Electronic transport calculations showing electron-phonon separation in directions transverse to high current

With a focus for a moment just on the three phonon processes then the textbook equation for the rate of change of n _q is

$$\begin{eqnarray}\begin{array}{rcl}{\left.\displaystyle \frac{\partial {n}_{{\boldsymbol{q}}}}{\partial t}\right|}_{3p} & = & \displaystyle \int \displaystyle \int \left[\left\{-{n}_{{\boldsymbol{q}}}{n}_{{{\boldsymbol{q}}}^{{\prime} }}(1+{n}_{{\boldsymbol{q}}^{\prime\prime} }){Q}_{{\boldsymbol{q}},{{\boldsymbol{q}}}^{{\prime} }}^{{\boldsymbol{q}}^{\prime\prime} }+(1+{n}_{{\boldsymbol{q}}})(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}){n}_{{\boldsymbol{q}}^{\prime\prime} }{Q}_{{\boldsymbol{q}}^{\prime\prime} }^{{\boldsymbol{q}},{{\boldsymbol{q}}}^{{\prime} }}\right\}\delta ({\boldsymbol{q}}^{\prime\prime} -{\boldsymbol{q}}-{{\boldsymbol{q}}}^{{\prime} })\right.\\ & & \left.+\displaystyle \frac{1}{2}\left\{-{n}_{{\boldsymbol{q}}}(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }})(1+{n}_{{\boldsymbol{q}}^{\prime\prime} }){Q}_{{\boldsymbol{q}}}^{{{\boldsymbol{q}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }+(1+{n}_{{\boldsymbol{q}}}){n}_{{{\boldsymbol{q}}}^{{\prime} }}{n}_{{\boldsymbol{q}}^{\prime\prime} }{Q}_{{{\boldsymbol{q}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }^{{\boldsymbol{q}}}\right\}\delta ({\boldsymbol{q}}-{{\boldsymbol{q}}}^{{\prime} }-{\boldsymbol{q}}^{\prime\prime} )\right]{{\boldsymbol{dq}}}^{{\prime} }{\boldsymbol{dq}}^{\prime\prime} ,\end{array}\end{eqnarray} \tag{ 2.4 }$$

where the $\tfrac{1}{2}$ is to avoid double-counting. By the principle of detailed balance, this rate is zero if all occupation factors n _q take the equilibrium forms ${n}_{{\boldsymbol{q}}}^{0}$. Because of electron-phonon scattering and the production of phonons q '', n _{q ''} will not take the equilibrium form. For this specific factor a form ${n}_{{\boldsymbol{q}}^{\prime\prime} }={n}_{{\boldsymbol{q}}^{\prime\prime} }^{0}+{n}_{{\boldsymbol{q}}^{\prime\prime} }^{* }$ is used. The physical reasoning is that one focuses only on 3-phonon scattering that occurs subsequently to the production of a phonon q '' by (first order) electron phonon scattering via ${W}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }$.

For the remainder of this subsection it suffices to consider just one of the four terms in (2.4). It will be shown below in this section that the second term dominates the others so this is the term discussed immediately, which leaves:

$$\begin{eqnarray}&&{\left.\displaystyle \frac{\partial {n}_{{\boldsymbol{q}}}}{\partial t}\right|}_{3p,2}=\int \int (1+{n}_{{\boldsymbol{q}}}^{0})(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0}){n}_{{\boldsymbol{q}}^{\prime\prime} }^{* }{Q}_{{\boldsymbol{q}}^{\prime\prime} }^{{\boldsymbol{q}},{{\boldsymbol{q}}}^{{\prime} }}\delta ({\boldsymbol{q}}^{\prime\prime} -{\boldsymbol{q}}-{{\boldsymbol{q}}}^{{\prime} }){{\boldsymbol{dq}}}^{{\prime} }{\boldsymbol{dq^{\prime\prime} }}.\end{eqnarray} \tag{ 2.5 }$$

As mentioned above, ${n}_{{\boldsymbol{q}}^{\prime\prime} }^{* }$ is produced by electron-phonon scattering. The phonons produced from these events will decay with a lifetime τ_q'' that is the lifetime of hot phonons produced from the electron-phonon scattering. The distribution ${n}_{{\boldsymbol{q}}^{\prime\prime} }^{* }$ is described by

$$\begin{eqnarray}&&\displaystyle \frac{\partial {n}_{{\boldsymbol{q}}^{\prime\prime} }^{* }}{\partial t}={\left.-\displaystyle \frac{{n}_{{\boldsymbol{q}}^{\prime\prime} }^{* }}{{\tau }_{q^{\prime\prime} }}+\displaystyle \frac{\partial {n}_{{\boldsymbol{q}}^{\prime\prime} }^{* }}{\partial t}\right|}_{{ep}},\end{eqnarray} \tag{ 2.6 }$$

where the first order electron-phonon scattering rate is given by

$$\begin{eqnarray}&&{\left.\displaystyle \frac{\partial {n}_{{\boldsymbol{q}}}^{* }}{\partial t}\right|}_{{ep}}=\int \int {g}_{{\boldsymbol{k}}}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }})(1+{n}_{{\boldsymbol{q}}^{\prime\prime} }^{0}){W}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }\delta ({\boldsymbol{k}}-{{\boldsymbol{k}}}^{{\prime} }-{\boldsymbol{q}}^{\prime\prime} ){\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }.\end{eqnarray} \tag{ 2.7 }$$

