Theory of magnetoresistance due to lattice ...

PHILOSOPHICAL MAGAZINE, 2016 VOL. 96, NO. 17, 1832–1860 http://dx.doi.org/10.1080/14786435.2016.1178861

Theory of magnetoresistance due to lattice dislocations in face-centred cubic metals Q. Bian and M. Niewczas Department of Materials Science and Engineering, McMaster University, Hamilton, Canada

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ABSTRACT

ARTICLE HISTORY

A theoretical model to describe the low temperature magnetoresistivity of high purity copper single and polycrystals containing different density and distribution of dislocations has been developed. In the model, magnetoresistivity tensor is evaluated numerically using the effective medium approximation. The anisotropy of dislocation-induced relaxation time is considered by incorporating two independent energy bands with different relaxation times and the spherical and cylindrical Fermi surfaces representing open, extended and closed electron orbits. The effect of dislocation microstructure is introduced by means of two adjustable parameters corresponding to the length and direction of electron orbits in the momentum space, which permits prediction of magnetoresistance of FCC metals containing different density and distribution of dislocations. The results reveal that dislocation microstructure influences the character of the field-dependent magnetoresistivity. In the orientation of the open orbits, the quadratic variation in magnetoresistivity changes to quasi-linear as the density of dislocations increases. In the closed orbit orientation, dislocations delay the onset of magnetoresistivity saturation. The results indicate that in the open orbit orientations of the crystals, the anisotropic relaxation time due to small-angle dislocation scattering induces the upward deviation from Kohler’s rule. In the closed orbit orientations Kohler’s rule holds, independent of the density of dislocations. The results obtained with the model show good agreement with the experimental measurements of transverse magnetoresistivity in deformed single and polycrystal samples of copper at 2 K.

Received 19 February 2016 Accepted 10 April 2016 KEYWORDS

Low temperature magnetoresistivity; effective medium theory; lattice dislocations; magnetotransport; electron orbits; electron scattering; relaxation time; Kohler’s rule

1. Introduction Dislocations stored in a crystal lattice influence its electrical properties [1,2]. Studies of electrical resistivity in pure metals containing dislocations reveal that the transport of conduction electrons is determined by current-controlled events occurring by high-angle scattering of electrons from the dislocation core, with the radius of the scattering crosssection comparable to the length of the Burgers vector of dislocations, ∼ 0.26 nm in copper [3]. In de Haas-van Alphen(dHvA) measurements of Dingle temperature, the scattering of electrons is determined by phase-coherent events on the Landau levels. The dephasing analysis of reduction of dHvA amplitude shows that the dislocation-induced scattering CONTACT M. Niewczas

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PHILOSOPHICAL MAGAZINE

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depends on the microstructure and occurs by small- or high-angle scattering from the long-range strain field of dislocations [4–6]. Considerable efforts have been directed to understand these processes, but no theory capable of describing magnetotransport of electrons in the material containing often complex dislocation arrangements has been developed so far. In recent paper [7], we have reported experimental studies on the influence of lattice dislocations on transverse magnetoresistivity (TMR) in single and polycrystals of copper at 2 K. The results show that dislocations strongly affect the dependence of TMR on the magnetic field and suppress magnetoresistivity (MR) anisotropy. In the orientations of closed belly orbits, TMR follows Kohler’s rule, independent on the density of dislocations stored in the single crystals. In the orientations of open orbits, TMR of deformed crystals violates Kohler’s rule and shows upward deviation from TMR of reference virgin crystals in crystals containing low density of dislocations and downward deviation in crystals containing high density of dislocations. The transition between upward and downward deviation from the Kohler’s plot occurs smoothly at the early Stage II of work hardening, when the density of stored dislocations approaches ∼1 × 1014 m−2 and dislocations start to form three-dimensional network of densely packed dislocation walls [8–14]. The dislocation network destroys the periodicity of the lattice and limits the length of open orbits in the momentum space. The topology of closed orbits is influenced considerably less by this kind of dislocation arrangement and the closed orbits are preserved during microstructure development, resulting in Kohler’s rule holding for a wide range of dislocation densities stored in the crystals. Our measurements suggest that the magnetotransport of electrons is dominated by small-angle scattering in single crystals with average density of dislocations, below ∼1014 m−2 , and high-angle scattering in crystals with average dislocation density above ∼1014 m−2 level. The experimental observations have been explained qualitatively by a two-band model, involving a belly band formed by electrons moving on the belly orbits and a neck band formed by electrons moving near the neck regions, each band characterised by an individual ⃗ is relaxation time [15]. For noble metals, the dislocation-induced relaxation time τdis (k) anisotropic and has a constant value τb when the electron wave vector k⃗ is localised in the belly region of the Fermi surface [16]. The relaxation time assumes a much smaller value τn (τn ≪ τb ) when k⃗ is localised in the neck regions of the Fermi surface due to small-angle scattering of electrons by dislocations [16]. Using Bergman’s theory [16], we have argued that in single crystals with low dislocation content and in arbitrary orientation of the magnetic field, the electrons move on the open orbits formed by ⟨1 1 1⟩ necks and TMR increases, because τn in the neck region is small due to the small-angle scattering of electrons on dislocations. This causes upward deviation from Kohler’s rule with respect to dislocation-free crystals. However, when the electrons move in the belly orbits, the relaxation time τb due to dislocation and impurity scattering is isotropic, leading to conservation of Kohler’s rule in distorted crystals [7]. The present work is the continuation of our previous studies [7]. The aim is to provide a theoretical foundation for the experimental results reported there and to gain insight into the effect of dislocation microstructure on low temperature transverse magnetoresistance in copper.

