2 September 2002

Physics Letters A 301 (2002) 490–495

www.elsevier.com/locate/pla

Diffusion of off-center impurities in a double-well potential

Alex Gordona,∗, Simon Dorfmanb, David Fuksc

a Department of Mathematics and Physics, Faculty of Science and Science Education and Center for Computational Mathematics andScientific Computation, University of Haifa at Oranim, 36006 Tivon, Israel

b Department of Physics, Technion, Israel Institute of Technology, 32000 Haifa, Israelc Materials Engineering Department, Ben-Gurion University of the Negev, PO Box 653, 84105 Beer Sheva, Israel

Received 26 June 2002; accepted 24 July 2002

Communicated by J. Flouquet

Abstract

Diffusion by jumps of off-center impurities in a double-well potential is considered in metals. It is shown that successivejumps of the impurity atom caused by the long-range interactions constitute diffusive motion occurring in random manner astransfer of a “defect” resembling a domain wall in ferromagnetic or ferroelectric materials. We consider the conditions of thedominating contribution of the off-center mechanism in the impurity transfer. We show that the crossover from the thermalactivated diffusion to Brownian motion may occur at high temperatures, when it is accompanied by the enhancement of thediffusion. 2002 Elsevier Science B.V. All rights reserved.

PACS: 66.30

The classical diffusive motion of a particle in ex-ternal potential, either random or periodic, has alwaysconstituted explicitly or implicitly an integral part ofmaterial science [1,2]. Migration of vacancies and in-terstitial or substitutional impurities are now well un-derstood as diffusion processes. The main diffusionmechanisms are the migration of atoms on vacancysites and interstitial motion of impurity atoms [1,2]. Innumerous metallic allows the atoms of minority com-ponent diffuse orders of magnitude faster than the hostatoms (fast diffusion) [3]. It is supposed that mecha-nisms vary from system to system. There are a num-

* Corresponding author.E-mail address: algor@math.haifa.ac.il (A. Gordon).

ber of explanations of this phenomenon. For exam-ple in [4], hetero-phase fluctuations induced by low-frequency lattice vibrations around the solute atomsare proposed to create a local environment resemblinginterstitial and vacancy-bound positions of the solutes.These configurations act as media for fast diffusion. In[5] the enhancement of diffusion is analyzed by a spe-cial mechanism driving non-equilibrium phase transi-tion. The absence of a universal mechanism explainingthe enhancement of diffusion demands a further con-sideration of the problem and a search for new mech-anisms. We propose that the presence of off-centerimpurities in the host lattice may form a low energymigration barrier. Even a small off-center displace-ment of any impurity is enough to stimulate diffusionin metal or alloy. To our knowledge the problem has

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A. Gordon et al. / Physics Letters A 301 (2002) 490–495 491

not been examined enough. Therefore, it is interest-ing and deserves a study dealing with the role of theoff-center impurities in the diffusion enhancement inmetals and alloys.

Recently an off-center position for interstitial car-bon in Cu–C has been calculated [6]. The carbon ionwas displaced from the center of the octahedral in-terstitial position to obtain a measure of the variationof interaction energy, bonding and hybridization withcopper, with optimized local geometry of surroundingCu atoms [6–9]. In some cases local expansion or con-traction of the copper host lattice was explored. At therange of carbon concentrations under investigation theenergy balance of the repulsion and attraction forcesof the system caused the off-center carbon positiontherefore. The analyses of Mulliken charges for inter-stitial carbon in Cu–C performed in [6–9] has provedthe charge transfer from Cu to C and lead to the con-clusion that in the relaxed geometry carbon charge isnegative and is about−1e. It is also known that, whencarbon diffuses in Fe, there is a tendency for it to benegatively charged [10]. The energy barriers for thecarbon jump between the two off-center positions aremuch lower than that required for the jumps on vacan-cies or on interstitial lattice positions. Consequently,the jump between two off-center positions is the mostprobable process than any other jump including thetransition of the carbon ion from one interstitial siteto another one. Due to the long-range elastic interac-tions between the nearest interstitials their successivejumps may be movement of a charged defect transfer-ring along the lattice through possible locations. Thelow energy barrier between two off-center positionsfor carbon interstitials and the long-range interactionbetween them lead to the transfer of the charged car-bon defect.

