3.40 Electron-Phonon Coupling and Electronic Friction

A common practical expression for the vibrational linewidth of a phonon, ${\gamma }_{𝐪\nu }$, of frequency ${\omega }_{𝐪\nu }$, induced by first-order EPC is defined in Eq 3.165:

${\gamma }_{𝐪\nu }(\mathrm{\hslash }{\omega }_{𝐪\nu })=$

$2\pi {\displaystyle \sum _{\sigma mn}}{\displaystyle \int \frac{d𝐤}{{\mathrm{\Omega }}_{BZ}}{|g_{mn\nu }(𝐤,𝐪)|}^2(f_{n𝐤}-f_{m𝐤+𝐪})\delta ({ϵ}_{m𝐤+𝐪}-{ϵ}_{n𝐤}-\mathrm{\hslash }{\omega }_{𝐪\nu })}$

(3.165)

This expression can be obtained from the imaginary part of the first-order nonadiabatic phonon self-energy [105, 232] or can be derived from Fermi’s golden rule. [4, 214] The EPC matrix elements, $g$ are electronically screened, see Ref [105] for more information. The Fermi occupation factors for a given state $f({ϵ}_{n𝐤})$ (written as $f_{n𝐤}$) describe the electronic temperature. Eq 3.165 covers only interband excitations for the case of $𝐪=0$ ($m>n$) and interband and direct intraband excitations for the case of $𝐪\ne 0$ ($m\ge n$).

The matrix elements that feature in Eq 3.165 have units of energy and are defined as follows:

$$g_{mn\nu }(𝐤,𝐪)={\left(\frac{\mathrm{\hslash }}{2M_{\nu }{\omega }_{𝐪\nu }}\right)}^{1/2}{\tilde{g}}_{mn\nu }(𝐤,𝐪),$$

(3.166)

with

$${\tilde{g}}_{mn\nu }(𝐤,𝐪)=⟨m𝐤+𝐪|{\partial }_{𝐪\nu }V|n𝐤⟩.$$

(3.167)

The EPC matrix elements in Eq 3.167 have units of energy per length. They describe the excitation of an electron from a state $n𝐤$ to a state $m𝐤+𝐪$ by absorption of a phonon $𝐪\nu$. $V$ in Eq 3.167 is the self-consistent ("screened") effective potential from a Kohn-Sham DFT calculation. Its derivative with respect to atomic displacement ${\partial }_{𝐪\nu }V$, is defined as

$${\partial }_{𝐪\nu }V=\sum _{a\kappa }{u}_{𝐪\nu }^{a\kappa }\frac{\partial }{\partial R_{𝐪,a\kappa }}V(𝐫).$$

(3.168)

Eq 3.168 introduces the elements of the phonon displacement eigenmode $u_{𝐪\nu }^{a\kappa }$ associated with phonon $𝐪\nu$. Here, $a$ is an atom index and $\kappa$ indicates the 3 Cartesian directions $x$, $y$, and $z$. Note that we do not include mass weighting as it is considered in Eq 3.166 via $M_{\nu }$. By inserting Eq 3.168 into Eq 3.167 we find

$${\tilde{g}}_{mn\nu }(𝐤,𝐪)=\sum _{a\kappa }{u}_{𝐪\nu }^{a\kappa }⟨m𝐤+𝐪|\frac{\partial }{\partial R_{𝐪,a\kappa }}V(𝐫)|n𝐤⟩={𝐮}_{𝐪\nu }\cdot {\widetilde{𝐠}}_{mn}(𝐤,𝐪).$$

(3.169)

On the right-hand side of Eq 3.169 we introduce a vector notation to denote the multiplication of the normal mode displacement vector ${𝐮}_{𝐪\nu }$ with the vector of atomic contributions to the perturbation potential ${\widetilde{𝐠}}_{mn}(𝐤,𝐪)$.

The continuum of electronic states in metals can already be excited by the vibrational motion of the adsorbate atoms due to resonance at similar energy scales. As a result adsorbate atoms exhibit additional frictional forces. Another way to view this is that the vibrations or phonons are being screened by electronic excitations giving them a finite lifetime. Assuming that the ground and excited state PES are parallel and that these adiabatic effects are only weakly perturbing the adsorbate nuclear motion, we can apply perturbation theory to this problem.