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2. Overview of effective medium theory for crystals with internal defects The methodology of the present model is based on the effective medium theory used extensively to evaluate transport properties of macroscopically inhomogeneous media. The central concept of the theory was developed by Maxwell-Garnett [17] and further by Bruggeman [18] and Landauer [19]. Herring [20] formulated effective medium theory in mathematically rigorous form and applied it to describe the effect of random inhomogeneities on galvanomagnetic properties of materials. He used perturbation theory to obtain the effective conductivity tensor, based on the assumption that the spatial variation in electrical properties caused by the inhomogeneities is not large and is continuous throughout the volume of the material. Cohen and Jortner [21] developed a method to treat the Hall effect in disordered two-component systems by assuming that the material behaves like spherical dielectrics embedded in a homogeneous medium in a magnetic field with an effective MR. Their method permits dealing with disordered materials characterised by large spatial fluctuations in electrical properties. Based on the same idea, Stachowiak [22,23] calculated the effective MR tensor of polycrystalline metals with an open Fermi surface. His results showed that in the presence of open orbits, the MR as a function of the applied field tends to be quasi-linear. However, the spherical shape of crystallites assumed in Stachowiak’s calculations does not always approximate the microstructural inhomogeneities in real materials. By employing the Green’s function method to solve the boundary problem, Stroud and Pan [24–26] generalised the theory to a degree that one can use arbitrary shapes of crystallites in modelling galvanomagnetic properties of polycrystalline materials. In Stroud and Pan’s work, the effective MR tensor is represented in a self-consistent form convenient to numerical calculations. An application of their method to free-electron metals containing voids reveals that even a small volume fraction of voids present in the lattice leads to a linear variation of MR with the applied field. These results violate the Lifshitz theory which predicts a saturating MR behaviour in homogeneous uncompensated metals with no open orbits in k space [27,28]. Conclusions of Stroud and Pan have been regarded in the literature as evidence that the anomalous behaviour of MR showing a linear dependence with the applied field in simple metals with closed Fermi surface [29–33] results from lattice inhomogenities [34,35]. Amundsen has suggested that this effect might be caused by lattice dislocations stored in the samples [36]. Apart from these studies, no other results on the effect of dislocations on MR are available in the literature. In the present paper, we develop an effective medium algorithm to predict galvanomagnetic properties of copper with dislocations. The theory relies on the above-mentioned two-band scattering model with different relaxation times for belly and neck electrons, enabling us to reproduce the TMR behaviour of single crystals in arbitrary orientation of the magnetic field and in polycrystals at different stages of microstructure evolution, as reported in Ref. [7] for copper. Nevertheless, the present theory is applicable to all metals with an open Fermi surface. The remainder of this paper is arranged as follows: Section 6 provides the theoretical development of the effective medium approximation used in numerical studies. Sections 4 and 5 develop the model of MR of single crystals containing different density and distribution of lattice dislocations, including low density of dislocations in planar arrangement (Section 4) and high density of dislocations forming cell-like dislocation network (Section 5). Section 6 gives the summary of the results.


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3. Effective conductivity of crystal with dislocations We consider an inhomogeneous medium consisting of randomly distributed crystallites or grains of pure components and assume that the linear dimensions of individual crystallites are large enough compared to the mean free path characteristic of a current carrier and significantly small compared to the dimension of the sample itself. The effective conductivity σ˜ eff for such inhomogeneous medium is defined as:


⟨⃗J (⃗r )⟩ = σ˜ eff · ⟨E⃗ (⃗r )⟩,

(1)

where angular brackets denote spatial averages and the current density ⃗J (⃗r ) and electrical field E⃗ (⃗r ) at any position ⃗r are related by the local conductivity σ˜ (⃗r ) of the pure component at that position i.e. ⃗J (⃗r ) = σ˜ (⃗r ) · E⃗ (⃗r ). (2)

Let the medium be composed of n components, the conductivity σ˜ i for the ith component be constant and the crystallites in each component be of the same shape. In the effective medium approximation, each crystallite may be regarded as being embedded in a uniform medium of the conductivity σ˜ m instead of its real environment. According to Stroud and Pan [24–26], if the crystallites have an ellipsoidal shape, the effective conductivity can be expressed as: σ˜ eff = σ˜ m +

!

n " p=1

˜p

δ σ˜ p (1˜ − $ · δ σ˜ p )

−1

# ! ·

n " p=1

˜p

(1˜ − $ · δ σ˜ p )

−1

#−1

,

(3)

where: 1˜ is the unit tensor, δ σ˜ p = σ˜ p − σ˜ m , and $˜ p is the depolarisation tensor determined by: $ ⃗ ′ G(⃗r ′ )nˆ ′ dS′ . (4) ∇ $˜ p = S′

In Equation (4), integration is performed on the surface S′ of the crystallite of pth component centred at the origin ⃗r = 0; nˆ ′ is a unit normal vector pointing outward from the surface, and G is a Green’s function for the medium. G is the solution of the boundary problem [24]: ⃗ r − ⃗r ′ ) = −δ(⃗r − ⃗r ′ ), ⃗ · σ˜ m · ∇G(⃗ ∇ lim

|⃗r −⃗r ′ |→∞

′

G(⃗r − ⃗r ) −→ 0.

(5) (6)

Equation (3) provides convenient way to calculate the effective conductivity of inhomogeneous medium by iteration. ⃗ , the conductivity tensor of crystals is a In the presence of an external magnetic field B ⃗ . Let the applied magnetic field coincide with z axis of a Cartesian frame with function of B % & the basis set xˆ , yˆ , zˆ and the current density be on the xoy plane. The conductivity tensor σ˜ m for the uniform host medium can be written as: ⎡ xx xy ⎤ σm σm 0 xy (7) σ˜ m = ⎣ −σm σmxx 0 ⎦ . 0 0 σmzz

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⃗ ) = σji ( − B ⃗ ), (i, j = xˆ , yˆ , zˆ ). Substituting The tensor fulfils the Onsager’s relation σij (B Equation (7) into Equation (5) and solving the boundary problem generates the Green’s function of the form [24]: G(⃗r − ⃗r ′ ) =

1 +

4πσmxx ,

σmzz


(x − x ′ )2 (x − x ′ )2 (z − z ′ )2 × + + σmxx σmxx σmzz

-− 12

. (8)

Now we turn to the case of our interest, i.e.: single crystals containing different density and distribution of lattice dislocations. Studies of the dislocation microstructure in ⟨541⟩ copper single crystals deformed in tension indicate that during plastic flow single crystals develop an initially planar arrangement of parallel edge dislocations with the same Burgers vector, lying on primary {1 1 1} planes [9,11,13,37,38]. As more dislocations are accumulated in the lattice, dislocations arrange themselves in cell-like network consisting of dislocation-free crystallites separated by dislocation-dense walls [9,11,13,14,37,38]. Considering these features of the dislocation microstructure in copper, we use different models to calculate MR of crystals with dislocations. For a slightly deformed sample with a planar arrangement of edge dislocations, we represent segments of dislocation as a linear defect of cylindrical shape and assume the crystal be composed of a mixture of cylindrical crystallites representing undistorted lattice and cylindrical dislocation segments in the same orientation. For single crystals deformed to large strains, containing a well-developed cell-like dislocation network, we assume the dislocation cells are represented by spherical crystallites with certain distributions of spacial orientations. The details of the models are given in Sections 4.2 and 5, respectively. To obtain the conductivity of a crystal with dislocations by means of applying the selfconsistent Equation (4), one has to compute the depolarisation tensor for crystallites. For a cylindrical crystallite, the centre of the cylinder is placed in the origin of xyz frame. For the arbitrary orientation of the cylinder, in which the axis of the cylinder is oriented along a direction denoted by inclination angle θo and azimuth angle φo , the components of the depolarisation tensor . can/ be conveniently calculated in the spherical coordinate system ˆ φˆ through the combination of Equations (4) and (8). The results with the basis set rˆ , θ, are: $rˆ′ rˆ $θ′ˆ θˆ