The steady-state solution to (2.6) is given by

$$\begin{eqnarray}&&{n}_{{\boldsymbol{q}}^{\prime\prime} }^{* }={\left.{\int }_{0}^{\infty }\displaystyle \frac{\partial {n}_{{\boldsymbol{q}}}^{* }}{\partial t}\right|}_{{ep}}\exp (-t/{\tau }_{q^{\prime\prime} }){dt},\end{eqnarray} \tag{ 2.8 }$$

which can be substituted into (2.5) to give:

$$\begin{eqnarray}\begin{array}{rcl}{\left.\displaystyle \frac{\partial {n}_{{\boldsymbol{q}}}}{\partial t}\right|}_{3p,2} & = & \displaystyle \int \displaystyle \int \displaystyle \int \displaystyle \int {e}^{-t/{\tau }_{q^{\prime\prime} }}{g}_{{\boldsymbol{k}}}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }})(1+{n}_{{\boldsymbol{q}}}^{0})(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0})\\ & & \times (1+{n}_{{\boldsymbol{q}}^{\prime\prime} }^{0}){W}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }{Q}_{{\boldsymbol{q}}^{\prime\prime} }^{{\boldsymbol{q}},{{\boldsymbol{q}}}^{{\prime} }}\delta ({\boldsymbol{k}}-{{\boldsymbol{k}}}^{{\prime} }-{\boldsymbol{q}}-{{\boldsymbol{q}}}^{{\prime} }){\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{{\boldsymbol{dq}}}^{{\prime} }{dt}.\end{array}\end{eqnarray} \tag{ 2.9 }$$

Comparison to the (dominant) third term in (2.3) leads to the conclusion that ${\tau }_{1}=(1+{n}_{{\boldsymbol{q}}^{\prime\prime} }^{0}){\tau }_{q^{\prime\prime} }$ and

$$\begin{eqnarray}&&{A}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}},{{\boldsymbol{q}}}^{{\prime} }}={\int }_{0}^{\infty }{e}^{-t/{\tau }_{q^{\prime\prime} }}(1+{n}_{{\boldsymbol{q}}^{\prime\prime} }^{0}){W}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }{Q}_{{\boldsymbol{q}}^{\prime\prime} }^{{\boldsymbol{q}},{{\boldsymbol{q}}}^{{\prime} }}\delta ({\boldsymbol{k}}-{{\boldsymbol{k}}}^{{\prime} }-{\boldsymbol{q}}-{{\boldsymbol{q}}}^{{\prime} }){dt}.\end{eqnarray} \tag{ 2.10 }$$

Focusing on a single ${Q}_{{\boldsymbol{q}}^{\prime\prime} }^{{\boldsymbol{q}},{{\boldsymbol{q}}}^{{\prime} }}$ vertex, such as depicted in the right part of figure 1e), one derives an equation similar to (2.6) for τ_q'' as:

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\tau }_{q^{\prime\prime} }}\equiv \int \int (1+{n}_{{\boldsymbol{q}}}^{0})(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0}){Q}_{{\boldsymbol{q}}^{\prime\prime} }^{{\boldsymbol{q}},{{\boldsymbol{q}}}^{{\prime} }}\delta ({\boldsymbol{q}}^{\prime\prime} -{\boldsymbol{q}}-{{\boldsymbol{q}}}^{{\prime} }){\boldsymbol{dq}}{{\boldsymbol{dq}}}^{{\prime} }.\end{eqnarray} \tag{ 2.11 }$$

This expression for the decay rate will be useful in the analysis which follows.

Since the main contribution to the k sums comes from electronic states near the Fermi level where 2E_k − μ ≈ E_F , and the Fermi energy E_F always greatly exceeds _q'', the leading term in (3.9) comes from the ${\epsilon }_{q^{\prime\prime} }\left[2{E}_{k}-\mu \right]{\cos }^{2}\theta ^{\prime}$ term. In the results below only this term is considered:

$$\begin{eqnarray}\begin{array}{rcl}{f}_{{ph},x} & \approx & \displaystyle \frac{{\hslash }{\gamma }_{2}{\beta }^{2}{E}_{F}}{2}\displaystyle \int \displaystyle \int \displaystyle \int \displaystyle \int {g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0})(1+{n}_{{\boldsymbol{q}}}^{0})(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0})\\ & & \times {\epsilon }_{q^{\prime\prime} }{\left(\cos \theta ^{\prime} \right)}^{2}{A}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}},{{\boldsymbol{q}}}^{{\prime} }}{q}_{x}{\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{\boldsymbol{dq}}{{\boldsymbol{dq}}}^{{\prime} }.\end{array}\end{eqnarray} \tag{ 3.10 }$$

The third and fourth terms of (2.3) produce the major contributions to the phonon transverse force f_ph,x . These main contributions occur because in the third and fourth terms ${E}_{{\boldsymbol{k}}}\gt {E}_{{{\boldsymbol{k}}}^{{\prime} }}$ always holds. This is not the case in the first and second terms of (2.3) since in these processes the virtual phonons q '' can go both into and out of the W vertex. The plus sign in the front of (3.10) means that the third term in (2.3) represented by the scattering diagram in figure 1(e) is the dominant one. The physical interpretation of this is that the process emitting two phonons into the heat bath creates the most entropy and receives an extra statistical weight for this.

Bringing back the explicit integral over q '':

$$\begin{eqnarray}\begin{array}{rcl}{f}_{{ph},x} & \approx & \displaystyle \frac{{\hslash }{\gamma }_{2}{\beta }^{2}{E}_{F}}{2}\displaystyle \int \displaystyle \int \displaystyle \int \displaystyle \int \displaystyle \int {g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0}){A}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}},{{\boldsymbol{q}}}^{{\prime} }}\\ & & \times {\epsilon }_{q^{\prime\prime} }{\left(\cos \theta ^{\prime} \right)}^{2}({q}_{x}^{{\prime\prime} }-{q}_{x}^{{\prime} })(1+{n}_{{\boldsymbol{q}}}^{0})(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0})\\ & & \times \delta ({\boldsymbol{k}}-{{\boldsymbol{k}}}^{{\prime} }-{\boldsymbol{q}}^{\prime\prime} ){\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{\boldsymbol{dq}}{{\boldsymbol{dq}}}^{{\prime} }{\boldsymbol{dq}}^{\prime\prime} .\end{array}\end{eqnarray} \tag{ 3.11 }$$

In the next subsection 3.2, spatial variation in the transverse direction is treated. Leading terms have an extra factor of ${q}_{x}^{{\prime\prime} }$, producing a ${\left({q}_{x}^{{\prime\prime} }\right)}^{2}$ in the integrand and thus giving a non-zero result. A factor of ${q}_{x}^{{\prime} }{q}_{x}^{{\prime\prime} }$ is also produced that gives zero upon integration. This allows the ${q}_{x}^{{\prime\prime} }-{q}_{x}^{{\prime} }$ factor in (3.11) to be replaced by ${q}_{x}^{{\prime\prime} }$.

