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2.2 Quantum Mechanical Electromigration Description

The force due to EM is modeled in continuum mechanical problems as described above by

$(2.20) \{begin}{gather} \b {F}=Z^{*}e\rho \b {J}, \mathref {(2.20)} \{end}{gather}$

where the $Z^{*}$ is called effective valence or effective charge. The EM induced force on an atomic scale was theoretically studied by Huntington et al. [64]. They used some simplifications, such as the defects are decoupled from the lattice, the electrons are scattered by atoms only, and the creation or annihilation of phonons is neglected. Under these assumptions the $x$ -directional momentum transferred from the scattered electrons to the defects per time is given by

$(2.21) \{begin}{gather} \displaystyle \frac {dM_{x}}{dt}=-\Bigl (\displaystyle \frac {1}{4\pi ^{3}}\Bigr )^{2}\iint \Bigl (\frac {m_{0}}{h}\Bigr ) \Bigl (\displaystyle \frac {\partial E}{\partial \b {k}_{x}}-\frac {\partial E}{\partial \b {k}_{x}}\Bigr )f(\b {k})(1-f(\b {k}’))W_{d}(\b {k},\ \b {k}’)\mathrm {d}\b {k}\mathrm {d}\b {k}’, \mathref {(2.21)} \{end}{gather}$

where $f(\b {k})$ is the distribution function of the electrons and $W_{d}(\b {k},\ \b {k}’)$ is the transition probability per unit time of an electron in state $\b {k}$ to jump into state $\b {k}’$ By separating the two energies’ differentiation into two integrals, interchanging the primed and the unprimed variables of the second integration, and employing the substitution

$(2.22) \{begin}{gather} \displaystyle \frac {f(\b {k})-f_{0}(\b {k})}{\tau _{a}}=\int f(\b {k})(1-f(\b {k}’))W_{d}(\b {k},\ \b {k}’)-f(\b {k}’)(1-f(\b {k}’))W_{d}(\b {k}’,\ \b {k})\frac {\mathrm {d}\b {k}\mathrm {d}\b {k}’}{4\pi ^{2}} \mathref {(2.22)} \{end}{gather}$

equation (2.21) can be written in the form

$(2.23) \{begin}{gather} \displaystyle \frac {dM_{x}}{dt}=\ \Bigl (\frac {m_{0}}{\tau _{a}\hbar }\Bigr )\int \frac {\partial E}{\partial \b {k}_{x}}f(\b {k})\frac {\mathrm {d}\b {k}}{4\pi ^{3}}, \mathref {(2.23)} \{end}{gather}$

where $\tau _{a}$ is the relaxation time of the electrons. As a common assumption for metals, the relaxation time is taken to be constant over all states $\b {k}. f_{0}(\b {k})$ describes the electron distribution at equilibrium and integrates to zero. The current density in the $x$ -direction can be expressed by [3, 78]

$(2.24) \{begin}{gather} J_{x}=\displaystyle \frac {-e}{4\pi ^{2}}\int f(\b {k})\frac {\partial E(\b {k})}{\hbar \partial \b {k}_{x}}\mathrm {d}\b {k}. \mathref {(2.24)} \{end}{gather}$

By comparison of (2.21) and (2.24) the relation

$(2.25) \{begin}{gather} \displaystyle \frac {\mathrm {d}M_{x}}{\mathrm {d}t}=\frac {J_{x}m_{0}}{e\tau _{a}} \mathref {(2.25)} \{end}{gather}$

is obtained. With the density of defects $N_{a}$ , the density of the conducting electrons $n,$ and the contribution of the defects to the resistivity $\rho _{d} = |m^{*}|/ne^{2}\tau _{a}$ the force can be expressed by

$(2.26) \{begin}{gather} F_{\mathrm {w}\mathrm {i}\mathrm {n}\mathrm {d}}=-\displaystyle \frac {neJ_{x}\rho _{\mathrm {d}}m_{0}}{N_{d}|m^{*}|}=-eE_{x}z\frac {N\rho _{\mathrm {d}}}{\rho N}\frac {m_{0}}{|m^{*}|}, \mathref {(2.26)} \{end}{gather}$

where the density of the conducting electrons is substituted by $n = zN$ , where $N$ is the density of the lattice atoms, and $z$ is the number of conducting electrons per lattice atom. For an ion at a saddle point between two vacant position the interaction of the ion and the conducting electron is the strongest, whereas the interaction at lattice points is the weakest. On the way from one lattice point to an other the interaction varies and as this position dependent interaction is not known, Huntington et al. [64] chose a sinusoidal form leading to

