Electric features of dislocations and electric force between dislocations

Abstract

Dislocations and dislocation dynamics are the cores of material plasticity. In this work, the electric features of dislocations were investigated theoretically. An intrinsic electric field around a single dislocation was revealed. In addition to the well-known Peach–Koehler force, it was established that an important intrinsic electric force exists between dislocations, which is uncovered here for the first time and has been neglected since the discovery of dislocations. The electric forces may be large and sometimes could exceed the Peach–Koehler force for metals and some dielectric materials with large dielectric constant. Therefore, the electric force is anticipated to play a vital role in dislocation dynamics and material plasticity. Moreover, an external electric field could exert an electric force on dislocations and a threshold electric field was subsequently discovered above which this force enables dislocations to glide. Interestingly, it was found that some dislocations move in one direction, but others move in reverse in an identical electric field, which is in agreement with experimental observations. Despite dislocation motion under an electric field, to one’s surprise, both edge and screw dislocations do not carry net charges by themselves, which may tackle the long-standing puzzle on the charges of dislocations. These findings may supply people with new fundamental knowledge on dislocations as well as dislocation dynamics, and may assist people in understanding related phenomena.

Keywords

dislocation dislocation dynamics electric features

1. Introduction

Dislocations are a primary crystalline plastic mechanism and dominate plastic behaviors of crystalline materials. These dislocations have been found to display both mechanical and electric features. In addition to mechanical properties, electric features of dislocations are interesting and important, which has been indicated by the related investigations. To explore the electric properties of dislocations, several experimental methods have been developed and utilized, for example, dislocation motion driven by an externally applied electric field [1], electron-beam-induced current (EBIC) of dislocations [2, 3], electrical conductivity of nano-contact semiconductor wafer with dislocations [4], electrical conductivity of dislocation-stored thin films [5], electrical voltage during dislocation-based plastic deformations [6], motion of low-angle grain boundaries constructed by edge dislocations under an external electric field [7], and so on.

Based on these experimental methods, dislocations were found to present intriguing electric properties. The existence of dislocations usually detrimentally reduces the function of semiconductor-based devices [8], indicating that understanding the electric properties of dislocations is the key to the next-generation nano-devices. The majority of dislocations in pure crystal NaCl was experimentally observed to move opposite to the electric field, demonstrating that they are negatively charged [1]; however, a large proportion of dislocations was indeed found to move along the electric field, which was disturbing [1]. The application of a strong electric field enables dislocations to glide in bent LiF crystals [9], which was attributed to positively charged dislocations [9]. Experimental observations indicated that the dislocations are negatively charged in LiF and NaCl crystals [10]. Experimental results for single-crystal NaCl showed that negative charges appears on the concave side, whereas positive charges emerges on the convex sides [6], and only the edge dislocation components correlates with electrical charges [6]. The pure tilt low-angle boundary motion was observed under a threshold electric field, and the driving force for the boundary motion arises from the interaction between the electric field and the array of charged edge dislocations [7]. Experiments showed that for a single-crystal NaCl plastically deformed in an inhomogeneous manner, an electrical current flows through the crystal, suggesting that the dislocations must be positively charged [11]. It was indicated in experiments that edge dislocations carry negative charges in a nominally pure crystal NaCl but bear positive charges in an oxide-doped crystal [12]. Unlike edge dislocations, screw dislocations were not detected to exhibit charge effects [12]. In the vicinity of room temperature, dislocations in single-crystal NaCl were positively charged [13]. By means of dislocation photoconduction spectrum in alkali halide crystals, a strong internal electric field around an edge dislocation was established in ionic crystals such as NaCl, KCl, KBr, and KI crystals [14]. In a single-crystal KCl doped with CaCl₂, experimental observations showed that only edge dislocations can be driven by a large electric field and they are negatively charged [15]. In addition, in Ca²⁺-doped single-crystal KCl, only when an electric field is larger than a critical value can the edge dislocations be moved [16], and the charges carried by dislocations is initially negative but can change from negative to positive [16]. Both experimental and theoretical studies on charged dislocations in alkali halide crystals were reviewed [17], phenomenological models for charged dislocations were discussed [17]. In a review [18], it was shown that movement of dislocations may lead to the electric effects accompanying plastic deformations and the electric signals was ascribed to the motion of dislocations bearing electric charges. Experimental observations on electroplasticity showed that when mechanical plasticity happens for a material, flow stress is usually reduced noticeably by an applied electric field [19]. It was attributed to enhanced dislocation mobility, which arises from an electric field applying a driving force on dislocations [20]. Experimental investigations on charge effects of dislocations in ionic crystals revealed that charges carried by dislocations may depend on temperature and impurities [21]. Upon sign of charges possessed by dislocations, many of the experimental results, even in the same type of crystals, were still heavily debated [21].

Theoretical effort has also been spent to clarify the related phenomena. Electric interaction between a dislocation and a solute atom was considered in copper-based alloys, and calculations suggested that it is between 1/3 and 1/6 of the corresponding elastic interaction [22]. The electric field effect on the movement of edge dislocations in smectic layers was discussed [23]. For a one-dimensional quasicrystal, the electric–elastic field caused by a straight dislocation was addressed [24]. For some piezoelectric semiconductors and ceramic materials, a charged three-dimensional dislocation loop model was developed for novel layered structures [25].

As mentioned previously, despite the many experimental and theoretical attempts having been made to clarify the electric features of dislocations, the underlying physical origin is still puzzling.