Substituting in for ${A}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}},{{\boldsymbol{q}}}^{{\prime} }}$ from (2.10) into (3.11), while also implementing (2.11), gives

$$\begin{eqnarray}\begin{array}{rcl}{f}_{{ph},x} & \approx & \displaystyle \frac{{\hslash }{\gamma }_{2}{\beta }^{2}{E}_{F}}{2}\displaystyle \int \displaystyle \int \displaystyle \int \displaystyle \int {g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0})(1+{n}_{{\boldsymbol{q}}^{\prime\prime} }^{0}){\epsilon }_{q^{\prime\prime} }{\left(\cos \theta ^{\prime} \right)}^{2}{q}_{x}^{{\prime\prime} }\\ & & \times {W}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }{e}^{-t/{\tau }_{q^{\prime\prime} }}\displaystyle \frac{1}{{\tau }_{q^{\prime\prime} }}\delta ({\boldsymbol{k}}-{{\boldsymbol{k}}}^{{\prime} }-{\boldsymbol{q}}^{\prime\prime} ){\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{\boldsymbol{dq}}^{\prime\prime} {dt}.\end{array}\end{eqnarray} \tag{ 3.12 }$$

The factor of q_x in (3.10) will make the integral zero by symmetry unless another q_x is produced. This can be accomplished if the transition rate ${A}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}},{{\boldsymbol{q}}}^{{\prime} }}$ varies in the (transverse) x-direction. By (3.12) one way this can happen if the matrix element ${W}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }$ varies in the x-direction. This can also happen if occupation factors ${g}_{{\boldsymbol{k}}}^{0}$, ${g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0}$, ${n}_{{\boldsymbol{q}}^{\prime\prime} }^{0}$ vary in the x-direction, and lastly, if τ_q'' varies in the x-direction.

The two scattering vertices do not occur at the same time and place. The temporal separation is τ_q'' and the spatial separation is c τ_q''. If r is the location of the midpoint of the electron-phonon collision and the 3-phonon collision then ${W}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }$ must be evaluated at ${\boldsymbol{r}}-\tfrac{1}{2}c{\tau }_{q^{\prime\prime} }\tfrac{{q}_{x}^{{\prime\prime} }}{q^{\prime\prime} }\hat{x}$. So also must ${g}_{{\boldsymbol{k}}}^{0}$, ${g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0}$, and ${n}_{{\boldsymbol{q}}^{\prime\prime} }^{0}$ be evaluated at ${\boldsymbol{r}}-\tfrac{1}{2}c{\tau }_{q^{\prime\prime} }\tfrac{{q}_{x}^{{\prime\prime} }}{q^{\prime\prime} }\hat{x}$. In contrast, since the lifetime τ_q'' involves phonon-phonon scattering with vertex ${Q}_{{\boldsymbol{q}}^{\prime\prime} }^{{\boldsymbol{q}},{{\boldsymbol{q}}}^{{\prime} }}$, it must be evaluated at position ${\boldsymbol{r}}+\tfrac{1}{2}c{\tau }_{q^{\prime\prime} }\tfrac{{q}_{x}^{{\prime\prime} }}{q^{\prime\prime} }\hat{x}$. Taking the first order expansion:

$$\begin{eqnarray}\begin{array}{rcl}{f}_{{ph},x} & \approx & \displaystyle \frac{{\hslash }{\gamma }_{2}{\beta }^{2}{E}_{F}}{4}\displaystyle \int \displaystyle \int \displaystyle \int \displaystyle \int \left[-\displaystyle \frac{\partial }{\partial x}\left\{{g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0})(1+{n}_{{\boldsymbol{q}}^{\prime\prime} }^{0}){W}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }\right\}\right.\\ & & \left.+{g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0})(1+{n}_{{\boldsymbol{q}}^{\prime\prime} }^{0}){W}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }\left(\displaystyle \frac{-1}{{\tau }_{q^{\prime\prime} }}\right)\displaystyle \frac{\partial {\tau }_{q^{\prime\prime} }}{\partial x}\right]\\ & & \times \displaystyle \frac{c{\epsilon }_{q^{\prime\prime} }}{q^{\prime\prime} }{\left({q}_{x}^{{\prime\prime} }\right)}^{2}{e}^{-t/{\tau }_{q^{\prime\prime} }}{\left(\cos \theta ^{\prime} \right)}^{2}\delta ({\boldsymbol{k}}-{{\boldsymbol{k}}}^{{\prime} }-{\boldsymbol{q}}^{\prime\prime} ){\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{\boldsymbol{dq}}^{\prime\prime} {dt}.\end{array}\end{eqnarray} \tag{ 3.13 }$$