It should be noted that the transport of this defectalong the lattice occurs in random manner as aresult of thermal fluctuations. In the presence ofthe low-energy barrier for the off-center interstitialits thermally activated mechanism of a jump fromone octahedral interstice to another one has lowprobability due to the sufficiently high value of thediffusion barrier for this jump. In the case of thecarbon in copper the ratio of these energy barriersis about 10 [6–9,11]. Hence the contribution of thehigh barrier mechanism seems to be negligible. Werefer to diffusion by using the well-known ideology

of the Arrhenius behavior of the activated diffusion[1,2]. The Arrhenius form of activated temperaturedependence is usually considered as self-evident inthe interpretation of the experimental data. However,at high temperatures, in particular, in the systemsincluding off-centers, deviations from the Arrheniusbehavior may take place (see, for example, [1,2]).

In this communication we consider conditions ofthe dominating contribution of the off-center mech-anism transfer over the contribution of the intersti-tial jump mechanism to diffusion impurities in met-als. We also examine the collective nature of thediffusive motion in metals caused by the long-rangeelastic interactions between off-center impurities andbetween them and the lattice. One of the goals ofthe communication is also to show that the crossoverfrom the thermal activated diffusion to Brownian mo-tion may occur at high temperatures. The crossover isaccompanied by the enhancement of the diffusion ofoff-center interstitials. The very existence of the lowenergy barrier for the off-center atom makes the mech-anism of the off-center diffusion dominant over the in-terstitial diffusion mechanism of the transfer throughthe large energy barrier. We show that the Brownian-type motion of impurity atoms in double-well po-tential with elastic interactions of these atoms mayresult in formation of kink-type excitations of off-center impurities, which move as Brownian particles.It should be noted that the pure Arrhenius behaviordoes not exist at any temperatures due to the pres-ence of the linear temperature in the pre-exponentialfactor of the diffusion coefficient. The collective mo-tion of the off-center impurities gives rise to kink-type solution for atomic excitations. Thus, in thiscommunication we predict a possible effect of thetransfer of the charged defect without the atom trans-port along the lattice. Such a transport mechanism mayoccur, for example, in extremely dilute Cu–C solidsolutions, which are formed at the copper–carbon in-terface, as proved experimentally in [6,7]. The en-hancement of diffusion occurs as result of the crossoverfrom the activated regime to the Brownian one anddue to the large difference between the energy bar-riers for the off-center transfer and the interstitialjump.

In the case, in which the existence of the double-well potential for impurities has been proven (see,for instance, [6,7]), we propose here the following

492 A. Gordon et al. / Physics Letters A 301 (2002) 490–495

effective Hamiltonian:

(1)

H =∑

i

[mi

2

(dui

dt

)2

+ V (ui) + K

2(ui+1 − ui)

2].

ui is the displacement of theith impurity ion, whichmoves in the double-well potentialV (ui) , and inter-acts with nearest neighbor by the harmonic coupling,andK is the positive constant of this interaction. Thedisplacement is due to the collective motion of impuri-ties, which can be described as the movement of effec-tive particles of massm, i.e.,m is the effective mass ofthe impurities whose motion is correlated. The modelis one-dimensional but further we will discuss possiblegeneralizations of our model for a three-dimensionalcase. In the continuum approximation we have

(2)H =∫

dx

l

[m

2

(∂u

∂t

)2

+ V (u) + mc20

2

(∂u

∂x

)2],

where

(3)V (u) = −A

2u2 + B

4u4,

and coefficientsA and B are positive,mc20 = l2K,

andl is the distance between the nearest carbon ions,c0 is the characteristic velocity associated with theharmonic coupling between the impurities resultingfrom the elastic forces. It may be approximatelyestimated as the sound velocity in the direction ofthe excitation motion. In the continual approximationone can assume thatui+1 − ui = l ∂u

∂x. Taking into

account a damping we obtain the following equationof motion:

(4)mc20∂2u

∂x2− m

∂2u

∂t2− mΓ

∂u

∂t− dV

du= 0.

The coefficientΓ designates the damping effect.The dissipative term is introduced to describe theinteraction with the other degrees of freedom inthe crystal, which are taken into account as a heatbath producing a random fluctuating force related tothe damping coefficient by the fluctuation–dissipationtheorem (see below). Thus, to consider the diffusionof impurities we include a random fluctuating forceR(x, t) due to the remaining lattice into (4). Then wearrive at the generalized Langevin equation in spaceand in time

(5)mc20∂2u

∂x2 − m∂2u

∂t2 − mΓ∂u

∂t− dV

du= R(x, t).