$φ′ˆ φˆ

0 1 1 1 D 2 + , =− zz 2 L σm (1 − χ)(1 − χ cos2 θo ) + 1 − χ sin2 θo + =− σ zz (1 − χ)(1 − χ cos2 θo ) 2m 3 0 1 + 1 D 2 1 − χ sin2 θo 1 + − × , 4 L 1 − χ cos2 θo 1 + 1 − χ sin2 θo 1 + =− zz σm (1 − χ)(1 − χ cos2 θo )


×

2

3 0 1 1 1 D 2 + − , 4 L 1 − χ cos2 θo 1 + 1 − χ sin2 θo 1

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(9)


where χ = 1 − σmxx /σmzz and D/L is the ratio of the diameter D and length L of the cylinder. The remaining non-diagonal components of the tensor vanish after the integration of the odd integrands over the surfaces symmetrical about the three axes in the spherical coordinate system. The components of the second rank depolarisation tensor are subjected to the transformation rule in the xyz frame as: (10) $˜ = R$˜ ′ R−1 ,

where $˜ is the tensor in xyz frame and R is a real orthogonal transformation matrix in the form: ⎤ ⎡ sin θo cos φo cos θo cos φo − sin φo (11) R = ⎣ sin θo sin φo cos θo sin φo cos φo ⎦ . cos θo − sin θo 0 For a spherical crystallite, the depolarisation tensor can be written as [26]: 3 2 √ + sin−1 χ 1 −1 , 1−χ √ $zz = m χσzz χ $xx = $yy

2 √ 3 sin−1 χ 1 $zz + + m m =− 2 χσxx σzz

(12)

with other components being zero. It should be emphasised that the self-consistent Equation (3) has been developed initially for ellipsoidal crystallites, but it is also applicable to the cylindrical-shaped crystallites, since a cylinder or a rod can be approximated by a corresponding ellipsoid with the appropriate shape factor [24–26].

4. Magnetoresistivity of crystals with low density of dislocations 4.1. Model development We deal first with single crystals strained to less than 3% in tension, containing low density of dislocations up to ∼ 1014 m−2 [7]. Electron microscopy observations show that the dislocation arrangement at this stage of deformation is very inhomogeneous. The dislocation microstructure consists of long-edge dislocation segments in the primary glide plane {1 1 1}, forming characteristic dislocation bundles separating large volumes of material free of dislocations. The dislocation bundles exhibit small misorientation across the bundle, which increases as the dislocation density increases in deformed single crystals [9,11,13,14,37,38]. To describe this kind of microstructure in the framework of effective medium approximation, we simplify it by assuming that it contains a large volume of dislocation-free crystallites of cylindrical shape in the same crystallographic orientation, and a very small volume of crystallites containing dislocations. From the perspective of effective conductivity, dislocations can be regarded as the lines of voids [39] treated as

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Figure 1. (colour online) The geometry of a modelling framework for a crystal with dislocations. The cylindrical crystallites representing undistorted lattice are parallel to each other in approximately the same crystallographic orientation and they separate rodlike dislocations.

cylindrical cavities with associated strain energy. Hence they can be considered as electronscatterers with zero conductivity. Both volumes are mixed together in a form of a binary composite with one component being undistorted lattice and the other component being dislocations. For simplicity, the two components are assumed to be parallel to each other. Figure 1 shows schematically the geometry of the model used in the calculations. ⃗ = Bˆz , conduction electrons in the crystallites move on certain In the magnetic field B orbits. We follow Overhauser and Huberman [40–42] and assume that the deformed single crystal is represented by two independent energy bands with the spherical and cylindrical Fermi surfaces, respectively. The conduction electrons in these two bands have different relaxation times denoted by τs and τc , respectively. For simplicity, τs and τc are assumed to be field-independent. Closed orbits occur when electrons move on the spherical Fermi surface characteristic of free electrons. The conductivity tensor for this spherical band is given by [43]: ⎤ ⎡ 1/(1 + λ2 ) −λ/(1 + λ2 ) 0 σ˜ s = σ0s ⎣ λ/(1 + λ2 ) 1/(1 + λ2 ) 0 ⎦ , (13) 0 0 1

where λ = ωc τs , ωc = eB/m is the cyclotron frequency, e and m are conventional notations for electron charge and mass (the electron cyclotron effect on mass is neglected), σ0s = ns e2 τs /m is the conductivity in the absence of the magnetic field and ns is the density of electrons in the spherical band. In the present model, the open or extended orbits are generated when electrons move on the cylindrical Fermi surface and the length of the orbits depends on the orientation of the cylinder in the magnetic field. Let d⃗c be the directional vector of the cylindrical axis of the Fermi surface determined by the polar angle α between d⃗c and yˆ -axis and the azimuth angle β between the projection of d⃗c on the xoz plane and zˆ -axis. Extended orbits are formed by the intersection of the Fermi surface with the planes normal to the magnetic field, as shown in Figure 2. It is clear that β = π/2 corresponds to open orbits, whereas other β values correspond to extended orbits. The components of the conductivity tensor σ˜ c (α, β) for the cylindrical band in xyz frame are [41]: σxˆcxˆ =

5 σ0c 4 1 − sin2 α sin2 β , 2 1+-



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Figure 2. (colour online) Schematic graph showing the orientation of the cylindrical Fermi surface and the formation of extended orbits.

σxˆcyˆ = σxˆczˆ = σyˆcxˆ = σyˆcyˆ = σyˆczˆ = σzˆcxˆ = σzˆcyˆ = σzˆczˆ =

σ0c 1 + -2 σ0c 1 + -2 σ0c 1 + -2 σ0c 1 + -2 σ0c 1 + -2 σ0c 1 + -2 σ0c 1 + -2 σ0c 1 + -2

4 5 − sin α cos α sin β − - sin α cos β , 5 4 − sin2 α sin β cos β + - cos α ,

4 5 − sin α cos α sin β + - sin α cos β , sin2 α,

4 5 − sin α cos α cos β − - sin α sin β , 5 4 − sin2 α sin β cos β − - cos α ,

4 5 − sin α cos α cos β + - sin α sin β , 4 5 1 − sin2 α cos2 β ,

(14)

where - = ωc τc sin α cos β, σ0c = nc e2 τc /m and nc denotes the density of electrons moving on the cylindrical Fermi surface. For n conduction electrons in the system, with a fraction fe of electrons moving on the cylindrical Fermi surface and the remaining (1 − fe ) electrons moving on the spherical Fermi surface, the total conductivity σ˜ t from both spherical and cylindrical bands is given by the rule of mixture [42]: σ˜ t (α, β) = (1 − fe )σ˜ s + fe σ˜ c with σ0s = ne2 τs /m and σ0c = ne2 τc /m.