In the other respect, the interaction between dislocations is also of paramount importance and it usually plays a pivotal role in dislocation dynamics, as indicated by numerous studies. Edge dislocations were found to accumulate to form sub-boundaries by means of dislocation interaction [26]. Interaction between dislocations was employed to study the dislocation spacing dependence of external shear stress that can drive them to intersect [27]. Interaction between screw dislocations and grain boundary edge dislocations was employed to understand impediment of dislocation gliding across low-angle grain boundaries [28]. The mutual interaction between dislocations in two-dimensional colloidal crystals was investigated by means of video microscopy analysis on particle trajectories [29]. Pair interaction of nearest-neighbor dislocations was found to cause a tail distribution of the dislocation velocity in a form of power law [30]. Discrete dislocation dynamics (DDD) simulation was carried out to analyze transport of dislocations underneath a sliding contact with a tip in some crystals [31]. During indentation and sliding, DDD was performed to study frictional properties of contact [32, 33]. A three-dimensional phase field dislocation dynamics (PFDD) model was utilized to explore the relationship between the Peierls barrier and line characters of dislocations [34]. Two-dimensional DDD simulations were used to elucidate the nano-indentation properties of a single crystal [35]. A geometrically projected DDD was proposed to simulate dislocation motions under high strains and long time scales [36]. Three-dimensional DDD was utilized to delve into the nano-indentation behaviors affected by grain boundaries for an aluminum bicrystal [37]. Based on DDD, dominant yielding mechanisms in single-crystalline copper nano-pillars was investigated [38], and for dislocation nucleation mechanism a transition from dislocation multiplication to surface nucleation was observed [38]. Two-dimensional DDD simulations were performed to minimally capture the phenomenology of nanocrystalline deformation [39]. In terms of three-dimensional DDD, screw dislocations were found to have an important effect on strain bursts at high strain rate [40]. Through DDD simulations, the creep behaviors of Ni-based single-crystal superalloys at high temperatures were investigated [41]. In addition, according to DDD simulations, correlation between diffraction peak broadening and dislocation density in relaxed dislocation networks was studied [42]. For these DDD simulations, the interaction between dislocations was key and can yield an internal stress field that influences dislocation dynamics heavily. For all the related research, only the Peach–Koehler force between dislocations was considered, however, the intrinsic electric force between dislocations was neglected.

Overall, knowledge on the electric features of dislocations, especially intrinsic electric force between dislocations, has been lacking.

In this work, the electric features of dislocations and an intrinsic electric force between dislocations was explored by an analytic theory. The structure of the paper is as follows. In Section 2, the basic theory is introduced. In Section 3.1, an intrinsic electric field around a single dislocation is revealed. Consequently, in Section 3.2, the discovery of an intrinsic electric force between dislocations is described. In Section 3.3, the motions of dislocations under an electric field and the related force are addressed. In Section 4, the conclusions are given.

2. Theory

The most important foundation in this work, i.e., the Yuheng Zhang equation [43], is introduced here. Many physical factors such as strain, temperature, doping, and so on can alter the Fermi surface of materials. Like water flowing from a higher place to a lower place, electrons in regions with high Fermi surface also drift to regions with low Fermi surface, creating an electric field. This electron drift process may be fast. According to the famous Born–Oppenheimer approximation, if a material is deformed and the related atoms move, the electrons are able to respond and approach their equilibrium state rapidly. Thus, the electric field yielded by Fermi surface alterations in different regions also achieves a steady state quickly. The physical relation between the electric field and the correlated Fermi surface alterations may be described by the Yuheng Zhang equation [43],

\nabla E_{F} = q \vec{E}

(1)

where E_F is the Fermi surface energy (FSE), namely, the electron chemical potential, and q is electron charge. Applications in some areas have been discussed previously [43, 44].

3. Results and discussion

3.1 Electric field of dislocations

By means of the Yuheng Zhang equation [43], i.e., equation (1), once the stress and strain field of a dislocation is known, the electric field surrounding this dislocation in a crystal with any crystalline symmetry is given by

\vec{E} = \frac{1}{q} \frac{δ E_{F}}{δ ξ_{ij}} \nabla ξ_{ij}

(2)

where $\vec{E}$ is the electric field around the dislocation, ξ_ij are strains, the indices are i, j=x, y, z and they obey the Einstein summation convention. In the following discussions, for simplicity and clarity, we take the cubic lattice as an example to present the vital role of this electric field.

The stress field generated by a straight edge dislocation [45] is given in Table 1 and the corresponding strain field [45] is given in Table 2. Thus, based on equation (2), the induced electric field surrounding an edge dislocation is given by

{\vec{E}}_{e} = \frac{1}{q} [\frac{δ E_{F}}{δ ξ_{xx}} \nabla ξ_{xx} + \frac{δ E_{F}}{δ ξ_{yy}} \nabla ξ_{yy} + \frac{δ E_{F}}{δ ξ_{zz}} \nabla ξ_{zz} + \frac{δ E_{F}}{δ ξ_{xy}} \nabla ξ_{xy} + \frac{δ E_{F}}{δ ξ_{yx}} \nabla ξ_{yx}]

Table 1.

Normal and shear stress components σ_ij (i, j = x, y, z) of an edge dislocation. Here G is the shear modulus, ν is Poisson’s ratio, b_e is the Burgers vector of the edge dislocation, and σ_ij are stress components.

Normal stresses	Shear stresses
$σ_{xx} = - \frac{G b_{e}}{2 π (1 - v)} \frac{y (3 x^{2} + y^{2})}{{(x^{2} + y^{2})}^{2}}$	$σ_{xy} = σ_{yx} = - \frac{G b_{e}}{2 π (1 - v)} \frac{x (y^{2} - x^{2})}{{(x^{2} + y^{2})}^{2}}$
$σ_{yy} = - \frac{G b_{e}}{2 π (1 - v)} \frac{y (y^{2} - x^{2})}{{(x^{2} + y^{2})}^{2}}$	$σ_{xz} = σ_{zx} = 0$
$σ_{zz} = ν (σ_{xx} + σ_{yy})$	$σ_{yz} = σ_{zy} = 0$

Table 2.

