The residual resistivity due to defect pairs

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The residual resistivity due to defect pairs

C.P. Flynn

To cite this version:

C.P. Flynn. The residual resistivity due to defect pairs. J. Phys. Radium, 1962, 23 (10), pp.654-658.

�10.1051/jphysrad:019620023010065401�. �jpa-00236656�

initial overlap appears ^to take place ^{across the}

Rame face. Such calculations yield several inte-

resting points. ^{The band} gaps ^across the ’( 10 . 0 )

faces in the 03B6 phases ^{in the} region ^of the ( 10. 0) } overlap appear ^to be less than 1 eV (AE(10.c) ^{- 0.73}

eV for the Ag-Sn system [6]) ^{and the band} gaps

across the 100. 21 ^{faces in the e phases in’the région} of 100*.21 overlapappear ^to be about ^{1 eV}

eV for the Ag-Zn system [7]). ^{If a} slight change

in the slope ^{of the c} spacings ^{in the} Ag-Sn system is interpreted to ^{be a} result of the {OO. 2} ^overlap

in that system (for ^{details see ref.} [7]). ^{The cor-} responding ^band gap is, equal ^to

.

0. 41 eV at ela == 1. 51.

The same band gap ^at the electron concentration value in the region ^{of the} (10.0) overlap ^{has been}

estimated to be AE(oo.2) ^{=1.32 eV at} e ja ^{= 2 .4} [6]. Therefore the lattice spacings ^{data indicate}

that the band gaps in the Brillouin ^zone of the

hexagonal phases ^{decrease with the increase of}

the solute content.

This work has been supported ⁱⁿ part by ^the

United States Atomic Energy Commission, Washington 25, ^{D. C.}

REFERENCES

[1] ^JONES (H.), ^Proc. Roy. Soc.,1934, ^{A 147, 396.}

[2] ^MASSALSKI (T. B.) ^{and KING} (H. W.), Prog. ^{Mat. Sci.,} 1961, 10, 1.

[3] ^JONES (H.), ^The Theory of Brillouin Zones and Elec- tronic States in Crystals, North Holland Publishing Co., Amsterdam, 1960.

[4] ^MASSALSKI (T. B.) and COCKAYNE (B.), ^{Acta Met.,} 1959, 7, 762.

[5] ^MASSALSKI (T. B.) ^{and KING} (H. W.), Acta Met., 1960, 8, ⁶⁷⁷ and 684.

[6] ^KING (H. W.) and MASSALSKI (T. B.), ^Phil. Mag., 1961, 6, 669.

[7] ^MASSALSKI (T. B.) ^{and KING} (H. W.), ^Acta Met., 1962, 10, in _press.

[8] GOODENOUGH (J. B.), Phys. Rev., 1953, 89, 282.

[9] ^HARRISON (W. A.) ^{and WEBB} (M. B.), The Fermi Sur- face, ^John Wiley ^and Sons, ^Inc.

THE RESIDUAL RESISTIVITY DUE TO DEFECT PAIRS (1) By ^{C. P.} FLYNN,

Department ^of Physics, University ^of Illinois, Urbana, ^Illinois.

Résumé. 2014 Des calculs ont été effectués dans le but de déterminer les contributions des phéno-

mènes de diffusions multiples cohérentes à la résistivité résiduelle des paires ^de défauts dans un

réseau métallique. ^{La théorie a été} appliquée ^aux paires de lacunes et d’impuretés d’étain ^dans

l’or où l’on trouve _que les échanges isotropes ^sont petits, de l’ordre de 5 % ^de la résistivité totale due aux défauts isolés. On trouve une énorme anisotropie ^{dans le cas des} paires de lacunes en

position ^de premiers et seconds voisins, ^{en accord avec} les résultats trouvés _{par Bross} ^et Seeger

avec un modèle différent. Les variations de résistivité isotrope ^et anisotrope ^{oscillent avec la} période kF 03B1/03C0 ^{avec une} décroissance en 03B120142, lorsque la distance a entre les défauts augmente.

Abstract.

Calculations have been performed to determine the contributions of coherent and

multiple scattering phenomena to the residual resistivity ^{of defect} pairs ^{in a} metal lattice. The

theory ^{has been} applied to vacancy and tin impurity pairs ⁱⁿ gold ^{where it} is found that the iso-

tropic changes ^are small, being ^{of the} order of five percent of the total resistivity ^{due to} the isolated defects. The anisotropy ^{in the case} of vacancy pairs at first and second neighbouring sites is found to be considerable, ⁱⁿ agreement with the results of calculations by ^{Bross and} Seeger using ^{a diffe-}

rent model. Both isotropic ^and anisotropic resistivity increments oscillate with period ^kF 03B1/03C0

and decays ^as 03B120142, ^with increasing separation a of the defects.