(15)

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Let the volume fraction of the pure crystallites be fv . Applying Equation (3) to this model, one can evaluate the effective conductivity tensor for the crystal with dislocations by the following equation: , 7−1 6 6 7−1 σ˜ eff = σ˜ m + (1 − fv )( − σ˜ m ) 1 + $˜ 1 · σ˜ m + fv δ σ˜ · 1 − $˜ 2 · δ σ˜ , 6 7−1 6 7−1 -−1 1 2 ˜ ˜ + fv 1 − $ · δ σ˜ , (16) · (1 − fv ) 1 + $ · σ˜ m


where,

δ σ˜ = σ˜ t (α, β) − σ˜ m .

(17)

$˜ 1 and $˜ 2 in Equation (16) are depolarisation tensors of dislocations and crystallites, respectively. The exact forms of the two tensors depend on the configurations and orientations of crystallites in the magnetic field. For simplicity, we assume that the dislocations and the undistorted crystallites are aligned along yˆ -axis and take θo = π/2 and φo = π/2 in Equation (11) to evaluate their Cartesian components. 4.2. Magnetoresistivity for open and extended orbits The self-consistent method is employed to solve Equation (16) numerically, starting from a free-electron conductivity tensor and letting the computed σ˜ eff be equal to a new σ˜ m in each new iterative loop until a self-consistence is achieved, typically after a few iterations. In calculations, the ratio D/L is taken to be 10−2 and 10−4 for undistorted crystallites and dislocations, respectively. The relaxation time τ0 (2 K) corresponding to the zero-field resistivity ρ0 (2 K) or zerofield conductivity σ0 (2 K) = ρ0−1 (2 K) at temperature T = 2 K can be obtained from the experimental measurement of the resistivity ratio RRR of single crystals, defined as the ratio of the resistivity at room temperature T = 298 K over the resistivity at T = 2 K [7]. The dislocation density is roughly proportional to the inverse of RRR, provided that the specific dislocation resistivity is constant. τ0 (2 K) and RRR are related as: τ0 (2 K) =

mσ0 (298 K) × RRR, ne2

(18)

where σ0 (289 K) = 5.96 × 107 S/m is the electrical conductivity of copper at T = 298 K. In real metal samples, τ0 (2 K) represents the average relaxation time of several scattering mechanisms, with impurities and lattice dislocations being two main scattering sources in the crystals used [7]. The relaxation time due to uncharged impurities with the same valence as the host atoms is isotropic [44]. On the other hand, the relaxation time due to lattice dislocations is anisotropic [15,16]. For the undeformed virgin samples, the impurity scattering may dominate the galvanomagnetic properties and therefore we assume that the relaxation time is uniform on the Fermi surface of copper, implying that τs = τc = τ0 (2 K). This assumption is not a necessary condition but it is used for convenience of calculations. Belly and neck electrons experience different scattering on dislocation and the dislocation ⃗ is a function of the electron wave vector k⃗ [15,16]. Two constants relaxation time τdis (k)



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Figure 3. (colour online) Effective MR ratio [ρxx (B) − ρxx (0)]/ρxx (0) against the magnetic field for different fraction of fe electrons moving on the cylindrical Fermi surface. In calculations, the volume fraction fv = 0.99, the residual resistivity ratio RRR = 250, polar angle β = 90◦ and α = 45◦ .

τb and τn represent the relaxation time for belly and neck regions in k-space. What follows is that 1/τ0 (2 K) for crystals with low dislocation content, corresponding to the measured ⃗ over the Fermi surface and it can zero-field resistivity at T = 2 K, is the average of τ −1 (k) be expressed as: 9 8 1 δ 1−δ 1 = = + , (19) ⃗ τ0 (2 K) τb τn τ (k) where brackets ⟨⟩ denote the average in k-space and δ is the fraction of belly area over the whole Fermi surface of one Brillouin zone. Defining an anisotropic factor η as τn = ητb , Equation (19) becomes: 1 0 1−δ . (20) τb = τ0 (2 K) δ + η

In the present two-band model, we assume that τc = τ0 (2 K) and τs = τb since the spherical band consists of only the belly electrons. In calculations, δ and η are used as adjustable parameters to match the experimental measurements. η = 1.0 corresponds to the two bands having the same relaxation time. Other parameters in the calculations are based on the physical properties of copper single crystals. To compare the present calculations against the experimental measurements of TMR reported in Ref. [7], the magnetoconductivity tensor σ˜ eff is inverted to obtain the effective MR tensor ρ˜ eff .

4.2.1. Parametric studies We start with parametric studies of the model for the TMR in the orientation of open and extend-orbits, as a function of dislocation density. The MR of real crystals is represented by MR ratio defined as [ρxx (B)−ρxx (0)]/ρxx (0), where ρxx (B) is the resistivity of crystals in the magnetic field B and ρxx (0) is the zero-field resistivity of deformed crystals, characterised by their resistivity ratio RRR. In the present section, we are concerned with qualitative changes in MR, therefore we neglect the anisotropy of the relaxation time by setting η = 1.0 in the calculations.


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Figure 4. (colour online) Effective MR ratio [ρxx (B) − ρxx (0)]/ρxx (0) against the magnetic field for different RRR values. In calculations, the volume fraction fv = 0.99, the electron fraction in the cylindrical band fe = 0.5, polar angle β = 90◦ and α = 45◦ .