Completing the integral over t gives

$$\begin{eqnarray}\begin{array}{rcl}{f}_{{ph},x} & = & -\displaystyle \frac{{\gamma }_{2}{\beta }^{2}{E}_{F}}{4}\displaystyle \frac{\partial }{\partial x}\displaystyle \int \displaystyle \int \displaystyle \int {\tau }_{q^{\prime\prime} }{g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0})(1+{n}_{{\boldsymbol{q}}^{\prime\prime} }^{0}){W}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }\\ & & \times {\epsilon }_{q^{\prime\prime} }^{2}\displaystyle \frac{{\left({q}_{x}^{{\prime\prime} }\right)}^{2}}{{\left(q^{\prime\prime} \right)}^{2}}{\left(\cos \theta ^{\prime} \right)}^{2}\delta ({\boldsymbol{k}}-{{\boldsymbol{k}}}^{{\prime} }-{\boldsymbol{q}}^{\prime\prime} ){\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{\boldsymbol{dq}}^{\prime\prime} .\end{array}\end{eqnarray} \tag{ 3.14 }$$

As mentioned, one way this force can be non-zero is if the matrix element ${W}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }$ varies in the x-direction. The derivative $\partial {W}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }/\partial x$ represents an explicit variation in the material composition in the x-direction. A simple example of this is an interface between two conventional metals, for example a gold-aluminum interface, either created by thin film growth in the x-direction or as a metallic superlattice [15]. In many models for electron-phonon interaction, using the deformation potential (including the Bardeen model), the square of the matrix element scales inversely with the lattice mass density [9, 13]. Near any of these interfaces one will find a sharp change in W in the direction perpendicular to the interface.

The following definition is made for an important time scale:

$$\begin{eqnarray}&&{\tau }^{* }\equiv \displaystyle \frac{3\int \int \int {\tau }_{q^{\prime\prime} }{G}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }{\left({q}_{x}^{\prime\prime} \right)}^{2}\delta ({\boldsymbol{k}}-{{\boldsymbol{k}}}^{{\prime} }-{\boldsymbol{q}}^{\prime\prime} ){\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{\boldsymbol{dq^{\prime\prime} }}}{\int \int \int {G}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }{\left(q^{\prime\prime} \right)}^{2}\delta ({\boldsymbol{k}}-{{\boldsymbol{k}}}^{{\prime} }-{\boldsymbol{q}}^{\prime\prime} ){\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{\boldsymbol{dq^{\prime\prime} }}}\end{eqnarray} \tag{ 3.15 }$$

where ${G}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }\equiv {g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0})(1+{n}_{{\boldsymbol{q}}^{\prime\prime} }^{0}){W}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}}^{\prime\prime} }{\cos }^{2}\theta ^{\prime}$. The time τ^* can be thought of as a nonequilibrium transport mean value of τ_q'' for a virtual phonon. The denominator in (3.15) closely resembles the Joule heating rate J _el · E which increases the phonon gas energy at rate ${\dot{u}}_{{ph}}$. Indeed, making use of (B.9) as derived in the appendix B allows (3.14) to be rewritten in a relatively concise form for the force per unit volume on the phonons as:

$$\begin{eqnarray}&&{f}_{{ph},x}=-\displaystyle \frac{1}{6}\displaystyle \frac{\partial }{\partial x}\left\{{\tau }^{* }{\dot{u}}_{{ph}}\right\}.\end{eqnarray} \tag{ 3.16 }$$

The force f_ph,x only exists while the electronic system is shifted away from equilibrium. In this case one may think of the phonons as being embedded in a nonequilibrium electronic system. This extra force term while being embedded will add onto any other force terms to dictate the phonon dynamics.

As expected from momentum conservation the transverse force on electrons is equal and opposite,

$$\begin{eqnarray}&&{f}_{{el},x}=+\displaystyle \frac{1}{6}\displaystyle \frac{\partial }{\partial x}\left\{{\tau }^{* }\,{\dot{u}}_{{ph}}\right\}.\end{eqnarray} \tag{ 3.17 }$$

This force is not to be confused with the drag force in the $-\hat{z}$ direction. The force f_el,x can also be explicitly derived by setting up equations similar to (3.2.a )–(3.2.f ) while anticipating the ${q}_{x}^{{\prime\prime} }$ factor from the spatial variation, and following the minus signs carefully.

Together (3.16) and (3.17) constitute the main result presented here.

The stationary state solution for (4.3) is readily found when U_d explicitly depends on transverse coordinates. In this case the solution is

$$\begin{eqnarray}&&{T}_{{ss}}-{T}_{a}=\displaystyle \frac{{U}_{d}}{2(1-{\alpha }_{{ph}}){c}_{V}}.\end{eqnarray} \tag{ 4.7 }$$

The result is a non-flat temperature profile superimposed on top of an otherwise uniform ambient temperature. Joule heating will raise the ambient temperature but in practice, effective external cooling can keep T_a close to what ambient is when there is no electrical flow. The profile exists even when α_ph = 0. In this case the profile is driven completely by any explicit spatial variation in U_d . The temperature profile becomes flat when the electrical current is turned off and U_d = 0. This profile is entirely a nonequilibrium effect and the effect will be small with the system near equilibrium. A trickle of current will not suffice; Rather, the system will likely have to pushed quite hard before the profile becomes detectable. For example, if an amount of energy equal to about 1% of the internal thermal energy is dissipated in what is typically a short time τ^*, then a profile amplitude of about 1% of ambient could be observed. Thin samples with good heat-sinking would avoid excessive temperature rise from Joule heating and allow the opportunity for such profiles to be observed. Indeed, dissipated power levels need not be so high if ambient temperature is well below the Debye temperature where c_V is small.

A good example system is an alternating ABAB electrically conductive superlattice with the A layers having larger u_d than the B layers. Electrical current running along the conducting layers will produce the temperature and charge profiles. The A layers will be warmer than the B layers and as will be shown below, electrons will tend to be pushed into the B layers. An applied bias to this superlattice results in the same electric field E in each layer. Reexpressing (4.7) explicitly in terms of E gives

$$\begin{eqnarray}&&{T}_{{ss}}-{T}_{a}=\displaystyle \frac{{\tau }^{* }\,{\sigma }_{{el}}}{2(1-{\alpha }_{{ph}}){c}_{V}}{E}^{2},\end{eqnarray} \tag{ 4.8 }$$

where it is understood that σ_el , c_V , and α_ph have a spatial dependence, changing sharply at each AB interface.