LE JOURNAL DE PHYSIQUE ET LE RADIUM TOME 23, ^OCTOBRE 1962,

Introduction.

In the theory ^of resistivity ^due

to point ^{defects in a} metal lattice, ^{it is} commonly

assumed that the various .deviations of the lattice from petfect periodicity ^scatter electrons inde-

pendently. ^This being ^the case, the observed linear dependence ^{of residual} resistance on defect (1) This work has been supported ⁱⁿ part by ^{the U. S.}

Atomic Energy Commission.

concentration is automatically ^{built into the theo-}

retical description ^{of the} expriments. Conversely,

the linearity ^{of the} resistivity-concentration ^rela-

tionship up ^to concentrations _{of many} atomic peur-

cent of defects, ^{ànd the} reasonable agreement

between the rate of change ^of resistivity ^{with con-}

centration observed and that predicted ^{on theore-}

tical grounds, may be taken ^to imply ^that point

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:019620023010065401

655

defects do indeed scatter electrons in a sensibly

independent fashion, ^even when situated ooat neigh- bouring lattice sites.

With the advance of experimental techniques ⁱⁿ

the field of defect physics making it appear possible

that defect pairs may be isolated experimentally,

it is interesting ^{to cônsider} further the interaction

terms which appear ⁱⁿ the theoretical description

of the residual resistivity ^for non-isolated defects,

and to determine ^the magnitude ^and angular ^de- pendance ^{of the excess} contributions for possible comparison ^with experiment. ^{In this} paper ^we will discuss the isotropic ^and angular dependent

eff ects of coherent and of multiple scattering pheno-

mena on the scattering ^{cross section of defect} pairs.

In general, ^the isotropic ^{effects will be found to be}

small as would be anticipated ^{from the} arguments ,given above, ^{but a} considerable anisotropy ^results

from the interaction terms in the scattering ^cross

section.

2. Components ^{of the} scattering ^{cross section.}

If we consider a Bloch wave of wave vector k, traveling through ^a metal lattice; then the presence of point ^defects provokes ^{a set} of scattered waves

where k is the wave vector of the scattered wave

along ^{the scattered} path rI’, and the subscripts p

identify ^the particular scattering ^centre. Asymp- totically, ^{the scattered wave functions take the}

form

with lfp(k, k’ )12 ^the scattering ^cross section of the defect _p. However, the incident wave at a parti-

cular defect consists not only ^of ’Yk(r), but also of the waves scatterèd by ^{the other defects in the}

lattice. We may then write the ^wave incident on a particular defect q ^as

where the waves scattered from’p, ^observed at q,

have been analyzed ^into plane ^{waves of wave}

vector k’ with coefficients ypq(k, k’), ^{and we have} assigned responsibility ^{for self} consistericy ^{to these} coefficients, ^{in that} they ^are interrelated in a compli-

cated fashion owing ^{to the} présent ^of multiple ^scat- tering phenomena. ^{The scattered wave of wave}

vector k’ from def ect q then ^{takes the form}

The power scattered from thé incident Bloch

wave may be computed from the square modulus

of the sum of the amplitudes ^{of the scattered}

waves. In addition ^{to terms} of the type

which correspond ^to independent incoherent scat-

tering by ^the defects, ^one finds the following ^con- tributions ^{to the} scattering :

(a) ^Coherent scattering ^terms. These take the form

with

and give ^the contribution of interférence terms between waves scattered from différent defects to the total scattered power.

1

(b) Multiple scattering ^termes. Elements of’the total cross section of the type

give ^the magnitude of the power scattered from the incident Bloch wave by multiple scattering pheno-

mena.

(c) Multiple incoherent scattering ^{terms. The}

fourth category of contributions to the square modulus of the sum of the- scattered amplitudes

may be written

These terms correspond ^{to the} rescattering ^of

waves already ^{lost from thé} incident Bloch wave

’¥k(r), and, therefore, ^{do not} contribute directly ^to

the §cattering ^{cross section} of the defects presented

to this ^incident wave. Indirectly, ^the gradual ^dis- sipation ^{of the scattered Waves into an} isotropic distribution, ^{which is} represented by ^these terms,

contributes to the resistivity by restricting ^the

deféct separations from which coherent scattering

effects may be acknowledged ^{to distances of the}

order of the mean free path. ^These considerations

are not significant ^{in what} follows, ^{and terms of the}

form given by equation (9) will not’be used.