Figure 3 shows the TMR ratio as a function of the magnetic field for different fractions of electrons moving in the cylindrical band. The polar angle α indicating the orientation of the cylindrical Fermi surface on x–y plane is taken to be 45◦ . It can be seen that MR ratio shows a quadratic behaviour with the magnetic field, but it falls off as the fraction fe of electrons moving on the cylindrical Fermi surface decreases. The quadratic dependence may be lost at sufficiently low values of fe , as shown in Figure 3. To understand the effect of the density of dislocations on the TMR for open orbits, we have computed the field-dependent MR ratio for the deformed samples with different RRR, as shown in Figure 4. As the density of dislocation increases and RRR decreases, MR decreases and changes its quadratic dependence on the field to almost linear dependence in the field range considered (Figure 4), a trend observed also in the experiments [7]. Figure 5 shows the MR ratio as a function of the polar angle β for different RRR’s. The peak height at β = 90◦ in Figure 5 corresponds to the TMR in the open orbit orientation of the crystals with different dislocation densities represented by RRR’s. It is seen that the peak height is substantially reduced as the RRR decreases. The results show good qualitative agreement with the experimental results of the same effect [7]. Figure 6 shows TMR ratio as a function of the magnetic field for different polar angles β. As the β angle deviates continuously from 90◦ , the open orbits transition gradually to ellipsoidal extended orbits and in the extreme condition for β = 0 they transform to closed orbits. The length of the extended orbits depends on angle β. In general, TMR of crystals in extended orbit orientations is expected to be in the middle between the open and closed orbit orientations. Such a feature is observed in Figure 6 and it is seen that a small deviation of only a few degree of β from the exact orientation of open orbits at β = 90◦ leads to the disappearance of quadratic variation in MR with the magnetic field, whereas larger deviation of 10 degrees or more causes a saturation of MR at high fields. 4.2.2. Modelling transverse magnetoresistivity of real crystals To examine the applicability of the present theory for predicting TMR of real crystals, we focus first on the field-dependent MR in the open orbit orientation for the virgin



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Figure 5. (colour online) Effective MR ratio [ρxx (B) − ρxx (0)]/ρxx (0) as a function of the polar angle β for different RRR values. In calculations, volume fraction fv = 0.99, the electron fraction in the cylindrical band fe = 0.5, the polar angle α = 45◦ and the magnetic field B = 9 Tesla.

Figure 6. (colour online) Effective MR ratio [ρxx (B) − ρxx (0)]/ρxx (0) against the magnetic field for different polar angles β corresponding to the extended orbits. In calculations, the volume fraction fv = 0.99, the residual resistivity ratio RRR = 250, the electron fraction in the cylindrical band fe = 0.5 and polar angle α = 45◦ .

sample 733B1 and slightly deformed sample 733B2, which shows an upward deviation from Kohler’s plot [7]. In Figure 7, the solid blue points and half-solid olive points are the measured data for the virgin and deformed samples with RRR = 418 and RRR = 251, respectively. To simulate MR of the virgin sample 733B1, we assume that the relaxation time in two bands is the same, i.e. η = 1.0. The other parameters are: RRR = 418, fe = 0.53 and fv = 0.99. The results plotted by the solid red line in Figure 7, show excellent agreement with the experimental measurements of TMR. For the slightly deformed sample 733B2, we assume that the configuration of the open orbits is not affected by the low density of dislocations stored in the sample and that


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Figure 7. (colour online) Comparison of the calculated field-dependent TMR ratio in the open orbit orientation with the experimental data. The solid blue and half-solid olive points are the plots of the experimental data for the undeformed and deformed the samples in 733B group having the resistivity ratio 418 and 251 respectively [7]. The red solid, olive square and red circle curves are the theoretical values obtained with the volume fraction fv = 0.99 and electron fraction in cylindrical band fe = 0.53 but with the different RRR, η and δ values.

Figure 8. (colour online) The Kohler’s plots of the TMR data used in Figure 7.

the dislocations act as scattering sources and influence merely the relaxation time of conduction electrons. Therefore, we apply the same set of parameters used in calculations of magnetoresitivity for the virgin sample, but set RRR = 251 to account for the dislocations accumulated in the sample 733B2. The results are plotted in Figure 7 with the unfilled olive square points. It can be seen that computed MR is much smaller than the measured MR for 733B2 sample. To eliminate this discrepancy, one option is to increase the electron fraction fe (see Figure 3). This, however, has no physical ground because it would imply that dislocations increase the number of electrons moving on the open orbits of cylindrical



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Figure 9. (colour online) Comparison of the calculated field-dependent TMR ratio in open and extended orbit orientations with the experimental measurements of magnetoresistivity for sample fig9. Solid lines are the theoretical values obtained with the volume fraction fv = 0.99, the residual resistivity ratio RRR = 222, open orbit electron fraction fe = 0.57 and polar angle α = 45◦ . Scattered points are the MR results of the sample fig9 with the same RRR, from Ref. [7].

Fermi surface. Hence, the difference must be caused by the same relaxation time used in the two-band model of dislocated metal. If one employs a different relaxation time by letting η = 0.1 and δ = 0.9, while keeping other parameters unchanged, the theoretical results plotted by the red circle points in Figure 7 show an excellent agreement with the experimental measurements. Figure 8 shows the Kohler’s plots of the TMR data from Figure 7. It is seen that when the scattering of conduction electrons on dislocations is uniform, or, in other words, when the dislocation-induced relaxation time is assumed to be isotropic, Kohler’s rule holds, as visible by the overlap of the red-solid and olive-square curves calculated with η = 1.0 in two-band model in Figure 8. However, when the relaxation time due to electron-dislocation scattering is assumed to be different in the spherical and cylindrical bands, the anisotropy of the relaxation time leads to the upward deviation of Kohler’s rule shown by the red circles in Figure 8, consistent with the experimental observations [7]. We have also computed the TRM ratio for deformed single crystals in the extended orbit orientations, using different relaxation times in the two band model. Figure 9 shows the comparison of the calculated TMR with the experimental TMR measurements for the crystal with RRR = 222 [7]. Solid curves show the calculated MR ratio for open and extended orbits represented by different β’s figures; points in Figure 9 show corresponding experimental measurements. It is seen that the computed results exhibit very good agreement with the experiment. 4.3. Magnetoresistivity for closed orbits To model TMR in the closed orbit orientations, we assume that the conduction electrons in the cylindrical band move on the short extended orbits, which are formed in certain field orientations. In the undeformed and slightly deformed samples, the model crystallites are parallel to each order (Figure 1) and therefore these electron orbits have roughly the


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Figure 10. (colour online) Effective MR ratio [ρxx (B) − ρxx (0)]/ρxx (0) against the magnetic field for different β’s corresponding to the extended orbits. The volume fraction fv = 0.99, electron fraction fe = 0.5, residual resistivity ratio RRR = 400 and polar angle α = 45◦ .

