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Submitted on 1 Jan 1978
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TEMPERATURE DEPENDENCE OF THE
ELECTRICAL RESISTIVITY OF METALS WITH
DISLOCATIONS
V. Vinokur, V.Ya. Kravchenko
To cite this version:
JOURNAL DE PHYSIQUE
Colloque
C6,
suppldmentau
no8 ,
Tome39, aotit
1978,page
C6-1056TEMPERATURE
DEPENDENCE
OFME
ELECTRICAL RESISTIVITY
OFMETALS
W I T HDISLOCATIONS
V.M. Vinokur and V.Ya.Kravchenko
Solid State Physics Institute, ChemogoZovka, Moscot~ province, USSR.
Rdsum6.- On effectue un calcul de diffusion des dlectrons par les dislocations en prenant en consi- dbration l'influence de celles-ci sur le spectre de phonons d'un mdtal (perturbation lindaire de la matrice de forces) et en supposant que l'amplitude de diffusion des atomes situds sur les lignes de dislocation est diffbrente de celle des atomes non perturbbs. On obtient et analyse la ddpendance en tempdrature de la contribution apportde B la rdsistivitd par les dislocations.
Abstract.- The electron scattering on dislocations is calculated taking into account the influence of dislocations on the phonon spectrum of a metal (line a perturbation of force constant matrix) and on the assumption that the scattering amplitude of atoms located at the dislocation lines dif- fers from that of unperturbed atoms.
The temperature dependence of the dislocation extraresistivity derived is analised.
Some metals (cu,Ag~Au~Mo,Zn,Al) show a sharp tions which are introduced as usually/S/and are increase in the dislocation extra resistivity in determined from the Dyson equation, which is deri- quite a narrow temperature range (20-100 K)/1,4/. ved from the equation for atomic displacements. The effect observed is rather considerable. The Electroresistivity is determined by the follo-
d d
maximum values of the parameter r=p (T) /p (0)
-I
wing expression are sometimes as high as 2 or 3. We treat the pro-blem of electroresistivity of dislocated metal by taking into account the mechanism for electron scat- tering on phonon modes resulting from dislocations /3,4/.
We proceed from the assumption that the dislo- cation core makes a principal contribution to the electron scattering. Let the atoms of dislocation lines have the scattering amplitude a1,which is dif- ferent from the amplitude of other atoms ao. The dislocation is assumed to be a local line perturba- tion of the force constant matrix of the crystal 151.
The electron scattering probability Wkk, is calculated taking the total pseudopotential of the dislocated metal as ~erturbations :
- (2n)2sys =
;
I
a a ++
Wkk'-
q
I Snn (K-K: Ek-Ek 9 )CO n,n'
+
Snn, (,,U)=
1
<exp EiqRn(ta exp Lizn,(of1
>eIUtdt (1) -Q)+
Here m,K,ek are the mass, wave vector, and energy of the electron, respectively, V. is the volume of
+
the unit cell; Zn(t)= RO +Sn(t), Un(t) is the displa- n
cement operator in ~eisenberg's representation, and < > denotes the thermal average. Furthermore on- ly the second-order in U term is retained in expan- sion of correlation function Snn,.Fourier-components of correlators <Un(+)Un,> are conventionally expres- sed throught an imaginary partof lattice Green func-
1
P= d c dg'
($J~-@~,Y
$;(I-5;'
Wkk, (2) where j is the current density and the nonequili- brium addition to the Fermi distribution func-- + + -P
tion f a is taken in the form k E (E-electric field). k
d
Dislocation electroresistivity p (T) is deter- mined by three processes: direct scattering on dis- locations (which is proportional to (~a)~), scatte-
2
ring on the disturb phonon spectrum eao), and the interference of the ~revious effects +a Aa, Aa
0
=al -ao). For the residual dislocation resistivity we have :
d 24n4 P ~ ( A ~ ) ~
P 6 ) = CL 7 'eL
Where c = ~ bD ~is the concentration of disloca- ; tions; b is the lattice constant, and Cl involves
+
all nonparallel to E dislocations with their orien- tational factors.
d
The temperature-dependent part of p (T) may be easily analysed at high-temperature (T,?@;9 is the Debye temperature) and at the low-temperature
(T<< 0 ) limit. Analysis shows that considerable values of r(T) may be provided only by the distur- bed phonon spectrum and interference contributions.
The numerical estimates were made for Cu para- meters, with a being given by pseudopotential theory. The value of a] was determined from (3) by
d
using experimental value of p (0) / = 2x10-IQ .cm2 D
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786467
As a result we obtain
References
The first expression is resulting from the weak, and the second from the strong perturbation of of force matrix. The estimates (4) show that it possible to interprete the measurable (%I) values of r(T) in terms of the scattering processes in- volved.
At the low-temperature limit the main contri- butions to r(T) are also due to the phonon and in- terference processes. The temperature dependence is given both by the power functions (from first to sixth power in T B )
,
and exponential functions, suchB
as exp (-CLT 3 / 2 ) and exp(-
F).
The contributionsfrom different terms depend on the magnitude of for- ce constants perturbation potential U. The type of transition to the linear high-temperature limit is greatly sensitive to the value of U, namely the stronger U, the steeper transition.
It should be noted that r(T) is affected by dislocations parallel to the applied electric field
d in contrast to residual resistivity p (0).
/I/Basinski,Z.S., Dugdale,Z.S. and Howie,A. Phil.Mag.2 (1966) 289.
/2/Grigorianz,N.A., Pervakov,V.A. and Hotkevich,V.I. Fizika Tverdogo tela
I
(1969) 229./3/Gantmakher,V.F., and Kulesko,G.