Normal and shear strains ξ_ij (i, j = x, y, z) of an edge dislocation. Here G is the shear modulus, ν is Poisson’s ratio, b_e is the Burgers vector of the edge dislocation, and σ_ij (i, j = x, y, z) are stress components.

Normal strains	Shear strains
$ξ_{xx} = \frac{1}{2 G (1 + ν)} [σ_{xx} - ν (σ_{yy} + σ_{zz})]$	$ξ_{xy} = ξ_{yx} = \frac{1}{2 G} σ_{xy}$
$ξ_{yy} = \frac{1}{2 G (1 + ν)} [σ_{yy} - ν (σ_{xx} + σ_{zz})]$	$ξ_{xz} = ξ_{zx} = 0$
$ξ_{zz} = \frac{1}{2 G (1 + ν)} [σ_{zz} - ν (σ_{xx} + σ_{yy})]$	$ξ_{yz} = ξ_{zy} = 0$

For a cubic lattice such as simple cubic (SC) lattice, body-centered cubic (BCC) lattice, face-centered cubic (FCC) lattice (for example, diamond crystal structure, zinc blende crystal structure) and so on, owing to the crystalline symmetry, the following relations may exist:

\frac{V}{V_{0}} = (1 + ξ_{xx}) (1 + ξ_{yy}) (1 + ξ_{zz}), \frac{δ E_{F}}{δ ξ_{xx}} = \frac{δ E_{F}}{δ ξ_{yy}} = \frac{δ E_{F}}{δ ξ_{zz}} \approx V \frac{δ E_{F}}{δ V},

and

\frac{δ E_{F}}{δ ξ_{xy}} = \frac{δ E_{F}}{δ ξ_{yx}} = \frac{δ E_{F}}{δ ξ_{xz}} = \frac{δ E_{F}}{δ ξ_{zx}} = \frac{δ E_{F}}{δ ξ_{yz}} = \frac{δ E_{F}}{δ ξ_{zy}} = \frac{δ E_{F}}{δ ξ}

where V, V₀, and ξ denote the unit cell volume, unstrained unit cell volume, and shear strain. Hence, the electric field around an edge dislocation is

{\vec{E}}_{e} = \frac{V}{q} \frac{δ E_{F}}{δ V} \nabla (ξ_{xx} + ξ_{yy} + ξ_{zz}) + \frac{2}{q} \frac{δ E_{F}}{δ ξ} \nabla ξ_{xy}

Substitute the strain components in Table 2, and we have

{\vec{E}}_{e} = \frac{V}{q} \frac{δ E_{F}}{δ V} \frac{1 - 2 ν}{2 G (1 + ν)} \nabla (σ_{xx} + σ_{yy} + σ_{zz}) + \frac{2}{q} \frac{δ E_{F}}{δ ξ} \frac{1}{2 G} \nabla σ_{xy}

According to the stress component σ_zz in Table 1, we have

{\vec{E}}_{e} = \frac{V}{q} \frac{δ E_{F}}{δ V} \frac{1 - 2 ν}{2 G} \nabla (σ_{xx} + σ_{yy}) + \frac{2}{q} \frac{δ E_{F}}{δ ξ} \frac{1}{2 G} \nabla σ_{xy}

Substituting expressions for the stress components in Table 1, one may easily obtain the electric field around an edge dislocation

{\vec{E}}_{e} = - \frac{V}{q} \frac{δ E_{F}}{δ V} \frac{1 - 2 ν}{2 G} \frac{G b_{e}}{2 π (1 - ν)} \nabla \frac{2 y}{x^{2} + y^{2}} - \frac{2}{q} \frac{δ E_{F}}{δ ξ} \frac{1}{2 G} \frac{G b_{e}}{2 π (1 - v)} \nabla \frac{x (y^{2} - x^{2})}{{(x^{2} + y^{2})}^{2}}

In terms of simple calculations, the electric field around an individual edge dislocation is

{\vec{E}}_{e} = \frac{m b_{e}}{2 π (1 - ν)} [\frac{2 xy}{{(x^{2} + y^{2})}^{2}} \vec{x} + \frac{y^{2} - x^{2}}{{(x^{2} + y^{2})}^{2}} \vec{y}] - \frac{n b_{e}}{2 π (1 - ν)} [\frac{{(x^{2} + y^{2})}^{2} - 8 x^{2} y^{2}}{{(x^{2} + y^{2})}^{3}} \vec{x} + \frac{2 xy (3 x^{2} - y^{2})}{{(x^{2} + y^{2})}^{3}} \vec{y}]

(3)

where parameters m and n are

m = \frac{V}{q} \frac{δ E_{F}}{δ V} (1 - 2 ν) and n = \frac{1}{q} \frac{δ E_{F}}{δ ξ}

ν is Poisson’s ratio, and $\vec{x}$ and $\vec{y}$ are unit vectors.