In order to make specific calculations and predictions for the induced temperature profile a good estimate for the phonon lifetime τ^* is required. The simplest approach is to use the lattice thermal conductivity expression ${\kappa }_{{latt}}=\tfrac{1}{3}{c}_{V}{c}^{2}{\tau }^{* }$, which changes (4.8) to

$$\begin{eqnarray}&&{T}_{{ss}}-{T}_{a}=\displaystyle \frac{{{aE}}^{2}}{2\left(1-\nu {{aE}}^{2}\right)},\end{eqnarray} \tag{ 4.9 }$$

where

$$\begin{eqnarray}&&a\equiv \displaystyle \frac{3{\kappa }_{{latt}}{\sigma }_{{el}}}{2{c}_{V}^{2}{c}^{2}}.\end{eqnarray} \tag{ 4.10 }$$

A good electrical conductor like silver would have a relatively large value for a and be more likely to give measurable values of T_ss − T_a . The lattice thermal conductivity for conductors such as Ag can be calculated from first principles [19, 20]. At room temperature (RT) the value is κ_latt = 5.2 W/mK and at 77 K the value is 12 W/mK. Making use of well-known values for the electrical conductivity, specific heat capacity and speed of sound, [8] allows for easy calculation of the parameter a and subsequent calculation of the curves shown in figure 2. If the AB superlattice consists of Ag and a relatively poor electrical conductor such as Bi, the result is square wave temperature profile with amplitude T_Ag − T_Bi . The RT plot in figure 2 (solid black curve) shows an upwards trend with T_Ag − T_Bi going as E². Any effects from the (1 − ν aE²) factor in the denominator of (4.9) are not seen. A reasonable value for the minimum discernible temperature shift is 1 mK. For such a shift to be produced a 12.2 kV/m field would be required at RT. This is not easily achieved in practice as this temperature profile would be superimposed on top of significant Joule heating that would be difficult to heat sink, even for very thin, planar samples.

Zoom In Zoom Out Reset image size

Moving to lower temperature does make the task easier as indicated by the 77 K curve (dashed blue). For most of the range of electric fields shown the trend is similar to RT but with larger values of T_Ag − T_Bi . At fields above 2 × 10⁵ V/m, the curve bends upwards because of the (1 − ν aE²) factor. A value of ν = 3 is used since σ_el goes approximately as T⁻¹ and τ^* is expected to go as T⁻². At 77 K, the minimal discernible temperature of 1 mK is predicted to occur at a field of 2.1 kV/m. This is about 6 times less than at RT. Though these calculations indicate that detection of this temperature profile would not be easy, there does not seem to be any fundamental reason ruling out this detection. To date no electrical transport measurements have shown the effect predicted here. These preliminary investigations suggest that such a predicted temperature profile would be more easily detected at lower temperatures.

It may also be worthwhile for researchers to investigate other dissipative systems as well. The form of (3.16) suggests that any system in which the dissipated energy has a spatial gradient will result in forces on some particles, with the forces on the phonons to ensure momentum conservation. It may be the case that in such systems these forces are more readily detected. Examples of such systems include chemical reactions (reaction-diffusion dynamics) and front propagation (with dynamics) during phase transitions.

In the case where there is no explicit spatial dependence for U_d the material properties would be assumed to be uniform and (4.3) becomes

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}T}{\partial {t}^{2}}=\displaystyle \frac{1}{3}{c}^{2}\left(1-{\alpha }_{{ph}}\right){{\rm{\nabla }}}^{2}T-\displaystyle \frac{{c}^{2}{c}_{V}}{3\kappa }\displaystyle \frac{\partial T}{\partial t}+\displaystyle \frac{3{\tau }_{N}{c}^{2}}{5}\displaystyle \frac{\partial {{\rm{\nabla }}}^{2}T}{\partial t}.\end{eqnarray} \tag{ 4.11 }$$

When the system is pushed very hard with high rates of dissipation such that at α_ph ≥ 1, (4.11) becomes unstable; There is a bifurcation, a solution would grow very quickly and would diverge if the system is strictly linear. Taking a spatial Fourier transform of ΔT to $\tilde{T}(k)$, and subsequently adding in an appropriate nonlinear term gives

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}\tilde{T}}{\partial {t}^{2}}=-\displaystyle \frac{1}{3}{c}^{2}\left(1-{\alpha }_{{ph}}\right){k}^{2}\tilde{T}-\displaystyle \frac{{c}^{2}{c}_{V}}{3\kappa }\displaystyle \frac{\partial \tilde{T}}{\partial t}-\displaystyle \frac{3{\tau }_{N}{c}^{2}{k}^{2}}{5}\displaystyle \frac{\partial \tilde{T}}{\partial t}-{\gamma }_{k}{\tilde{T}}^{2}.\end{eqnarray} \tag{ 4.12 }$$

Adding the nonlinear term ensures a finite stationary solution of $\tilde{T}(k)=({\alpha }_{{ph}}-1){c}^{2}/3{\gamma }_{k}$, valid when α_ph > 1. Nonlinear terms like ${\gamma }_{k}{\tilde{T}}^{2}$ can arise when the coefficient of thermal conductivity depends on temperature [21]. The details of these nonlinear terms, in particular how they depend on wavevector k , determine which wavelengths are the most prominent. It is possible that this stationary state solution is not spatially uniform, and would therefore constitute spontaneous symmetry breaking.