Figure 11. (colour online) Effective TMR ratio [ρxx (B) − ρxx (0)]/ρxx (0) against the magnetic field for different fe ’s. In calculations, the volume fraction fv = 0.99, residual resistivity ratio RRR = 400 and polar angles θ = 80◦ and α = 45◦ .

same configuration. In highly deformed single crystals and in polycrystals, the electrons in the cylindrical band move on the short extended orbits due to the dislocation walls and/or grain boundaries restricting the length of the open orbits. In the present model, TMR in the closed orbit orientation can be calculated assuming small β values. Figure 10 shows the field-dependent effective TMR ratio for different β ̸ = 90◦ corresponding to the extended orbits. It is seen that all curves show similar saturating characteristics in the range of applied field considered. One observes that with the decreasing polar angle β, the saturation TMR decreases. A saturating MR phenomenon is a universal feature of MR in metals with a closed Fermi surface, when the electrons are localised predominantly in closed orbits [34,45].



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Figure 12. (colour online) Effective TMR ratio [ρxx (B) − ρxx (0)]/ρxx (0) against the magnetic field for samples containing different dislocation density characterised by RRR values. In calculations, crystallite volume fraction fv = 0.99, electron fraction fe = 0.1 and the polar angles β = 80◦ and α = 45◦ .

Figure 13. (colour online) Comparison of the calculated field-dependent TMR ratio [ρxx (B) − ρxx (0)]/ρxx (0) for the closed orbit with the experimental measurements. The circle points are the experimental TMR measurements of the samples with different RRR values [7]. Symbol curves extended to large fields are theoretical calculations of TMR ratio for different RRR’s values. In calculations, the volume fraction fv = 0.99, the electron fraction fe = 0.12 and the polar angles β = 86◦ and α = 45◦ .

To examine the effect of the electron fraction fe on the closed orbit TMR, we have calculated the field-dependent TMR ratio with different fe ’s in the cylindrical band, as shown in Figure 11. The behaviour of MR shows a similar saturation pattern as in Figure 10, the TMR at the saturation decreases as fe decreases. To determine the variation in TMR on dislocation density, we have computed fielddependent effective TMR ratio for various RRR values, as shown in Figure 12. With more dislocations accumulated in the lattice, indicated by decreasing RRR, the approach to saturation and saturation MR are delayed to the higher fields in the closed orbit


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Figure 14. (colour online) Kohler’s plots of TMR data in the closed orbit orientation shown in Figure 13.

orientations. This indicates that dislocations change the saturation characteristics of TMR for closed orbits and cause TMR to be linearly dependent on the magnetic field at higher fields. To verify the theoretical predictions, Figure 13 shows the comparison of the calculated TMR with the measured TMR [7]. It is seen that both results show good agreement in the range of magnetic field up to 9 Tesla accessible with our system. We have also investigated the effect of the anisotropic relaxation time on the closed orbit MR for sample 733B, using different relaxation times (η = 0.1 and δ = 0.9) in a two-band model. The computed effective TMR ratio for sample 733B is plotted by red half-filled square points in Figure 13 and overlaps with the curve calculated using the same relaxation time in two bands or η = 1.0, shown by green half-filled square points. This implies that the anisotropic relaxation time due to electron-dislocation scattering has a uniform effect on belly electrons, consistent with the predictions of other authors [15,16]. Consequently Kohler’s rule holds for the TMR in the closed orbit orientations, where the galvanomagnetic properties are dominated by belly electrons, as shown in Figure 14.

5. Magnetoresistivity due to cell-like dislocation network 5.1. Model development In highly deformed single crystals, dislocations of various slip systems are accumulated in a three-dimensional network consisting of misoriented cells with low defect content, enclosed by high dislocation density walls [9,13,14,37,38]. The dislocation substructure in highly deformed metal can thus be regarded as a composite comprising dense dislocation walls and dislocation-free cell interiors, as shown diagrammatically in Figure 15. The misorientation of dislocation cells and dislocation arrangement in the walls may have a random distribution in different regions of the microstructure, which depends on the degree of deformation. In the present model, we use the spherical crystallites to represent the dislocation cells characteristic of highly deformed material. Following the treatment by other researchers ⃗ = Bˆz , the dislocation microstructure [23,25,42], we assume that in the external field B



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can be represented by a mixture of two-component spherical crystallites oriented in different directions. Spherical crystallites of the first component are oriented in such a way that only closed orbits exist on the plane perpendicular to the applied field in xoy plane. The magnetoconductivity tensor for this component is described by Equation (13). The crystallites of the second component are oriented in the other direction in which only the open or long extended orbits occur in xoy plane, with the corresponding magnetoconductivity tensor given by Equation (14). As in the case of slightly deformed crystals, the spherical and cylindrical bands are used in the model, with a certain fraction of electrons moving in two types of orbits. To take into account the misorientation of dislocation cells developed at large deformations, the model assumes that open orbit crystallites are oriented randomly in a given region of the microstructure, which results in an equivalent distribution of orientations of open orbits in xoy plane, with equal probability of orientations of open orbit crystallites in each direction. This means that every value of the polar angle α in Equation (14) is equally probable in a given region of the microstructure. The effective TMR of such model is the average of conductivity over all possible electron orbits. Let fv denote the volume fraction of open orbit crystallites and 0 the distributed region of the polar angle α on the plane perpendicular to the magnetic field, as shown in Figure 16. The effective TMR tensor for single crystal containing cell-like dislocation microstructure is given by:

Figure 15. (colour online) (a) Schematic diagram of dislocation-cell microstructure used in the model. (b) Weak-beam TEM observations of the real microstructure in ⟨100⟩ copper single crystals deformed to the stress of 300 MPa, with resistivity ratio RRR = 138.


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Figure 16. (colour online) Schematic diagram for the electron orbits in highly deformed crystals with dislocation-cell structure and the definition of distribution angle 0.

: , 6 7−1 76 1 0 σ˜ eff = σ˜ m + (1 − fv )δ σ˜ 1 1 − $˜ 1 · δ σ˜ 1 + fv δ σ˜ 2 · 1 − $˜ 2 · δ σ˜ 2 0 0 , 7−1 6 7−1 -−1 6 1 2 ˜ ˜ · (1 − fv ) 1 − $ · δ σ˜ 1 + fv 1 − $ · δ σ˜ 2 dα, (21) where: δ σ˜ 1 = σ˜ s − σ˜ m , c

δ σ˜ 2 = σ˜ (α, β) − σ˜ m .