$(2.27) \{begin}{gather} \b {F}(y)=\b {F}_{\mathrm {m}}\displaystyle \sin ^{2}\Bigl (\frac {\pi y}{a}\Bigr ), \mathref {(2.27)} \{end}{gather}$

where $a$ is the jump distance, and $\b {F}_{m}$ is the maximum force. For a jump path $j$ with an angle $\theta _{j}$ between the path and the force $\b {F}_{m}$ , the energy required for a jump can be calculated by

$(2.28) \{begin}{gather} \displaystyle \triangle V_{j}=\int \limits _0^{\frac {a_{j}}{2}}\mathbf {F}(y)\cdot \mathrm {d}\b {y}=\frac {1}{4}a_{j}F_{wind}\cos \theta _{j}. \mathref {(2.28)} \{end}{gather}$

The net flow of atoms due to EM in the current direction is the sum of the probabilities of jumps (along the paths j) times the jump length in the current direction [78].

$(2.29) \{begin}{gather} J_{\mathrm {w}\mathrm {i}\mathrm {n}\mathrm {d}}=\displaystyle \sum _{j}C_{lJ_{0}}\exp \Bigl (-\frac {V}{k_{\mathrm {B}}T}\Bigr )a_{j}\cos (\theta _{j})\sinh \Bigl (\frac {\triangle V_{j}}{k_{\mathrm {B}}T}\Bigr ) \mathref {(2.29)} \{end}{gather}$

$l/0$ is the atomic vibration frequency, $C$ the concentration of the ions in the metal, and $V$ the saddle point energy including the formation energy and the motion energy of vacancies. This equation can be linearized and rewritten to

$(2.30) \{begin}{gather} J_{\mathrm {w}\mathrm {i}\mathrm {n}\mathrm {d}}=\displaystyle \frac {cDF_{wind}}{k_{\mathrm {B}}T} \mathref {(2.30)} \{end}{gather}$

with $D$ being the diffusion coefficient.

$(2.31) \{begin}{gather} D=\displaystyle \frac {1}{2}v_{0}\exp \Bigl (-\frac {V}{k_{\mathrm {B}}T}\Bigr )\sum _{j}a_{j}^{2}\cos (\theta _{j}) \mathref {(2.31)} \{end}{gather}$

The resulting equation (2.30) differs form the Nernst-Einstein relation by the factor 2 in the denominator, which is the average of the chosen position dependent interaction of the conducting electrons and the ions on their path from one lattice point to an other. In addition to the force due to the electron wind also the force due to the electric potential gradient has to be included, leading to the effective charge

$(2.32) \{begin}{gather} e’=eZ^{*}=ez\displaystyle \ \Bigl (\frac {1}{2}\ \Bigl (\frac {\rho _{\mathrm {d}}N}{\rho N_{\mathrm {d}}}\frac {m_{0}}{|m^{*}|}\Bigr )\ -1\Bigr ) \mathref {(2.32)} \{end}{gather}$

and to the effective valence $Z^{*}$ Using the Einstein relation for field assisted diffusion in a potential the drift velocity can be expressed by [94]

$(2.33) \{begin}{gather} v_{\mathrm {E}\mathrm {M}}=\displaystyle \frac {DF}{k_{\mathrm {B}}T}=\frac {DZ^{*}e\rho J_{x}}{k_{\mathrm {B}}T}. \mathref {(2.33)} \{end}{gather}$

This was the first quantum mechanical expression of the EM induced ion velocity. Within the ballistic model [96] it was shown that there is a linear relation between the EM induced flux and the current density.

For a quantum mechanical force calculation another equation was developed and widely used, based on the scattering states of the conducting electrons [15, 117, 124, 132, 128] obtained from the linear response theory of Kubo [84],

$(2.34) \{begin}{gather} \b {F}_{\mathrm {w}\mathrm {i}\mathrm {n}\mathrm {d}}=\displaystyle \frac {e\Omega }{4\pi ^{3}}\bigg [\iint \limits _{\mathrm {FS}}\frac {\mathrm {d}^{2}\b {k}}{|\nabla E_{\b {k}}|}\tau _{E_{\b {k}}}\b {v}_{\mathrm {F}}(\b {k})\iiint \psi _{\b {k}}^{*}(r)\nabla _{\b {R}}V(|\b {R}-\b {r}|)\psi _{\b {k}}(\b {r})\mathrm {d}^{3}\b {r}\bigg ] \cdot {\it \b {E}}. \{end}{gather}$