The partial differential equation (4.12) bears some similarity to the Kuramoto-Sivashinsky (KS) equation, an important analysis tool used in the study of nonequilibrium systems that has been successfully utilized to model patterns formed in laminar flame fronts, certain types of Poiseuille flow, trapped ion modes in plasmas, and systems with Eckhaus instabilities such as Rayleigh-Bénard convection [4–6, 22–25]. Modeling with the KS equation can produce instabilities and interesting pattern formation when the diffusion coefficient is made negative. Though the KS equation has been used widely as a model, clear explanations are lacking in the literature for why a diffusion coefficient can be negative. The analysis presented here shows how a first principles approach can produce an effectively negative diffusion coefficient.

Observation of such an instability and/or pattern formation would likely require very high electrical current along the principal z-axis direction. Bringing the system near α_ph = 1 may dissipate so much energy such as to push the system near the point of irreparable damage, perhaps from melting, vaporization, or oxidation. Indeed, it may be the case that just before this threshold there is considerable channeling at short length scales with the higher temperature channels getting dangerously close to melting or vaporizing. This temperature channeling may be what happens just as the proverbial fuse blows. The above-mentioned positive feedback between channels may make it even more difficult to avoid blowing the fuse.

Comparing (4.7) to (5.5) for stationary states shows that the temperature profile is anticorrelated to the electron density profile. The occurs as the direct result of the opposing nature of the phonon and electron forces. This anticorrelation is stronger for electrons at shorter length scales, since the charge profile falls off at large length scales.

The interplay between phonon and electron displacements can be better understood in the case where a system that is initially uniform spatially, has enough electrical current applied such that the bifurcation point α_ph = 1 has been reached. The instability is expected to make some regions hotter than others. By the definition of α_ph this means that the relatively cooler regions will experience an increase in U_d , thus making U_d no longer uniform. This means that for the expected positive α_el the cooler regions will receive extra electrons. Not only is a non-flat charge profile created spontaneously, but the direction of the electronic relaxation is such as to increase U_d even more. This positive feedback for electrons works in concert with that of the phonons to increase and amplify any spatially non-uniform features in U_d .

This has interesting implications on research approaches that have been developed for the general treatment of nonequilibrium systems. One such approach is GENERIC (general equation for the nonequilibrium reversible-irreversible coupling), in which the dynamical equations for dissipative fluids is described by two distinct terms, one for reversible Poisson kinematics and one for dissipative Ginzburg-Landau kinematics [26–28]. Produced here is are dissipative terms which might be expected, except that they are forces perpendicular to the principle axis of conduction of the main variable. The dissipation potential energy density U_d may be closely related to the dissipative potential function Ψ, used in the GENERIC scheme.

The function U_d cannot be the same as Ψ because also produced here, and perhaps not as expected, are conservative terms that fit in perfectly well into the Poisson kinematics, and having dissipation as their fundamental source. More precisely, it is pure dissipation in the one principle direction that produces reversible terms in the Hamiltonian that are perpendicular to that direction. Thus the same function U_d winds up contributing to both sides, reversible and irreversible, of the GENERIC formalism.

What has happened is that the z and x directions have become coupled when double vertex scattering has been taken into account. The result of the perpendicular conservative forces is quite surprising.

Since states near the Fermi level are expected to make the most significant contributions to these integrals, one can assume b and $b^{\prime}$ are small and one arrives at the method for replacing nonequlibrium electron occupancies g _k and g _k with equlibrium functions:

$$\begin{eqnarray}&&-{g}_{{\boldsymbol{k}}}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}){n}_{{\boldsymbol{q}}}^{0}(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0})\approx {g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0}){n}_{{\boldsymbol{q}}}^{0}(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0}){a}_{1\gt }\end{eqnarray} \tag{ A.1 }$$

with

$$\begin{eqnarray}&&{a}_{1\gt }=-1+{\gamma }_{2}{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0}\left[-1+{\beta }^{2}{\tilde{E}}_{k}({\tilde{E}}_{k}-\mu ){\cos }^{2}\theta ^{\prime} \right]-{\gamma }_{2}(1-{g}_{{\boldsymbol{k}}}^{0})\left[-1+{\beta }^{2}{E}_{k}({E}_{k}-\mu ){\cos }^{2}\theta \right]\end{eqnarray} \tag{ A.2 }$$

noting that ${\gamma }_{1}^{2}=4{\gamma }_{2}{E}_{k}{\cos }^{2}\theta$. Thus

$$\begin{eqnarray}&&{f}_{1\gt }={\hslash }\int \int \int {\int }_{\gt }{g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0}){n}_{{\boldsymbol{q}}}^{0}(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0}){a}_{1\gt }{A}_{{\boldsymbol{k}},{\boldsymbol{q}}}^{{{\boldsymbol{k}}}^{{\prime} },{{\boldsymbol{q}}}^{{\prime} }}{q}_{x}{\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{\boldsymbol{dq}}{{\boldsymbol{dq}}}^{{\prime} }.\end{eqnarray} \tag{ A.3 }$$

For a_1< one simply replaces _q'' with − _q''. Proceeding similarly for the evaluation of f_2>, while using the principle of detailed balance [9] in the form:

$$\begin{eqnarray}&&{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0}(1-{g}_{{\boldsymbol{k}}}^{0}){n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0}(1+{n}_{{\boldsymbol{q}}}^{0})={g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0}){n}_{{\boldsymbol{q}}}^{0}(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0}),\end{eqnarray} \tag{ A.4 }$$

gives

$$\begin{eqnarray}&&{f}_{2\gt }={\hslash }\int \int \int {\int }_{\gt }{g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0}){n}_{{\boldsymbol{q}}}^{0}(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0}){a}_{2\gt }{A}_{{\boldsymbol{k}},{\boldsymbol{q}}}^{{{\boldsymbol{k}}}^{{\prime} },{{\boldsymbol{q}}}^{{\prime} }}{q}_{x}{\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{\boldsymbol{dq}}{{\boldsymbol{dq}}}^{{\prime} },\end{eqnarray} \tag{ A.5 }$$

with

$$\begin{eqnarray}&&{a}_{2\gt }=1-{g}_{{\boldsymbol{k}}}^{0}\left[-{\gamma }_{2}+\displaystyle \frac{1}{2}{\gamma }_{1}^{2}\tanh \displaystyle \frac{b}{2}\right]+(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0})\left[-{\gamma }_{2}^{\prime} +\displaystyle \frac{1}{2}{\gamma }_{1}{{\prime} }^{2}\tanh \displaystyle \frac{b^{\prime} }{2}\right]\end{eqnarray} \tag{ A.6 }$$