(22) (23)

$˜ 1 and $˜ 2 in Equation (21) represent the depolarisation tensors for the crystallites with closed and open orbits, respectively. For the spherical-shaped crystallites, the components of the depolarisation tensors, $˜ 1 and $˜ 2 , can be evaluated with Equation (12). It should be pointed out that the length of open orbits, characterised by β in Equation (14), depends on the size of the dislocation cells. In practice, β can be used as an adjustable parameter to match calculations with experimental measurements. Similarly 0, denoting the misorientation of dislocation cells, is also used as an adjustable parameter in the present model; its exact value may be determined by inspecting the microstructure using electron microscopy techniques [9,14]. In the case of polycrystals, the crystallites are oriented randomly and 0 = π. The relaxation time of electrons moving in the two bands (two types of orbits) is determined by the scattering on impurities and on dislocation walls. Unlike in the slightly deformed crystals, where dislocations are separated by a large distance and strain fields associated with individual dislocations do not overlap, the scattering potential of a dislocation wall is influenced by overlap of the strain field from entangled dislocations. Consequently, the electron scattering from the dislocation wall is determined by the average over the broad range of scattering angles, losing the features of smallangle scattering. Therefore, it is reasonable to assume that in the crystals with a cell-like dislocation network, the two bands have the same relaxation time determined by zero-field resistivity.



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Figure 17. (colour online) TMR ratio [ρxx (B) − ρxx (0)]/ρxx (0) against the magnetic field for different volume fraction of open orbit crystallites fv . In calculations, open orbit electron fraction fe = 0.5, residual resistivity ratio RRR = 118 and the polar angle β = π/2.

Figure 18. (colour online) TMR ratio [ρxx (B) − ρxx (0)]/ρxx (0) against the magnetic field for different fraction of electrons fe moving on the open orbits. In calculations, open orbit electron fraction fv = 0.3, residual resistivity ratio RRR = 118 and the polar angle β = π/2.

5.2. Parametric studies In what follows, the effective transverse magnetoconductivity tensor is calculated from Equation (21) by iterations and the corresponding MR tensor is obtained as an inverse of the magnetoconductivity tensor. Figure 17 shows the field-dependent TMR for different volume fractions of open orbit crystallites fv at the constant fe = 0.5. Similar results were obtained by other authors [26]. One observes the rather complicated influence of fv on TMR, as shown in Figure 17. TMR tends to saturate at high fields when fv is larger than 0.4, and as fv increases, the saturation occurs at lower field. TMR shows a quasi-parabolic dependence on the field when fv is smaller than 0.3, with the slope of the characteristic decreasing as fv decreases. Such a complex dependence of TMR on fv leads to the peculiar


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Figure 19. (colour online) Transverse MR ratio [ρxx (B) − ρxx (0)]/ρxx (0) against the magnetic field for different fraction of electrons moving on the open orbits. In calculations, open orbit electron fraction fv = 0.5, residual resistivity ratio RRR = 118 and the polar angle β = π/2.

Figure 20. (colour online) Transverse MR ratio [ρxx (B) − ρxx (0)]/ρxx (0) against the magnetic field for different values of RRR’s. In calculations, volume fraction fv = 0.5, open orbit electron fraction fe = 0.5 and the polar angle β = π/2.

response of TMR for different values of fe , as shown in Figures 18 and 19. The results reveal an approximately parabolic behaviour of TMR for fv = 0.3, independent of the variation in fe , which changes solely the slope of the curves at high field, as shown in Figure 18. On the other hand, the saturation of TMR observed at fv = 0.5, changes to the quasi-linear dependence of TMR on the field, as more electrons move in the open orbits (Figure 19). Figure 20 shows field-dependent TMR for the crystals with different RRR’s. In the model, it is assumed that dislocations exert no effect on the configuration and orientation of electron orbits, but determine the nature of the electron scattering through their influence on the relaxation time. Figure 20 reveals that dislocations decrease TMR and, interestingly, shows a similar behaviour as predicted in Figure 12 for closed orbit TMR.



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Figure 21. (colour online) field-dependent transverse MR ratio [ρxx (B) − ρxx (0)]/ρxx (0) for different integral regions in which the cylindrical Fermi surface orients randomly. In calculations, open orbit volume fraction fv = 0.5, open orbit electron fraction fe = 0.5, residual resistivity ratio RRR = 118 and the polar angle β = π/2

Figure 22. (colour online) field-dependent MR ratio [ρxx (B) − ρxx (0)]/ρxx (0) for different polar angle β’s. In calculations, open orbit volume fraction fv = 0.3, open orbit electron fraction fe = 0.3, residual resistivity ratio RRR = 118 and the integral region 0 = π.

The dislocation cells in the microstructure of highly deformed crystals develop certain misorientations between neighbouring cells, the misorientation angle depends on the degree of deformation. As mentioned before, the present model incorporates such an effect using the integral interval 0 as an adjustable parameter. Figure 21 shows fielddependent TMR ratios for different 0 values. It is seen that the integration region has a substantial effect on TMR. The higher density of stored dislocations corresponds to the larger misorientation between individual cells or a larger 0, because the orientation of the crystallites has a broader and more random distribution. In the extreme case of a highly


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Figure 23. (colour online) The effect of anisotropic relaxation time on the field-dependent TMR for polycrystals. TMR ratio [ρxx (B) − ρxx (0)]/ρxx (0) is calculated with different η and δ. In calculations, residual resistivity ratio RRR = 118, the integral region 0 = π, the electron fraction fe = 0.5 and volume fraction fv = 0.5 for filled and half-filled circle points; for filled and half-filled square curves the electron fraction fe = 0.3 and volume fraction fv = 0.3.

deformed single crystal, 0 = π, which represents a polycrystalline material with randomly oriented grains [7]. The configurations of open orbits in the real samples are restricted by the presence of grain boundaries and dislocation walls, which are controlled by different β values in the model. Figure 22 shows field-dependent TMR for the highly deformed samples for different polar angle β’s. As the polar angle β decreases, MR of the crystal decreases considerably. The effect of the anisotropic relaxation time on TMR for polycrystals has been studied by calculating TMR ratio for different η and δ in Figure 23. The results are also applicable to crystals with high dislocation content. The calculations shown in Figure 23 reveal that the different relaxation times in the two-band model induce the upward deviation of TMR from the Kohler’s plot, even though their individual behaviour is quite different due to the complex dependence of TMR on the parameter fe , as discussed earlier with reference to Figures 18 and 19. The upward deviation is much stronger when fe is smaller, similar to the behaviour observed previously in slightly deformed crystals (Figure 8). It should be mentioned that an upward deviation from the Kohler’s plot was observed by Jongenburger [46] in cold-drawn and annealed polycrystalline copper wires. Although the author did not characterise the dislocation microstructure of the samples, which determines the magneto-transport properties [7], the upward shift from the Kohler’s plot suggest that it is caused by recrystallisation of the wire microstructure, which typically produces low defect content single- or polycrystalline samples. In our experiments on highly deformed single crystals [7], we observe the exclusively downward deviation of the Kohler’s plot, attributed to the shortening of the length of electron orbits and the smaller relaxation time due to accumulation of dislocations and development of cell-like microstructure. The results suggest that the long-range strain potential of individual dislocations determining the small-angle scattering of electrons in slightly deformed crystals with low dislocation content, is averaged out in heavily deformed crystals containing high density of dislocations