In the case of screw dislocation in a cubic lattice, e.g., SC, BCC, and FCC lattices. The strain field components of a screw dislocation gives [45] $ξ_{xz} = ξ_{zx} = - b_{s} y / [4 π (x^{2} + y^{2})]$ , $ξ_{yz} = ξ_{zy} = b_{s} x / [4 π (x^{2} + y^{2})]$ , where b_s is the Burgers vector of this screw dislocation, and other strain components are zero. In addition, according to equations (1), (2), and the strain field, the induced electric field is given by $q {\vec{E}}_{s} = (\nabla ξ_{xz} + \nabla ξ_{zx} + \nabla ξ_{yz} + \nabla ξ_{zy}) δ E_{F} / δ ξ$ . Substituting the related strain components in this equation, one may obtain

{\vec{E}}_{s} = \frac{1}{q} \frac{δ E_{F}}{δ ξ} \frac{b_{s}}{4 π} \nabla \frac{x - y}{x^{2} + y^{2}}

After simple derivations, the electric field around an individual screw dislocation is

{\vec{E}}_{s} = \frac{n b_{s}}{2 π} [\frac{y^{2} - x^{2} + 2 xy}{{(x^{2} + y^{2})}^{2}} \vec{x} + \frac{y^{2} - x^{2} - 2 xy}{{(x^{2} + y^{2})}^{2}} \vec{y}]

(4)

For electric fields of an edge dislocation and a screw dislocation, the parameters m, n play a key role and should be discussed in some situations. As for some crystals at uniform temperature, strain differences may sometimes cause redistribution of constituents, e.g., impurity atoms, vacancies and interstitial atoms in crystals, ions in ionic crystals, and so on. Their transport and redistribution can also generate local alterations of FSE, so that FSE may be described by $E_{F} = E_{F} (ξ_{ij}, ρ_{k 1} (ξ_{ij}), \dots)$ , where ρ_k₁(ξ_ij) is a strain-dependent constituent density in the crystal. Thus, once an electric equilibrium state is reached, based on Equation (1), the electric field may be given by

\vec{E} = \frac{1}{q} (\frac{δ E_{F}}{δ ξ_{ij}} \nabla ξ_{ij} + \frac{δ E_{F}}{δ ρ_{k 1}} \frac{δ ρ_{k 1}}{δ ξ_{ij}} \nabla ξ_{ij} + \dots) = \frac{1}{q} \frac{d E_{F}}{d ξ_{ij}} \nabla ξ_{ij}

showing that equation (2) is changed from a partial differential form to be a complete differential form. The electric field of dislocations for an ionic crystal with cubic lattices such as NaCl crystal may also be described by equations (3) and (4) and the correlated results for electric forces in the following parts may also be valid, but the only alterations is that the important parameters m and n should take the complete differential form:

m = \frac{V}{q} \frac{d E_{F}}{dV} (1 - 2 ν) and n = \frac{1}{q} \frac{d E_{F}}{d ξ}

3.2 Electric force between dislocations

As far as is known, dislocation dynamics is mostly concerned with the realm of material plasticity. For plastic deformations of materials, most researchers only consider the mechanical force between dislocations, but neglect another vital force, namely, the intrinsic electric force between dislocations. In the following, we reveal this electric force.

Here the general electric force between any two dislocations is discussed first. The electric force between dislocations is equivalent to the force felt by a dislocation under the electric field produced by another dislocation. Also of note is that the electric field generated by a dislocation is always perpendicular to its dislocation line. Therefore, for a dislocation under a complex electric field, the related electric interaction energy in a plane perpendicular to the dislocation line is $ε_{0} ε \int d \vec{r} {\vec{E}}_{⊥} (\vec{r}) \cdot {\vec{E}}_{d} (\vec{r} - {\vec{r}}_{0})$ , where ${\vec{E}}_{⊥} (\vec{r})$ is the electric field component that is perpendicular to the dislocation line, ${\vec{E}}_{d} (\vec{r} - {\vec{r}}_{0})$ is the electric field yielded by the dislocation, and ${\vec{r}}_{0}$ is the position vector of dislocation in the plane perpendicular to dislocation line l. The related electric interaction energy for the whole dislocation can be derived by performing integral along the dislocation line

W_{12} = ε_{0} ε \int d \vec{l} \int d \vec{r} {\vec{E}}_{⊥} (\vec{r}) \cdot {\vec{E}}_{d} (\vec{r} - {\vec{r}}_{0})

(5)

Hence, the total electric force exerted on the whole dislocation line is

\vec{F} = - ε_{0} ε \int d \vec{l} \int d \vec{r} {\vec{E}}_{⊥} (\vec{r}) \cdot \nabla_{{\vec{r}}_{0}} {\vec{E}}_{d} (\vec{r} - {\vec{r}}_{0})

(6)

Of note is that equation (6) may apply for edge dislocations, screw dislocations, and mixed ones. Owing to the superposition principle of strain fields and stress fields [35], the electric fields generated by dislocations may also obey the superposition principle according to equation (2). Meanwhile, in analogy with the fact the strain energy is proportional to the square of strains, the electric energy is also proportional to the square of electric fields. Therefore, the electric energy and related electric force between a pair of any dislocations may not be influenced by other dislocations, and the superposition principle is available for the electric force of dislocations. In the following discussion, the electric forces between parallel straight dislocations will be taken as an introductory example.

3.2.1 Electric force between edge dislocations

Based on equation (3), the electric field around an individual edge dislocation could be written as ${\vec{E}}_{e} = b_{e} (λ \vec{x} + k \vec{y}) / 2 π (1 - ν)$ , where the parameters are

\begin{matrix} λ = 2 mxy / {(x^{2} + y^{2})}^{2} - n [{(x^{2} + y^{2})}^{2} - 8 x^{2} y^{2}] / {(x^{2} + y^{2})}^{3} \\ k = m (y^{2} - x^{2}) / {(x^{2} + y^{2})}^{2} - 2 nxy (3 x^{2} - y^{2}) / {(x^{2} + y^{2})}^{3} \end{matrix}

For most cubic materials, the normal strains could change the unit cell volume, thereby altering electron density, which may further induce lift of FSE. However, shear strains cannot cause variation of the electron density so that they may lead to much smaller lift of FSE than normal strains. Hence, the relation between parameters may fulfill m>>n. Based on equation (5), the electric interaction energy between two parallel straight edge dislocations is

W_{12} \approx \frac{ε_{0} ε m^{2} L}{4 π {(1 - ν)}^{2}} \frac{{\vec{b}}_{e 1} \cdot {\vec{b}}_{e 2}}{r^{2}}