Next, accounting for the phonon energy _q'':

$$\begin{eqnarray}&&{g}^{0}({E}_{k}-{\epsilon }_{q^{\prime\prime} })\approx {g}^{0}({E}_{k})+{g}^{0}({E}_{k})[1-{g}^{0}({E}_{k})]\beta {\epsilon }_{q^{\prime\prime} }.\end{eqnarray} \tag{ A.7 }$$

This is used with some algebra to give:

$$\begin{eqnarray}\begin{array}{rcl}{f}_{1\gt }+{f}_{2\gt } & = & {\hslash }\displaystyle \int \displaystyle \int \displaystyle \int \displaystyle \int {g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0}){n}_{{\boldsymbol{q}}}^{0}(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0})({a}_{1\gt }+{a}_{2\gt }){A}_{{\boldsymbol{k}},{\boldsymbol{q}}}^{{{\boldsymbol{k}}}^{{\prime} },{{\boldsymbol{q}}}^{{\prime} }}{q}_{x}{\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{\boldsymbol{dq}}{{\boldsymbol{dq}}}^{{\prime} }\\ & & -{\hslash }\displaystyle \int \displaystyle \int \displaystyle \int \displaystyle \int {\left[{g}_{{\boldsymbol{k}}}^{0}\right]}^{2}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0})\beta {\epsilon }_{q^{\prime\prime} }{n}_{{\boldsymbol{q}}}^{0}(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0})({a}_{1\gt }+{a}_{2\gt }){A}_{{\boldsymbol{k}},{\boldsymbol{q}}}^{{{\boldsymbol{k}}}^{{\prime} },{{\boldsymbol{q}}}^{{\prime} }}{q}_{x}{\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{\boldsymbol{dq}}{{\boldsymbol{dq}}}^{{\prime} },\end{array}\end{eqnarray} \tag{ A.8 }$$

where the arrows on the integrals are now dropped. With some more algebra the first four force terms add to:

$$\begin{eqnarray}&&\begin{array}{l}{f}_{1\gt }+{f}_{2\gt }+{f}_{1\lt }+{f}_{2\lt }=-2{\hslash }{\gamma }_{2}{\beta }^{2}\displaystyle \int \displaystyle \int \displaystyle \int \displaystyle \int {g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0}){n}_{{\boldsymbol{q}}}^{0}(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0}){E}_{k}({E}_{k}-\mu )\\ \,\ \times \ ({\cos }^{2}\theta -{\cos }^{2}\theta ^{\prime} ){A}_{{\boldsymbol{k}},{\boldsymbol{q}}}^{{{\boldsymbol{k}}}^{{\prime} },{{\boldsymbol{q}}}^{{\prime} }}{q}_{x}{\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{\boldsymbol{dq}}{{\boldsymbol{dq}}}^{{\prime} }+2{\hslash }{\gamma }_{2}{\beta }^{2}\displaystyle \int \displaystyle \int \displaystyle \int \displaystyle \int {g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0}){n}_{{\boldsymbol{q}}}^{0}(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0})\\ \,\ \times \ ({\epsilon }_{q^{\prime\prime} }^{2}{\cos }^{2}\theta ^{\prime} ){A}_{{\boldsymbol{k}},{\boldsymbol{q}}}^{{{\boldsymbol{k}}}^{{\prime} },{{\boldsymbol{q}}}^{{\prime} }}{q}_{x}{\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{\boldsymbol{dq}}{{\boldsymbol{dq}}}^{{\prime} }+2{\hslash }{\gamma }_{2}{\beta }^{3}\displaystyle \int \displaystyle \int \displaystyle \int \displaystyle \int {\left[{g}_{{\boldsymbol{k}}}^{0}\right]}^{2}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0}){n}_{{\boldsymbol{q}}}^{0}(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0})\\ \,\ \times \ ({E}_{k}-\mu ){\epsilon }_{q^{\prime\prime} }^{2}{\cos }^{2}\theta ^{\prime} {A}_{{\boldsymbol{k}},{\boldsymbol{q}}}^{{{\boldsymbol{k}}}^{{\prime} },{{\boldsymbol{q}}}^{{\prime} }}{q}_{x}{\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{\boldsymbol{dq}}{{\boldsymbol{dq}}}^{{\prime} }.\end{array}\end{eqnarray} \tag{ A.9 }$$

For the force terms f₃ and f₄, ${E}_{k^{\prime} }={E}_{k}-{\epsilon }_{q^{\prime\prime} }$. From (3.7)

$$\begin{eqnarray}&&{g}_{{\boldsymbol{k}}}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }})(1+{n}_{{\boldsymbol{q}}}^{0})(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0})\approx {g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0})(1+{n}_{{\boldsymbol{q}}}^{0})(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0}){a}_{3},\end{eqnarray} \tag{ A.10 }$$

with

$$\begin{eqnarray}\begin{array}{rcl}{a}_{3} & = & 1+(1-{g}_{{\boldsymbol{k}}}^{0}){\gamma }_{2}\left[-1+{\beta }^{2}{E}_{k}({E}_{k}-\mu ){\cos }^{2}\theta \right]\\ & & -{\gamma }_{2}{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0}\left[-1+{\beta }^{2}{E}_{k}({E}_{k}-\mu ){\cos }^{2}\theta ^{\prime} \right]\\ & & +{\gamma }_{2}{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0}{\beta }^{2}\left[{\epsilon }_{q^{\prime\prime} }({E}_{k}-\mu )+{\epsilon }_{q^{\prime\prime} }{E}_{k}-{\epsilon }_{q^{\prime\prime} }^{2}\right]{\cos }^{2}\theta ^{\prime} .\end{array}\end{eqnarray} \tag{ A.11 }$$