We therefore introduce a phenomenologicalLangevinequation of motion for the displacement. This equationhas terms deriving from an inharmonic double wellpotential, a damping and random fluctuating force,through which energy is dissipated, a spatial couplingarising from the long-range interaction. As has beenmentioned above, the dynamics of defect excitationsare those of a Brownian particle in a thermal reservoirwith 〈R(x, t)〉 = 0 and⟨R(x, t)R(x ′, t ′)

⟩(6)= 2Γ kBT mlδ(x − x ′)δ(t − t ′),

whereR describes the coupling of the lattice to a ther-mal reservoir. It has been shown that the diffusion con-stant for (5) coincides with one calculated accordingto the conventional theory of the Brownian motion[12,13]

(7)D =∫ ⟨

v(t)v(0)⟩dt = kBT

MΓ,

wherekB is the Boltzmann constant andM is the ef-fective mass of the impurity kink excitation describedby Eq. (4)

(8)M = (2A)3/2(m)1/2

3Blc0,

where M exhibits the essential dependence on theshape of the potential. The inclusion of energy dissipa-tion from the degree of freedomu(x, t) leads to a “do-main boundary”, whose average velocity vanishes inthe absence of any driving field. However, this averagevelocity takes no account of the random fluctuatingforce R(x, t), since〈R(x, t)〉 = 0. The effect of thisforce is to move the “domain boundary” (resemblingthe domain wall in ferromagnetic and ferroelectric ma-terials) in random fashion, so that while〈v〉 = 0, thevelocity auto-correlation function〈v(t)v(0)〉 does notvanish. The cooperative motion under study is char-acterized by the effective massM. Eq. (7) is actu-ally the result of the fluctuation–dissipation theorem[12,13]. The damping coefficientΓ is introduced ina phenomenological way. We assume that the damp-ing and random force act independently on each car-bon ion. The thermal motion of carbons is in equi-librium with the lattice. In Eq. (5) the displacementresults in energy dissipation into the remaining lattice.One of the ways of the damping estimation is the pre-sentation of the damping related to the displacement

A. Gordon et al. / Physics Letters A 301 (2002) 490–495 493

pattern motion. Thus, the damping is the damping ofthe displacement pattern motion. It may be estimatedas a value of the damping of the vibration mode asso-ciated with the displacement of the impurity into theoff-center position, i.e., the vibration of the ion in thedirection of the saddle point of the double-well po-tential and vice versa. Thus, the damping coefficientmay be determined from the mode line-shape. In spiteof the method of the damping estimation mentionedabove the coefficientΓ may depend on different fac-tors. Thus, the damping constant is an adjustable para-meter, which should be determined to fit the results toexperimental data.

Calculations of the mobility of the charged defectµ

by using the Eq. (4) give the following result [14–17]:

(9)µ = 3c0e∗

Γ A

(B

2m

)1/2

.

e∗ is the impurity effective charge transferred whenthe ion is displaced from one equilibrium position toanother. This charge differs from the charge of theimpurity due to the associated electron cloud move-ment. Using (7)–(9) we obtain the following relationbetween the mobility and the diffusion coefficient

(10)D = kBT µl

3e∗δ,

δ/2 is the distance from the potential maximum at thecenter of the double-well to one of two minima, thefactor 1/3 in (10) appears due to the averaging re-lated to the transition from the one-dimensional ap-proximation (4) to three-dimensional one. The defectundergoes a diffusive motion, for which we calcu-late the corresponding diffusion coefficient. The mo-bility and the diffusion coefficient are linearly related.As has been pointed out, this result follows from thefluctuation–dissipation theorem. This equation is dif-ferent from the Einstein one [12,13] by the dimension-less factorl/δ, by the three-dimensional averaging fac-tor 1/3 and by the presence of the effective charge andof the effective mass. Introducing the energy barrierE between the saddle point and the minimum of thepotential (3), we obtain

(11)D = kBT c0l

2Γ δ(2mE)1/2 .

By using the expression for the energy of the impurityinteractionEc = mc2

0/2 , we have

(12)D = kBT l

2Γ mδ

(Ec

E

)1/2

.

However, the result obtained for the diffusioncoefficient should be a particular case of a moregeneral behavior. In the presence of the double-wellpotential for the off-center impurities the diffusioncoefficient should have the following form:

D = kBT

mΓexp

(− E

kBT

)