(7)

where r is the distance between the two dislocations, L is the dislocation length, and b_e₁ and b_e₂ are Burgers vectors of the two edge dislocations. Thus, the related electric force per unit length is

{\vec{F}}_{ee} = \frac{ε_{0} ε m^{2}}{2 π {(1 - ν)}^{2}} \frac{{\vec{b}}_{e 1} \cdot {\vec{b}}_{e 2}}{r^{3}} \hat{\vec{r}}

(8)

where $\hat{\vec{r}}$ is a unit vector. This force sensitively depends on parameter m and the dielectric constant. According to equation (8), when the two Burgers vectors are parallel, the force is repulsive; and when they are antiparallel to each other, the force is attractive. The electric force may be a short-range force and could be ignored at large distance, because it decreases more quickly than the Peach–Koehler force [46] between edge dislocations. To show importance of the electric force, a comparison between the magnitude of the electric force and Peach–Koehler force was carried out, which is shown in Figure 1. As shown, a smaller distance and a larger product of dielectric constant and square of parameter m would cause a much stronger electric force than the Peach–Koehler force. For metals and materials with ultra-large dielectric constant, the electric force may usually exceed the Peach–Koehler force so that it must be considered seriously in the realm of dislocation dynamics, which has been neglected since the discovery of dislocations [47, 48].

3.2.2 Electric force between screw dislocations

Like the foregoing treatment, the electric force between two parallel screw dislocations may be derived in terms of electric interaction energy. Based on equations (4) and (5), the electric interaction energy may be given by

W_{12} \approx \frac{ε_{0} ε n^{2} L}{2 π} \frac{{\vec{b}}_{s 1} \cdot {\vec{b}}_{s 2}}{r^{2}}

(9)

where b_s₁ and b_s₂ are Burgers vectors of the two screw dislocations, L is the length of screw dislocations, and r is the distance between two dislocations. Thus, the electric force between two parallel screw dislocations of unit length at a distance r is

{\vec{F}}_{ss} = \frac{ε_{0} ε n^{2}}{π} \frac{{\vec{b}}_{s 1} \cdot {\vec{b}}_{s 2}}{r^{3}} \hat{\vec{r}}

(10)

Analogous to the situation of edge dislocations, electric force between screw dislocations decreases rapidly with distance and may be ignored at large distances. However, if the distance is only several nanometers and the dielectric constant is large, the electric force may be evident, which can affect plastic deformations of materials.

3.2.3 Electric force between screw and edge dislocations

In terms of electric fields of edge and screw dislocations, the electric interaction energy between an edge dislocation and a parallel screw dislocation (the dislocation line is parallel to each other) is obtained based on equation (5)

W_{12} \approx \frac{ε_{0} ε mnL}{4 π (1 - ν)} \frac{b_{e} b_{s}}{r^{2}}

(11)

where b_e and b_s are the magnitude of the Burgers vector for the edge dislocation and screw dislocation, respectively. Here b_e is always positive; b_s is positive if it is parallel to its dislocation line, but negative if it is antiparallel to dislocation line. Again L is the dislocation length and r is the distance between the two dislocation lines. Thus, the electric force between the two parallel dislocations of unit length at a distance r is

{\vec{F}}_{es} = \frac{ε_{0} ε mn}{2 π (1 - ν)} \frac{b_{e} b_{s}}{r^{3}} \hat{\vec{r}}

(12)

This equation indicates that the electric force depends on the dielectric constant and is also inversely proportional to the cube of distance. As demonstrated, an intrinsic electric force always exists between an edge dislocation and a parallel screw dislocation. Considering the fact that forces between an edge dislocation and a parallel screw dislocation has never been considered owing to the absence of the Peach–Koehler force [49], this electric force may be important.

3.2.4 Correction for image force

The image force of a dislocation near a planar grain boundary may be regarded as the interaction between a dislocation and its image dislocation. The classical image force of dislocations only considers the mechanical force and neglects electric force [45, 47]. Based on the previous discussions, the image forces should be corrected according to the electric force between dislocations, and the related corrections for an edge dislocation and a screw dislocation are

F_{e} = - \frac{μ b_{e}^{2}}{4 π (1 - ν) d} - \frac{ε_{0} ε m^{2}}{16 π {(1 - ν)}^{2}} \frac{b_{e}^{2}}{d^{3}}

(13)

F_{s} = - \frac{μ b_{s}^{2}}{4 π d} - \frac{ε_{0} ε n^{2}}{8 π} \frac{b_{s}^{2}}{d^{3}}

(14)

where d is the distance from the planar grain boundary, b_e and b_s are Burgers vectors of the edge dislocation and image dislocation, as shown in Figure 2, and F_e and F_s are the corrected image forces for an edge dislocation and a screw dislocation, respectively. In Equations (13) and (14), the first term is the mechanic force [45, 47] and the second term is the electric force. The electric forces are also attractive and are in the same direction as the mechanical forces. As pointed out previously, for metals and materials with high dielectric constant, the corrected electric forces may be of paramount importance.

For electric forces between dislocations, several points need to be emphasized here. First, electric forces between dislocations were discussed in piezoelectric materials [49], but the existence of electric forces between dislocations revealed in this work may be intrinsic, ubiquitous, and applicable for all crystals with any symmetry, not just restricted to piezoelectric materials. Second, for metals, the electric force may be so noticeable that it may dominate dislocation dynamics. Third, conclusions upon plastic deformations, as are built on classical dislocation dynamics where the electric force has been neglected [48], e.g., DDD [31 –42], may need to be re-examined carefully by adding the electric force. A comparison between this theory and associated experimental observations needs to be performed in the future.