3. The calculation of residual résistivity.

^Pro-

vided that the concept ^{of local} equilibrium ^{of eloc-}

trons in phase space is adéquate, ^a complète solu-

tion : of the Boltzman equation may be used ; to

compute ^an exact value of the electrical re3istivity

of a system, ^{In the} absence ^{of this} solution, ^the

exact result may be approached by ^the resistivity

variational method, ^{in terms of thé} probability

P(k, k’) ^{of an} electron of wave vector k being ^scat-

tered into k’. The essential arguments ^involved

are given by ^Ziman [1].

It has been found, however, ^{that even in the case}

of extremely anisotropie scattering potentigls, ^the approximation ^{of an} isotropic relaxation time which yields ^a simple displacement of the Fermi

sphere ^{on the} appZication ^{of an electric} field, may be used to give ^a good approximation ^{to the elec-}

trical resistivity. ^{In the} following calculation of the excess resistivity ^caused by ^the association of two point defects, ^we will therefore use only ^the isotropic trial function. This is in any ^event appro-

priate ^{tQ the} expérimental ^situation where the elec- tron mean free path ^{contains many defects or defect} pairs, ⁱⁿ positions ^or orientations which are effecti-

vely rapdom, and therefore the expectation ^value

of the relaxation time has no angular dependence.

By assuming ^{also an} isotropic ^Fermi distribution, isotropic individual scattering ^{centres, and no}

energy ^{transfer to} the lattice in the scattering pro- cess, the transition probability ^then re4qces ^{to a}

function only ^{of the} polar angles describing ^the change ^{in wave v-eotor on} scattering, ^{and of the} separation ^{a of the defects.} Throughout, ^{use will}

be made of the well-known angular scattering ^func-

tion for isotropic defects, namely

where _~ is thé phase shift induced in the lth partial

wave by ^the pf ^esence of the scattering ^centre.

4. The cohérent scattering contribution.

^If

we consider as shown in figure 1, ^{a Bloch wave}

FiG. 1. -

approaching ^{a pair} of defects at an angle ^{a to the}

axis a joining ^them, then it is easily found that at a

scattering angle 0, ^{the excess scattered wave due}

to interference effects between the waves scattered

separately ^{from the two} defects is asymptotically

where

cos ^{= cos oc cos 0} + ^{sin oc sin 0 cos} et> (12) and _qp is the azimuihal angle ^{between r and a.}

According ^to equation (7), ^{we should take}

acçount ^of multiple scattering effects in computing

the resistivity change ^{due to coherent} scattering,

but in practice we will ^{assume both} interaction

effects to pe small and ignore ^{the influence of one}

on the ^other. With equation (11) giving ^{the cohe-}

rent scattering effects, ^we may then note that ^the resistivity ^is derived ^from integrals ^over relatively

slowly varylng functions ^{of the} angle ^{o. At} large. ^r

the value ^of ’Ykl(r) ’Y;(r) then averages ^to unity ^{as a} normalization integral and may be ignored ^{in what}

follows. Finally, by manipulation ^{of the} expo- nential in equation (11), ^we may obtain the right

hand side in the form of an orthogonal expansion

in the angles defining ^{the axis} a, and since those terms dependent ^{on the 3zimuthal} angle ^{vanish in}

the integration ^over scattering angles, the inverse

relaxation time per defect per unit volume may be presented ^{in the form}

where

The isotropic ^{first term of this} expansion ^{is that} given previously by Flynn [2].

5. The multiple scattering contribution.

To

compute ^the effect of multiple scattering, ^{we need}

at each defect thé expansion ^{in Bloch waves} of ^the disturbance due to the other, for we will ignore the higher ^order multiple scattering ^terms occuring ⁱⁿ equation (8) ^{on the} grounds ^{that the} intensity ^{of a}

wave decreases rapidly ^{with the number of scat-} tering ^{events it} undergoes. ^{Since even the} direct

eff ect considered gives only ^{a small} change ^{in total}

cross section, ^{this is} evidently ^a reasonable approxi- mation.