(13)= kBT l

2mΓ δ

(Ec

E

)1/2

exp

(− E

kBT

),

whereE = A2/(4B). The Arrhenius behavior for dif-fusion holds atkBT � E, because the exponentialtemperature dependence dominates over the lineartemperature dependence of the pre-exponential fac-tor. At kBT E the barrier no longer plays dominantrole in the diffusion, since we arrive at Eqs. (7), (11)and (12) giving the typical Brownian motion. In thesecond case the linear temperature dependence of thediffusion coefficient should take place. The Arrheniusmechanism for the diffusive motion through the high-energy barrier, separating two interstitial positions, iscertainly much less effective than the off-center lowenergy jumps. Thus, the crossover occurs from the pre-dominantly Arrhenius behavior atkBT � E to non-activated motion atkBT E . Actually, essentialdeviations from the Arrhenius law temperature depen-dence should start atE ∝ kBT , when the tempera-ture is of order of magnitude of the energy barrier. InCu–C E = 0.07 eV and the interstitial energy bar-rier between two octahedral positions is about 0.7 eV[6–9]. Hence in this case the deviations from the Ar-rhenius exponential law should start at temperaturesclose to 1000 K. At high temperatures the motion ofthe off-center excitation is different from the trajec-tory of series of independent activated jumps acrossthe saddle point of the potential relief. The off-centermechanism dominates over the mechanism of the dif-fusion transport caused by jumps from one interstitialsite to another provided the energy barrier between thepotential bottom and the saddle point is much smallerthan the energy barrier separating two interstitial sites.Existence of both mechanisms allows presenting the

494 A. Gordon et al. / Physics Letters A 301 (2002) 490–495

diffusion coefficient in the form

(14)D = D1 + D2,

where D1 is given by Eq. (13) andD2 = D0 ×exp(− E′

kBT) [1,2], where E′ is the energy barrier

separating two interstitial sites, the pre-exponentialfactorD0 is essentially given by a “frequency factor”multiplied by the square of a typical “jumping length”.We consider the case, whenE � E′. Then theinterstitial mechanism is not effective. IfE ∝ E′a competition between the off-center and interstitialmechanisms may take place.

Thus, the diffusion of impurities may occur byjumps from one off-center position to another alongthe lattice, i.e., the migration of ionic defects. Thebasic idea is that the coupling between impuritiesgive rises to their correlated jumps between off-centerimpurity positions leading to diffusion of the inter-stitial defect along the host lattice. Charge transfertakes place at the expense of successive impuritydisplacements between the neighboring host atoms(the passage of the ionic defect). The induction ofneighbor impurity displacements is governed by theirlong-range elastic interactions but the movement oc-curs randomly as in the case of Brownian motion atsufficiently high temperatures. The jump of one inter-stitial impurity involves a considerable number of im-purities into jumping. Thus, the long-range interactionbetween the impurities serves a trigger of the impu-rity diffusion transport. The process is accompaniedby the effective mass transport. At low temperaturesthe Arrhenius thermally activated mechanism is dom-inant for the off-center located impurities. The excita-tion may be considered as the boundary between im-purities placed in, let say, the left position and thosesituated in the right site of the double-well minimum.This boundary migrates along the lattice leading to the“defect” transfer. In the case of the impurity motionbetween the two wells of the double-minimum po-tential relief the boundary moves from one impuritysite to another. The inharmonic double well potentialoriginates from the interaction between each impurityand the rest crystal lattice where it is immersed. Thetransfer of the defect takes place in random manner asa typical Brownian motion induced by thermal fluc-tuations at sufficiently high temperatures. We assumethat the density of such excitations is small enoughto neglect excitation–excitation interactions. Then the

“defect” interaction with an external heat bath causesrandom translations of the interstitials. The proposedmodification of the Einstein’s theory is related to thepresence of two equilibrium locations of an interstitialatom instead of one position in the framework of theusual Einstein consideration. The diffusion under con-sideration resembles the known diffusion of domainwalls in ferromagnetic and ferroelectric materials[18–20]. At low temperatures the defect motion occursby a thermally activated jump of the off-center impu-rity according to the Arrhenius temperature law. Es-sential deviations from the Arrhenius law start, whenthe temperature is of order of magnitude of the energybarrier. At high temperatures the diffusion coefficientexhibits the linear temperature dependence. The Ar-rhenius mechanism for the diffusive motion throughthe high-energy barrier, separating two interstitial po-sitions, is much less effective than the off-center lowenergy jumps. The off-center mechanism dominatesover the mechanism of the diffusion transport causedby jumps from one interstitial site to another providedthe energy barrier between the potential bottom andthe saddle point is much smaller than the energy bar-rier separating two interstitial sites. The usual intersti-tial mechanism is effective, if these energy barriers areclose or if the first barrier is larger than the latter one.

Acknowledgements

We are grateful to the German–Israeli ResearchFoundation, GIF, No. G-703-41.10/2001 for support.One of us (A.G.) is indebted to P. Wyder for hiscooperation in Ref. [17] leading to a part of the resultspresented here.

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