3.3. Dislocation motion under an external electric field

3.3.1 Edge dislocation motion under an external electric field

This electric field around an edge dislocation would influence electrical conductivity of crystals and can give rise to the electric energy of a dislocation, as discussed previously [43]. If an external electric field ${\vec{E}}_{2}$ was applied, ${\vec{E}}_{2} = E_{0} (\cos ψ \vec{x} + \sin ψ \vec{y})$ (where E₀ is the magnitude of the external field and ψ is the angle between the Burgers vector and electric field), according to equations (3) and (5) the electric interaction energy W₁₂ between the dashed edge dislocation and the external electric field is

W_{12} = \frac{ε_{0} ε L b_{e} E_{0}}{4 (1 - ν)} [n \cos (ψ - 4 ϕ) + 2 n \sin ψ \sin 2 ϕ - 2 m \sin (ψ - 2 ϕ)]

(15)

where ε₀ is the vacuum dielectric permittivity, ε is the relative dielectric constant, L is the dislocation length, and φ is the polar angle of dislocation. As pointed out previously, the magnitude of parameter m may be much larger than n for many materials. Thus, if the external field is parallel to the Burgers vector, i.e., ψ=0, the electric energy is

W_{12} \approx \frac{ε_{0} ε mL b_{e} E_{0}}{2 (1 - ν)} \sin 2 ϕ

It is easily seen that when m>0, this energy is the smallest at polar angles 3π/4 and 7π/4, meaning that dislocations (1) (in red color) may remain in the most stable state in the external electric field, as shown in Figure 3(a). However, at polar angles π/4 and 5π/4, W₁₂ is the highest, indicating that dislocations (2) (in blue color) may undergo an electric force pointing to dislocations (1) shown in Figure 3(a). The electric force component F in the gliding plane for a unit length dislocation may be

F = \frac{ε_{0} ε m b_{e} E_{0}}{(1 - ν) r_{0}} \cos 2 ϕ \sin ϕ

(16)

where r₀ is the distance of the dislocation away from the center axis of the crystalline grain. Under some conditions, e.g., high temperatures and large external electric field, the electric force may exceed the Peierls–Nabarro stress [45] and enables edge dislocations to glide. Thus, a threshold electric field E_th exists above which the edge dislocations can be driven to overcome motion barriers and glide,

E_{th} = \frac{2 G r_{0}}{ε_{0} ε m} \frac{\exp [- 2 π a / b_{e} (1 - ν)]}{\cos 2 ϕ \sin ϕ}

(17)

where G is the shear modulus and a is a crystalline lattice parameter. This threshold field is highly dependent on the distance r₀, dielectric constant ε, and polar angle of this edge dislocation. The existence of the threshold electric field was evidenced by experimental data in alkali halide crystals [7, 15, 16]. Based on analytical expressions of the flexoelectric coefficient [43], the parameter m can be written as $m = μ_{11} (1 - 2 ν) / ε_{0} (ε - 1)$ , where μ₁₁ is a flexoelectric coefficient. Thus, the related force and threshold electric field can be written as

F = \frac{ε μ_{11} b_{e} E_{0} (1 - 2 ν)}{(ε - 1) (1 - ν) r_{0}} \cos 2 ϕ \sin ϕ and E_{th} = \frac{2 G r_{0} (ε - 1)}{ε μ_{11} (1 - 2 ν)} \frac{\exp [- 2 π a / b_{e} (1 - ν)]}{\cos 2 ϕ \sin ϕ}

respectively.

If the Burgers vector is perpendicular to the electric field, i.e., ψ=π/2, the electric interaction energy is $W_{12} = - \frac{ε_{0} ε mL b_{e} E_{0}}{2 (1 - ν)} \cos 2 ϕ$ . If parameter m is also positive, W₁₂ is the lowest at polar angles 0 and π, demonstrating that dislocations (1) are the most stable, but at polar angles π/2 and 3π/2, W₁₂ is the largest, suggesting that an electric force is being exerted on these dislocations, as shown in Figure 3(b). Likely, the force component in the gliding plane for a unit length dislocation may be

F = \frac{ε_{0} ε m b_{e} E_{0}}{(1 - ν) r_{0}} \sin 2 ϕ \sin ϕ

(18)

In addition, a corresponding threshold electric field E_th exists and it is

E_{th} = \frac{2 G r_{0}}{ε_{0} ε m} \frac{\exp [- 2 π a / b_{e} (1 - ν)]}{\sin 2 ϕ \sin ϕ}

(19)

For many crystalline materials, the threshold fields at different conditions may be in the range 10⁴–10⁶ V/m, and it may decrease with increasing temperature because of material softening and increased relative dielectric constant. These points were verified by electric field-driving motion of small angle–tilt grain boundaries constructed by array of edge dislocations [7]. Furthermore, according to the expressions of electric forces in the above two cases, the forces are positive in some polar angle zones but negative in other angle zones. Therefore, under an electric field, some dislocations are expected to glide in a direction, but some other dislocations are anticipated to move in the reverse direction, which is indeed the experimental observations that the majority of dislocations move against the field, but a large proportion still move in the field direction in sodium chloride [1, 50, 51]. For some metals, owing to their ultra-large static dielectric constant, a small pulsed electric field may enable some dislocations to glide and a stronger electric field can drive more dislocations to glide, which may result in a noticeable reduction of flow stress in the plastic deformations. This is the observed for electroplasticity of metals [19] and may be the underlying physical origin. Despite the theory being in qualitative agreement with related experiments, a more detailed comparison between this theory and experiments is still needed in the future.

Edge dislocations can be driven to move by an electric field, but against previous numerous investigations on the debated charges of edge dislocation [9, 10, 20, 21, 47], here it is found that dislocations themselves do not actually carry net charges, so that mobile edge dislocations do not contribute to electrical conductivity, but they can affect electron mobility, thereby influencing the electrical conductivity of materials.