The calculation ^is facilitated by noting ^{that even}

at the nearest neighboring ^{site to a} primary ^scat- tering defect, ^the prinlary ^scattered waves near the centre of a second defect cell where the scattering potential is large, ^{bear a} strong resemblance to

plane ^{Bloch waves} travelling along ^the axis joining

the defects, ^{and with} phase appropriate ^{to thç}

phase ^{shifts of the} primary scatterer. ^{To the}

extent that the Hankel functions describing ^the primary ^{scattered waves} may be represented by

eikr /ikr, ^the totil scattered ^wave from the secon-

dary scatterer at an angle ^{0 to an incident} Bloch

wave vector k, ^is

657

where fi ^is given by equation (12) (1). ^The change

in incoherent scattering ^{from the} second defect is therefore

leading ^{to a} relaxation time due to multiple ^scatte- ring ^{from two} similar defects of

where

The inclusion of scattering ^{in the reverse direc-}

tion (i,e., reversing the roles of the scattering centres) ^{eliminate those} parts ^of equation (16)

which are odd in oc, and doubles the even terms.

The expansion ^{may if} desired be made orthogonal

in using Clebsh Gordon coefficients, ^{but the resul-} ting expression remains cumbersome and will not be given ^here.

.6. Applications.

Equations (14) ^and (17) ^have

been used with the phase ^shifts given by Flynn [3]

for relaxed vacancies in gold ^to compute ^the Po (cos oc) ^and F 3 (cos oc) ^{terms in the} resistivity ^of

divacancies in gold. Analogous computations

have been made for vacancy pairs ^{at second} neigh- bouring sites, ^{and the} isotropic resistivity ^incre-

TABLE 1

Coefficients of Po ^various Legendre polynomials ^{in the excess} resistivity of vacancy pairs ^{and of tin} impurity pairs ⁱⁿ gold, expressed ^as percentages of total resistivity ^due to the isolated defects. Values are given for defects at first and

second nearest neighbouring lattice sites using ^{first two} phase ^shifts only.

ment caused by ^tin impurities associating ^{at first}

and second neighbouring ^{sites in} gold ^{have also}

been determined using ^the phase shifts due to Blatt [4]. As commented previously, ^the validity.

of the vacancy phase ^{shifts is} contingent upon the relaxation of a divacancy being roughly ^{twice that}

of a monovacancy.

The calculated resistivity changes ^are presented

in Table I as percentages of the total resistivity ^due

to the pair ^of isolated defects. It will be seen

that the isotropic ^{effects are small, and in some}

(1) ^The approximation ^{made here is} reasonable owing ^to

the convenient characteristic of the phase ^shift analysis used, in that the first two partial ^{waves contain} virtually

all the scattered power. Consider the addition formula

where a

(r8 ^{+ r2}

2rro ^cos 03B8) 1/2 ; ro > r. For the va-

lues of l and n of 1 and 2 needed to describe the first two

partial ^{waves, the} expression eik1 J kr ^{is exact for} ho(kr) ^and

within 5 % ^of amplitude ^{and 150 and} phase ^for h1{kr) ^at

the first neighbouring ^{site in a} monovalent metal in the presence ^{of a} phase ^{shift of one radian.}

cases, sign changes between first and second neigh- bouring ^sites display ^the oscillatory ^{nature of the} resistivity contributions, ^{which in all cases are}

superimposed ^{on an a-2} asymptotic decay ^with increasing ^defect separation.

The angular dependant ^terms computed ^for

the _vacancy pairs ^{shows that the} anisotropy ^is

considerable. This is in agreement ^{with the}

results of Seeger ^{and Bross} [5], ^obtained by ^a quite ^différent method of calculations. ^While

the present ^model is, perhaps, ^more realistic, ^it

must be realized that the higher ^{even terms in the} expansion will also contribute to the anisotropy.

This will be particularly ^{evident on} remarking ^that

the multiple scattering contribution includes the effects of the " shadow " of the defect pair, ^and exceptional changes ⁱⁿ resistivity ^{would therefore}

be anticipated ^at values of oc near 0 and ^vc. These effects are readily ^{detected in} equation ^{(17), where}

it will be observed that at ^oc

0, ^the exponent vanishes and the various components ^{are found to}

add to produce ^{an a-1 rather than a--2} asymptotic decay ^of resistivity. It may also be noted that the inclusion of scattering ^{from lattice waves or from a}

uniform distribution of point ^defects through ^the

658

lattice would result in an exponential dissipation ^Discussion

of the partial waves with increasing ^{distance, and} B. CAROL!. - Dans un travail effectué ^{en 1961 (1)}

consequently an exponential ^{term in the} decay ^of _{on a étudié les effets des} interactions entre impu-

the resistivity contributions. ^{Finally, it may be} commented that while ^{thé the} retés dans un gaz d’électrons thode de perturbation poussée ^{au libres} second ordre. Le ^{par une mé-}

coherent and multiple scattering contributions to potentiel d’impureté est schématisé par un poten- the electron density distribution in the