From (3.8)

$$\begin{eqnarray}&&-{g}_{{{\boldsymbol{k}}}^{{\prime} }}(1-{g}_{{\boldsymbol{k}}}){n}_{{\boldsymbol{q}}}^{0}{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0}\approx -{g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0})(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0})(1+{n}_{{\boldsymbol{q}}}^{0}){a}_{4},\end{eqnarray} \tag{ A.12 }$$

with

$$\begin{eqnarray}\begin{array}{rcl}-{a}_{4} & = & 1-{g}_{{\boldsymbol{k}}}^{0}{\gamma }_{2}\left[-1+{\beta }^{2}{E}_{k}({E}_{k}-\mu ){\cos }^{2}\theta \right]\\ & & +{\gamma }_{2}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0})\left[-1+{\beta }^{2}{E}_{k}({E}_{k}-\mu ){\cos }^{2}\theta ^{\prime} \right]\\ & & +{\gamma }_{2}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0}){\beta }^{2}\left[-{\epsilon }_{q^{\prime\prime} }({E}_{k}-\mu )-{\epsilon }_{q^{\prime\prime} }{E}_{k}+{\epsilon }_{q^{\prime\prime} }^{2}\right]{\cos }^{2}\theta ^{\prime} .\end{array}\end{eqnarray} \tag{ A.13 }$$

Noting

$$\begin{eqnarray}\begin{array}{rcl}{a}_{3}+{a}_{4} & = & {\gamma }_{2}\ {\beta }^{2}{E}_{k}({E}_{k}-\mu )\left[{\cos }^{2}\theta -{\cos }^{2}\theta ^{\prime} \right]\\ & & +{\gamma }_{2}{\beta }^{2}{\epsilon }_{q^{\prime\prime} }\left[2{E}_{k}-\mu \right]{\cos }^{2}\theta ^{\prime} -{\gamma }_{2}{\beta }^{2}{\epsilon }_{q^{\prime\prime} }^{2}{\cos }^{2}\theta ^{\prime} ,\end{array}\end{eqnarray} \tag{ A.14 }$$

one concludes

$$\begin{eqnarray}&&{f}_{3}+{f}_{4}=\displaystyle \frac{{\hslash }}{2}\int \int \int \int {g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0})(1+{n}_{{\boldsymbol{q}}}^{0})(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0})({a}_{3}+{a}_{4}){A}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}},{{\boldsymbol{q}}}^{{\prime} }}{q}_{x}{\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{\boldsymbol{dq}}{{\boldsymbol{dq}}}^{{\prime} }.\end{eqnarray} \tag{ A.15 }$$

Bringing (A.9) and (A.15) together, f_1> + f_2> + f_1< + f_2< + f₃ + f₄ = f_ph,x with

$$\begin{eqnarray}\begin{array}{rcl}{f}_{{ph},x} & = & 2{\hslash }{\gamma }_{2}{\beta }^{2}\displaystyle \int \displaystyle \int \displaystyle \int \displaystyle \int {g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0}){n}_{{\boldsymbol{q}}}^{0}(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0})\\ & & \times \left\{{E}_{k}({E}_{k}-\mu )({\cos }^{2}\theta ^{\prime} -{\cos }^{2}\theta )+{\epsilon }_{q^{\prime\prime} }^{2}{\cos }^{2}\theta ^{\prime} +{g}_{{\boldsymbol{k}}}^{0}\beta ({E}_{k}-\mu ){\epsilon }_{q^{\prime\prime} }^{2}{\cos }^{2}\theta ^{\prime} \right\}\\ & & \times \ {A}_{{\boldsymbol{k}},{\boldsymbol{q}}}^{{{\boldsymbol{k}}}^{{\prime} },{{\boldsymbol{q}}}^{{\prime} }}{q}_{x}{\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{\boldsymbol{dq}}{{\boldsymbol{dq}}}^{{\prime} }+\displaystyle \frac{1}{2}{\hslash }{\gamma }_{2}{\beta }^{2}\displaystyle \int \displaystyle \int \displaystyle \int \displaystyle \int {g}_{{\boldsymbol{k}}}^{0}(1-{g}_{{{\boldsymbol{k}}}^{{\prime} }}^{0})(1+{n}_{{\boldsymbol{q}}}^{0})(1+{n}_{{{\boldsymbol{q}}}^{{\prime} }}^{0})\\ & & \times \ \left\{{E}_{k}({E}_{k}-\mu )\left[{\cos }^{2}\theta -{\cos }^{2}\theta ^{\prime} \right]+{\epsilon }_{q^{\prime\prime} }\left[2{E}_{k}-\mu \right]{\cos }^{2}\theta ^{\prime} -{\epsilon }_{q^{\prime\prime} }^{2}{\cos }^{2}\theta ^{\prime} \right\}\\ & & \times {A}_{{\boldsymbol{k}}}^{{{\boldsymbol{k}}}^{{\prime} },{\boldsymbol{q}},{{\boldsymbol{q}}}^{{\prime} }}{q}_{x}{\boldsymbol{dk}}{{\boldsymbol{dk}}}^{{\prime} }{\boldsymbol{dq}}{{\boldsymbol{dq}}}^{{\prime} }.\end{array}\end{eqnarray} \tag{ A.16 }$$