3.3.2 Screw dislocation motion under an external electric field

Based on equations (4) and (5), owing to rotation symmetry of a screw dislocation, the electric interaction energy between a screw dislocation and an external electric field ${\vec{E}}_{2} = E_{0} (\cos ψ \vec{x} + \sin ψ \vec{y})$ which is perpendicular to the screw dislocation line is given by

W_{12} = - \frac{ε_{0} ε nL b_{s} E_{0}}{\sqrt{2}} \sin (ψ + \frac{π}{4})

(20)

where ψ is angle between the electric field and transverse axis where a screw dislocation is situated. In the case of positive parameter n, this energy is the smallest at angle π/4, meaning that dislocations (1) (in red color) remain in the most stable state in the external electric field, as shown in Figure 4. However, at angle 5π/4, W₁₂ is the largest, indicating dislocations (2) (in blue color) may undergo an electric force. The magnitude of electric force for a unit length dislocation may be

F = \frac{ε_{0} ε n b_{s} E_{0}}{\sqrt{2} r_{0}} \cos (ψ + \frac{π}{4})

(21)

and its direction is tangentially pointing to dislocations (1) shown by the arrows in Figure 4. This force sensitively relies on the angle and distance from grain center axis, and may drive some dislocations into zones of dislocations (1), thereby forming dense zones and sparse zones of screw dislocations. If the sign of parameter n or the Burgers vector possesses a negative value, the electric interaction energy and the electric force may also be reversed.

Interestingly, like edge dislocations, screw dislocations can be driven to move by an external electric field, but they do not carry net charges by themselves.

In the analysis given here, only the strain field of a dislocation is considered, and its image dislocation and grain boundary relaxation effect on the strains and electric field in the grain have been ignored.

4. Conclusions

In summary, we have mainly explored electric features of dislocations and revealed an electric field around a dislocation, which leads to an important intrinsic ubiquitous electric force between dislocations. The electric force was uncovered here first, and it was found to be important and could surpass the classically known Peach–Koehler force for metals and materials possessing ultra-large dielectric constant. Thus, the electric force may be important for dislocation dynamics and related material plasticity. On the other hand, dislocations were found to be subject to an electric force under an external electric field. A threshold electric field was discovered, above which the electric force for dislocations under the external electric field enables dislocations to glide. Owing to the electric field-induced electric force, some dislocations move in one direction, but others move in reverse, which may explain the experimental results that have puzzled people for a long time. The electric properties of dislocations and the intrinsic electric forces between dislocations is anticipated to help people re-understand dislocation dynamics and related plastic deformations of materials.

Footnotes

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Yuanjie Huang

References

Caffyn

Goodfellow

Hamdani

AJ.

The movement of dislocations in an electric field in sodium chloride. Mat Res Bull 1967; 2: 1067–1074.

Kittler

Ulhaq-Bouillet

Higgs

Influence of copper contamination on recombination activity of misfit dislocations in SiGe/Si epilayers: Temperature dependence of activity as a marker characterizing the contamination level. J Appl Phys 1995; 78: 4573.

Kveder

Kittler

Schröter

Recombination activity of contaminated dislocations in silicon: A model describing electron-beam-induced current contrast behavior. Phys Rev B 2001; 63: 115208.

Ishida

Kakushima

Mizoguchi

Fujita

Role of dislocation movement in the electrical conductance of nanocontacts. Scientific Rep 2012; 2: 623.

Chauhan

Bastawros

AF.

Probing thickness-dependent dislocation storage in freestanding Cu films using residual electrical resistivity. Appl Phys Lett 2008; 93: 041901.

Hikata

Elbaum

Chick

Truell

Electrical charge study in sodium chloride during plastic deformation. J Appl Phys 1963; 34: 2154–2158.

Schwensfeir

Elbaum

Electric charge on dislocation arrays in sodium chloride. J Phys Chem Solids 1967; 28: 597–606.

Chen

Dai

Lee

, et al. Smart, effect of dislocations on electrical and optical properties of N-type Al0.34Ga0.66N. Appl Phys Lett 2008; 93: 192108.

Sproull

RL.

Charged dislocations in lithium fluoride. Philos Mag 1960; 5: 815–831.

10.

Vennik

Remaut

Dekeyser

Observations on charged dislocations in ionic crystals. Philos Mag 1961; 6(68): 997–1001.

11.

Fischbach

Nowick

AS.

Deformation-induced charge flow in NaCl crystals. Phys Rev 1955; 99: 1333–1334.

12.

Davidge

RW.

The sign of charged dislocations in NaCl. Philos Mag 1963; 8(92): 1369–1377.

13.

Strumane

De Batist

Amelinckx

Charged dislocations in sodium chloride single crystals. Solid State Commun 1963; 1: 1–4.

14.

Turchánnyi

Mátrai

Janszky

Tarján

Experimental data on the dislocation electric field in alkali halides. Acta Phys Acad Sci Hungar Tomus 1974; 35(1–4): 193–199.

15.

Colombo

Kataoka

JCM

. Movement of edge dislocations in KCl by large electric fields. Philos Mag A 1982; 46(2): 211–215.

16.

Kataoka

Colombo

JCM

. Direct measurements of dislocation charges in Ca²⁺-doped KCl by using large electric fields. Philos Mag A 1984; 49(3): 395–407.

17.

Tyapunina

Belozerova

EP.

Charged dislocations and properties of alkali halide crystals. Usp Fiz Nauk 1988; 156: 683–717.

18.

Urusovskaya

AA.

Electric effects associated with plastic deformation of ionic crystals. Sov Phys Usp 1969; 11(5): 631–643.

19.