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Figure 24. (colour online) Comparison of the calculated field-dependent TMR ratio [ρxx (B) − ρxx (0)]/ρxx (0) for different polar angles β’s with the experimental data. Open-point lines are theoretical TMR values for the deformed sample with RRR = 104. In calculations, open orbit volume fraction fv = 0.65, open orbit electron fraction fe = 0.42 and the integral region 0 = 0.45π. The solid points are the experimental TMR results for the sample with the same RRR = 104 value [7].

entangled in walls and cells. This in turn promotes isotropic scattering of electrons dominated by high-angle scattering process. Similar to the behaviour of TMR in single crystals, the violation of Kohler’s rule in polycrystals occurs when the anisotropy of the relaxation time is introduced to the calculation, as shown in Figure 23. Our model provides a theoretical foundation for the experimental verification of this effect. It has to be emphasised however, that employing Kohler’s plot analysis to examine the effect of anisotropic relaxation time on TMR is justified under the assumption that lattice defects do not change the configuration of electron orbits. This requires that parameters α and β, which control the length of electron orbits, are constant. These conditions are satisfied in slightly deformed polycrystals and highly deformed single crystals since the application of a small deformation does not change substantially the grain size of polycrystals or the dislocation cell size in single crystals.

5.3. Modelling TMR of real crystals To examine the applicability of the above model in predicting TMR of highly deformed material, we have calculated field-dependent TMR for copper single crystal with RRR = 104 and a copper polycrystal with RRR = 67 and compared it against experimental TMR measurements [7], as shown in Figures 24 and 25. It is found that the theoretical calculations are in good agreement with the experimental results and indicate that the microstructure exerts a significant effect on MR and has to be considered in the model to describe the galvanomagnetic properties of metals.


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Figure 25. (colour online) Comparison of the calculated field-dependent TMR ratio [ρxx (B) − ρxx (0)]/ρxx (0) with the experimental TMR of polycrystal. Open-point line is the theoretical TMR of the deformed sample with RRR = 67. In calculations, open orbit volume fraction fv = 0.55, open orbit electron fraction fe = 0.45 and the integral region 0 = π. The solid points represent experimental TMR results for polycrystalline copper with RRR = 67 [7].

6. Summary and conclusions A theoretical model has been developed to predict TMR of deformed FCC single and polycrystals within the framework of the effective medium approximation. The model incorporates two electron bands with the spherical and cylindrical Fermi surfaces and all types of electron orbits present in FCC lattice. The model incorporates features of dislocation microstructure in single- and polycrystals developed during plastic deformation to different levels of stress and permits the prediction of the behaviour of TMR for arbitrary density and distribution of lattice dislocation in the material. It is found that the proposed theory agrees well with the experimental results of transverse magnetoresitivity of copper at 2 K. Based on the results obtained following conclusions can be drawn: (1) Density and distribution of dislocations influence in a complex way the TMR of copper. In open orbit orientations, the quadratic variation of TMR with the magnetic field changes to quasi-linear dependence as the density of dislocations increases. (2) Dislocations delay TMR saturation in closed orbit orientation and change the character of TMR-field dependence to a linear dependence in highly deformed single crystals. (3) The anisotropic relaxation time due to electron dislocation scattering (τn ≪ τb ) is responsible for the upward deviation from the Kohler’s plot of TMR against the applied field in the open orbit orientation of the crystals with low density of dislocations. The upward deviation from Kohler’s rule is also predicted to occur in polycrystals, provided that the relaxation time is anisotropic. (4) In the closed orbit orientations, TMR is determined by isotropic dislocation scattering of electrons moving on the belly orbits and the field-dependent TMR obeys Kohler’s rule. Kohler’s rule holds and is independent of the degree of deformation


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provided that the dislocation relaxation time τb is constant. In general, Kohler’s rule will hold in a given field direction, as long as the conduction electrons experience the same relaxation time when moving on the orbits perpendicular to that direction. (5) The scattering of electrons on dislocation walls and grain boundaries determines the magneto-transport properties of single- and polycrystals containing dislocation microstructure. (6) The theory permits to predict magnetoresistance behaviour of FCC metals with arbitrary internal microstructure in the high field region difficult to achieve with standard laboratory equipment.


Abbreviations and Nomenclature For the convenience of the reader, the following lists the character notations and/or abbreviations used throughout the paper. Fundamental physical quantities conventionally labelled by letters are omitted. ⃗ τdis (k) τn τn ⃗j(⃗r ) σ˜ eff ρ˜ eff θ0 , φ0 E⃗ (⃗r ) ⟨··⟩ σ˜ m σ˜ p $˜ p G(⃗r − ⃗r ′ ) σ˜ s σ˜ c ns nc τs τc σ˜ t fv fe RRR δ η α, β

⃗ dislocation-induced relaxation time at k. relaxation time in neck region of the copper Fermi surface. relaxation time in belly region of the copper Fermi surface. electrical current density at displacement ⃗r . effective conductivity tensor. effective resistivity tensor (inverse of effective conductivity tensor). polar angles for the axial direction of cylindrical crystallites. electrical field at displacement ⃗r . weighted average in r- or k-space. conductivity tensor of the medium. conductivity tensor of the pth component. depolarisation tensor of the pth component. Green’s function of the medium. conductivity tensor of the spherical band. conductivity tensor of the cylindrical band. electron density in the spherical band. electron density in the cylindrical band. relaxation time in the spherical band. relaxation time in the cylindrical band. total conductivity tensor from the spherical and cylindrical bands. volume fraction of crystallites with open or extended electron orbits. fraction of electrons moving on open or extended electron orbits. resistivity ratio. fraction of the belly area over the whole Fermi surface in one Brillouin zone. anisotropic factor of the relaxation time. polar angles for the axial direction of cylindrical Fermi surface.

Acknowledgements Financial support of the Natural Sciences and Engineering Research Council of Canada under Discovery Grants Program is gratefully acknowledged.

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Disclosure statement No potential conflict of interest was reported by the authors.

Funding This work was Financially supported by Natural Sciences and Engineering Research Council of Canada under Discovery Grants Program.

ORCID M. Niewczas

http://orcid.org/0000-0002-8529-7194


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