lattice are tiel de Yukawa. On établit les formules donnant les

oseillatory in nature, and may not contribute variations de densité électronique ^{et de} dépla- greatly to the predicted Knight shift in dilute cement de Knight dues aux interactions. ^{On obtient}

alloys, the oscillatory character will nevertheless

des formules analogues à celles de M. Flynn, et des

result in a contribution to the mean square value of

résultats comparables. La résistivité résiduelle due

the density at nudear sites of électrons at the ^aux interactions entre impuretés est évaluée et Fermi surface, ^{and thus to the width} of nuclear

dans le cas particulier des bilacunes les résultats resonance lines. These effects are therefore of inte- suivants sontobtenus : des bilacunes les resultats rest as a possible explanation ^{of the anomalous} _{- Pour courant} d’électrons parallèle à ^l’axe

width obtained experimentally [6] ^{for the} ma- de la bilacune, ^{courant d} électrons parallèle a 1 ^sur

gnetic ^résonance lines of solvent nuclei in dilute

l’autre est telle que la résistivité totale est à peu

alloys. près la moitié de ce qu’elle ^{serait sans} interférence.

- Par contre, lorsque le courant électronique Acknowledgement. - ^{It is a} pleasure ^{to thank est} perpendiculaire ^à l’axe de la bilacune, ^{les deux}

Professor D. Lazarus for his encouragement ^and lacunes diffusent _{à peu} près indépendamment.

Mr. R. Odel for his assistance in the computations ₍₁₎ B. CAROLI, ^{Thèse de 3e} cycle, Faculté des Sciences

1nvolved in this work. d’Orsay, ^1961.

REFERENCES

[1] ^ZIMAN (J. M.), Electrons and Phonons, Oxford, 1960.

[2] ^FLYNN (C. P.), Phys. Rev., 1962,126.

[3] ^FLYNN (C. P.), Phys. Rev., 1962, 125, 881.

[4] ^BLATT (F. J.), Phys. Rev., 1959, 108.

[5] ^BROSS (H.) and SEEGER (A.), ^J. Phys. ^Chem. Solids, 1959, 11, ^115.

[6] ^ROWLAND (T. J.), Phys. Rev., 1962, 126.

NUCLEAR MAGNETIC RESONANCE IN ALLOYS

By ^N. BLOEMBERGEN,

Harvard University.

Résumé. 2014 On étudie d’abord l’effet d’une impureté ^non magnétique ^sur la résonance nucléaire dans une matrice métallique. ^Au voisinage ^de l’impureté, la densité d’électrons est modifiée et cette perturbation ^{s’étend à d’assez} longues distance. Il en résulte : 1) ^{Un effet sur} la resonance

quadrupolaire des noyaux de spin ^I > 1/2 qui ^{est en} général beaucoup plus fort que l’effet des distorsions du réseau. 2) Une modification du déplacement ^de Knight moyen, accompagnée (en

phase solide) ^d’un élargissement de la raie. On rappelle ensuite les propriétés ^anormales (signe ^et

variation thermique) ^du déplacement ^de Knight ^dans certains matériaux purs ^avec des bandes d

ou f incomplètes, et enfin les résultats récents relatifs aux alliages ferromagnétiques.

Abstract.

The effect of a non magnetic impurity ^on the nuclear resonance of the host metal is considered first. The impurity ^is responsible ^{for a} change in the electron density, ^{which has a} long range in space. ^This gives : 1) ^a strong ^{effect on the} quadrupole ^{resonance of} nuclei with

spin ^I > 1/2 (the effect of lattice distortions being usually ^much scaller) ; 2) ^a change ^{in the} Knight shift, ^and (in ^{the solid} state) ^a corresponding ^line broadening. ^{In a second} part the anomalous behaviour (sign ^and

tem erature variation) ^{of the} Knight shift in pure metallic systems ^with incomplete d ^or f ^shells (e.g. V3Ga, Pt, etc...), ^{is recalled and} finally tho recent results on ferro-

magnetic alloys ^{are reviewed.}

LE JOURNAL DE PHYSIQUE ET LE RADIUM TOME 23, ^OCTOBRE 1962,

When the present author reviewed this same

topic eight ^{years ago} [1], only experimental ^results

of Rowland’s early ^{work were available} [2]. ^It

was already ^{clear at that time} that nuclear spins

are very suiialle Lü probe ^thé local distribution of internal magnetic fields and electric field gradients

in the vicinity ^{of a solute atom in} alloy systems.

At that time thé existence of électron coupled