Conrad

Electroplasticity in metals and ceramics. Mater Sci Eng A 2000; 287: 276–287.

20.

Kulichenko

Smirnov

Influence of an electric field on the resistance of alkali halide crystals to deformation. Sov Phys Solid State 1981; 23: 595.

21.

Menezes

Nix

WD.

High temperature dislocation mobility in LiF part I. Charged dislocations in ionic crystals. Mater Sci Eng 1974; 16: 57–66.

22.

Cottrell

Hunter

Nabarro

FRN

. Electrical interaction of a dislocation and a solute atom. Lond Edinburgh Dublin Philos Mag J Sci Ser 7 1953; 44(357): 1064–1067.

23.

Holyst

Oswald

Influence of the electric field on edge dislocations in smectics. J Phys II France 1995; 5: 1525–1532.

24.

Yang

L-Z

Gao

Pan

Waksmanski

Electric–elastic field induced by a straight dislocation in one-dimensional quasicrystals. Acta Phys Polonica A 2014; 126(2): 467–470.

25.

Han

Pan

Sangghaleh

Albrecht

Charged dislocations in piezoelectric biomaterials. Int J Solids Struct 2014; 51: 2648–2655.

26.

Dekhtyar

AI.

Interaction between dislocations in the process of creep. Phys Metals Metallography 2006; 101(6): 596–599.

27.

Xia

Zhang

Yan

, et al. Simulation of the interaction between two different 1/2<111>screw dislocations in body-centered-cubic metal niobium. Computat Mater Sci 2020; 174: 109503.

28.

Kondo

Mitsuma

Shibata

Ikuhara

Direct observation of individual dislocation interaction processes with grain boundaries. Sci Adv 2016; 2: e1501926.

29.

Eisenmann

Gasser

Keim

Maret

Pair interaction of dislocations in two-dimensional crystals. Phys Rev Lett 2005; 95: 185502.

30.

Jafarpour

Angheluta

Goldenfeld

Velocity statistics for interacting edge dislocations in one dimension from Dyson’s Coulomb gas model. Phys Rev E 2013; 88: 042123.

31.

Gagel

Weygand

Gumbsch

Discrete dislocation dynamics simulations of dislocation transport during sliding. Acta Materialia 2018; 156: 215–227.

32.

Deshpande

Balint

Needleman

Van der Giessen

Size effects in single asperity frictional contacts. Modell Sim Mater Sci Eng 2006; 15(1): 97–108.

33.

Sun

van der Giessen

Nicola

Dry frictional contact of metal asperities: A dislocation dynamics analysis. Acta Materialia 2016; 109: 162–169.

34.

Peng

Mathew

Beyerlein

Dayal

Hunter

A 3D phase field dislocation dynamics model for body-centered cubic crystals. Computat Mater Sci 2020; 171: 109217.

35.

Song

Yavas

Van der Giessen

Papanikolaou

Discrete dislocation dynamics simulations of nanoindentation with pre-stress: Hardness and statistics of abrupt plastic events. J Mech Phys Solids 2019; 123: 332–347.

36.

Akhondzadeh

Sills

Papanikolaou

Van der Giessen

Cai

. Geometrically projected discrete dislocation dynamics. Modell Sim Mater Sci Eng 2018; 26(6): 065011.

37.

Zhang

, et al. Grain boundary effect on nanoindentation: A multiscale discrete dislocation dynamics model. J Mech Phys Solids 2019; 126: 117–135.

38.

Song

Liu

Zhuang

Liu

Sandfeld

Predicting the flow stress and dominant yielding mechanisms: analytical models based on discrete dislocation plasticity. Sci Rep 2019; 9: 20422.

39.

Song

Dimiduk

Papanikolaou

Universality class of nanocrystal plasticity: Localization and self-organization in discrete dislocation dynamics. Phys Rev Lett 2019; 122: 178001.

40.

Cui

Srivastava

Jiang

Gupta

Ghoniem

The role of slow screw dislocations in controlling fast strain avalanche dynamics in body-centered cubic metals. Int J Plast 2020; 124: 117–132.

41.

Gao

Ali

Hartmaier

Influence of rafted microstructures on creep in Ni-base single crystal superalloys: A 3D discrete dislocation dynamics study. Modell Simul Mater Sci Eng 2020; 28: 025001.

42.

Bamney

Tallman

Capolungo

Spearot

DE.

Virtual diffraction analysis of dislocations and dislocation networks in discrete dislocation dynamics simulations. Comput Mater Sci 2020; 174: 109473.

43.

Huang

Strain induced electric effect in condensed matters. J Mater Sci Applicat 5(3), 44-57(2019).

44.

Huang

Strain-induced electric effect in metals. J Mater Sci Applicat 2019; 5(3): 58–62.

45.

Messerschmidt

Dislocation Dynamics During Plastic Deformation. Berlin: Springer-Verlag, 2010.

46.

Peach

Koehler

JS.

The forces exerted on dislocations and the stress fields produced by them. Phys Rev 1950; 80(3): 436–439.

47.

Hirth

Lothe

Theory of Dislocations. 2nd edn. New York: John Wiley & Sons, 1982.

48.

Lubarda

VA.

Dislocation Burgers vector and the Peach–Koehler force: A review. J Mater Res Technol 2019; 8(1): 1550–1565.

49.

Pak

YE.

Force on a piezoelectric screw dislocation. J Appl Mech 1990; 57: 863–869.

50.

Caffyn

Goodfellow

Hamdani

AJ.

The movement of dislocations in an electric field in sodium chloride. Mat Res Bull 1967; 2: 1067–1074.

51.

Zagaruiko

NV.

Effect of an electrostatic field and a pulsed magnetic field on the movements of dislocations in sodium chloride. Sov Phys Cryst1965; 10: 63.