The considered atom is located at position $\b {R}$ and $V(|\b {R}-\b {r}|)$ is the effective one-electron potential. $\b {E}$ is the electric field, $\Omega$ is the volume of the unit cell, $\tau _{E}$ is the relaxation time of the scattered electron, $v_{\mathrm {F}}(\b {k})$ is the $\b {k}$ -dependent Fermi velocity, and $\psi _{\b {k}}$ is the wave function of the electron, which can be calculated with the Schröedinger equation [37]

$(2.35) \{begin}{gather} - \displaystyle \frac {1}{2}\nabla ^{2}\psi _{\b {k}}+V(\b {r})\psi _{\b {k}}(\b {r})=E_{\b {k}}\psi _{\b {k}}(\b {r}) \mathref {(2.35)} \{end}{gather}$

In (2.34) the first part on the right hand side has the meaning of the effective charge, has the form of a tensor of second order, and reflects the possible dependence of the crystal orientation on the current direction of the EM force especially for non-cubic crystal metals (e.g. zinc) [59]. For periodic structures the integral (2.34) is always equal 0. The reason is the symmetry of the wave functions regarding the crystal wave vector

$(2.36) \{begin}{gather} \psi _{\b {k}}(\b {x})\ =\psi _{-\b {k}}^{*}(\b {x}), \mathref {(2.36)} \{end}{gather}$

making the result of the integration over space an even function in $\b {k}$ due to the fact that the potential is a real valued function,

$(2.37) \{begin}{gather} \displaystyle \iiint \psi _{\b {k}}^{*}(r)\nabla _{R}V(|R-r|)\psi _{\b {k}}(r)\mathrm {d}^{3}\b {r}=\iiint \psi _{-\b {k}}^{*}(r)\nabla _{R}V(|R-r|)\psi _{-\b {k}}(\b {r})\mathrm {d}^{3}\b {r}. \mathref {(2.37)} \{end}{gather}$

Furthermore, the Fermi velocity is an odd function in $\b {k}$ leading to a vanishing result of the integration over the Fermi sphere. This causes the necessity of calculations of nonperiodic problems. For bulk materials the calculations were performed using the pseudo potential and the KKR method [129, 130, 131, 144, 145]. $Bly$ et al. [15, 111, 112] showed how a calculation for a single adatom can be carried out by employing the LKKR method [97, 98]. They used the muffin-tin approximation by confining the atomic potential to non-overlapping spheres with a constant interstitial potential [101]. The advantage of this method is the simplicity and the computational economy payed by an insufficient description of valence electron potentials in covalent open structures [147] compared to full potential calculations [105] The electron wave function was defined by

$(2.38) \{begin}{gather} \displaystyle \psi _{\b {k}}(\b {r})=\frac {4\pi }{\sqrt {\Omega }}\sum _{lm}i^{l}A_{lm}(\b {k})R_{l}(\b {r})Y_{lm}(\b {r}), \mathref {(2.38)} \{end}{gather}$

Figure 2.1.: The dependence of the effective valence on the distance of an adatom to a semi-infinite metal surface for two different location above the crystal lattice shown on the right [15]. $\tau$ is the time scale of relaxation of the electronic charge density.

$ \tau $ — Figure 2.1.: The dependence of the effective valence on the distance of an adatom to a semi-infinite metal surface for two different location above the crystal lattice shown on the right [15]. $\tau$ is the time scale of relaxation of the electronic charge density.

where $Y_{lm}(\b {r})$ is a spherical harmonic [150], $A_{lm}(\b {k})$ is the coefficient from the spherical wave expansion, evaluated by the LKKR calculation, and $R_{l}(\b {r})$ is the spherical solution of the Schrödinger equation [141], which can be expressed as

$(2.39) \{begin}{gather} R_{l}(\b {r})=\ (j_{l}(\kappa \b {r})+h_{l}^{(1)}(\kappa \b {r}))ie^{i\delta _{l}^{a}}\sin (\delta _{l}^{a}). \mathref {(2.39)} \{end}{gather}$

Here $\kappa = \sqrt {2E_{k}}, j_{l}$ is a spherical Bessel function [150], $h_{l}^{(1)}$ is the spherical Henkel function [150] of the first kind, and $\delta _{l}^{a}$ characterizes the scattering phase shift of each atom. The results show that the effective valance of an adatom is strongly dependent on the height of the atom relative to the metal surface (c.f. Figure 2.1). This dependence is quite well described by a simple ballistic model, if the reduced electron density relative to the bulk is taken into account [15]. This calculation was extended to islands of adatoms on a substrate modeled by the jellium model showing that the distance to islands has a huge impact on the effective valance of single adatoms [113].

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