example [82], ^{- remains as} yet without adequate explanation. Transport ^measu-

rements on dilute alloys suggest ^{that the} rigid

band model is a good approximation ^{to the elec-}

tronic structure of dilute alloy. ^{and that} transport properties, particularly thermoelectric effects,

reflect changes ^{of the band structure and Fermi}

surface with increasing ^solute concentration.

Thus, transport measurements in substances where the ^more exact and more powerful ^{resonance tech-} niques ^{cannot be} emplôyed, ^{will continue to} pro- vide valuable and interesting information.

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