The picture in Fig. 3.11 holds if:

• •

  the coupling is weak compared to the individual contributions of electrons and nuclei, and if

• •

  electron-hole pair excitations do not lead to a qualitative change in the nuclear dynamics.

One method to go beyond the Born-Oppenheimer approximation is the molecular dynamics with electronic friction (MDEF) method, by including the effect of electron-hole pair excitations on nuclear dynamics through inclusion of a friction force. In practice, MDEF is always employed by taking the Markov limit, where the memory of the system is ignored by assuming instantaneous response, as recited here from Ref [129]:

$$M{\ddot{R}}_{a\kappa }=-\frac{\partial V(𝑹)}{\partial R_{a\kappa }}-\sum _{a^{\prime }{\kappa }^{\prime }}{\mathrm{\Lambda }}_{a\kappa ,a^{\prime }{\kappa }^{\prime }}{\dot{R}}_{a^{\prime }{\kappa }^{\prime }}+{ℛ}_{a\kappa }(t).$$

(3.170)

The electronic friction tensor ($𝚲$) in Eq 3.170 does not have an explicit time dependence, but rather an implicit one. Expressions for the electronic friction tensor are given in the next section. $ℛ$ is a force associated with random white noise from the bath of electrons:

$$⟨{ℛ}_{a\kappa }(t){ℛ}_{a^{\prime }{\kappa }^{\prime }}(0)⟩=k_{\mathrm{B}}T{\mathrm{\Lambda }}_{a\kappa ,a^{\prime }{\kappa }^{\prime }}\delta (t-t^{\prime }).$$

(3.171)

Below, we show several further approximations that can be used to evaluate $𝚲$ to employ MDEF for gas-surface dynamics with atomistic simulations with electronic structure calculated at the DFT level.

We find an expression for the Cartesian friction tensor by inserting Eq 3.166 and Eq 3.169 into Eq 3.165 and we bring the normal mode displacement vectors ${𝐮}_{𝐪\nu }$ outside of the sum over states,

$${\gamma }_{𝐪\nu }=\mathrm{\hslash }{\mathrm{\Gamma }}_{𝐪\nu }=\mathrm{\hslash }\left[{\widetilde{𝐮}}_{𝐪\nu }{𝚲}^{𝐪}(\mathrm{\hslash }{\omega }_{𝐪}){\widetilde{𝐮}}_{𝐪\nu }^T\right],$$

(3.172)

and define the mass-weighted normal mode displacement vectors ${\widetilde{𝐮}}_{𝐪\nu }={𝐮}_{𝐪\nu }/\sqrt{M_{\nu }}$. Herein, we have defined the Cartesian friction tensor [214]

	${\mathrm{\Lambda }}_{a\kappa ,a^{\prime }{\kappa }^{\prime }}^{𝐪,\mathrm{Ia}}(\mathrm{\hslash }{\omega }_{𝐪})=$	$\pi \mathrm{\hslash }{\displaystyle \sum _{\sigma mn}}{\displaystyle \int \frac{d𝐤}{{\mathrm{\Omega }}_{BZ}}{\tilde{g}}_{mn}^{a\kappa }(𝐤,𝐪){({\tilde{g}}_{mn}^{a^{\prime }{\kappa }^{\prime }})}^{\ast }(𝐤,𝐪)}$		(3.173)
		$(f_{n𝐤}-f_{m𝐤+𝐪}){\displaystyle \frac{\delta ({ϵ}_{m𝐤+𝐪}-{ϵ}_{n𝐤}-\mathrm{\hslash }{\omega }_{𝐪})}{\mathrm{\hslash }{\omega }_{𝐪}}}.$

A similar result for the phonon linewidth can be generated on the basis of atomic response correlation functions. [50, 129] Additionally, we define a related expression for the Cartesian friction tensor by assuming $\mathrm{\hslash }{\omega }_{𝐪}$ is equivalent to the eigenvalue energy difference in the denominator of the right hand term:

	${\mathrm{\Lambda }}_{a\kappa ,a^{\prime }{\kappa }^{\prime }}^{𝐪,\mathrm{Ib}}(\mathrm{\hslash }{\omega }_{𝐪})=$	$\pi \mathrm{\hslash }{\displaystyle \sum _{\sigma mn}}{\displaystyle \int \frac{d𝐤}{{\mathrm{\Omega }}_{BZ}}{\tilde{g}}_{mn}^{a\kappa }(𝐤,𝐪){({\tilde{g}}_{mn}^{a^{\prime }{\kappa }^{\prime }})}^{\ast }(𝐤,𝐪)}$		(3.174)
		$(f_{n𝐤}-f_{m𝐤+𝐪}){\displaystyle \frac{\delta ({ϵ}_{m𝐤+𝐪}-{ϵ}_{n𝐤}-\mathrm{\hslash }{\omega }_{𝐪})}{{ϵ}_{m𝐤+𝐪}-{ϵ}_{n𝐤}}}.$

This expression is the one employed by default within the code, and was shown to be the most stable and general. [40] We can follow a similar derivation starting from the low temperature double delta expression by Allen [5] and arrive at:

$${\mathrm{\Lambda }}_{a\kappa ,a^{\prime }{\kappa }^{\prime }}^{𝐪,\mathrm{II}}=\pi \mathrm{\hslash }\sum _{\sigma mn}\int \frac{d𝐤}{{\mathrm{\Omega }}_{BZ}}{\tilde{g}}_{mn}^{a\kappa }(𝐤,𝐪){({\tilde{g}}_{mn}^{a^{\prime }{\kappa }^{\prime }})}^{\ast }(𝐤,𝐪)\delta ({ϵ}_{n𝐤}-{ϵ}_{\mathrm{F}})\delta ({ϵ}_{m𝐤+𝐪}-{ϵ}_{\mathrm{F}}).$$

(3.175)

These expressions of electronic friction are generalizations of the expressions defined by Maurer et al [214] for arbitrary $𝐪$ values and both have units of [mass/time] or [(energy$\times$time)/length²] or [action/length²].

The EPC matrix elements are provided by the following expression in a local orbital basis:

	${\tilde{g}}_{mn}^{a\kappa }(𝐤,𝐪)=$	${\displaystyle \sum _{ij}}{\left(C_{m𝐤+𝐪}^j\right)}^{\ast }{C}_{n𝐤}^i$		(3.176)
		$\left(H_{ij}^{a\kappa ,(1)}(𝐤,𝐪)-{ϵ}_{n𝐤}{S}_{ij}^{a\kappa ,L}(𝐤,𝐪)-{ϵ}_{m𝐤+𝐪}{S}_{ij}^{a\kappa ,R}(𝐤,𝐪)\right).$

Comparing with the findings of Maurer et al [214], we can relate the potential derivative (deformation potential matrix elements) and nonadiabatic coupling matrix elements as follows:

$$⟨{\psi }_{m𝐤+𝐪}|\frac{\partial V}{\partial R_{𝐪,a\kappa }}|{\psi }_{n,𝐤}⟩=({ϵ}_{m𝐤+𝐪}-{ϵ}_{n𝐤})\cdot ⟨{\psi }_{m𝐤+𝐪}|\frac{\partial }{\partial R_{𝐪,a\kappa }}|{\psi }_{n,𝐤}⟩.$$

(3.177)

As shorthand, we define a EPC matrix $G_{ij}^{a\kappa }(𝐤,𝐪)$ (contents of brackets in Eq 3.176) in local orbital representation as

$$G_{mn,ij}^{a\kappa }(𝐤,𝐪)=\left(H_{ij}^{a\kappa ,(1)}(𝐤,𝐪)-{ϵ}_{n𝐤}{S}_{ij}^{a\kappa ,L}(𝐤,𝐪)-{ϵ}_{m𝐤+𝐪}{S}_{ij}^{a\kappa ,R}(𝐤,𝐪)\right).$$

(3.178)

In previous works, for computational convenience, approximate versions of these matrix elements have been proposed by Head-Gordon and Tully [129, 214], where the dependence on individual electronic states is dropped and the linear response of the full overlap matrix is used instead:

$$G_{ij}^{a\kappa ,\mathrm{HGT}}(𝐤,𝐪)=\left(H_{ij}^{a\kappa ,(1)}(𝐤,𝐪)-{ϵ}_{\mathrm{F}}{S}_{ij}^{a\kappa ,(1)}(𝐤,𝐪)\right).$$

(3.179)

Maurer et al have assessed the impact of this approximation on vibrational relaxation rates for aperiodic systems and have found it to affect final predictions by only few percentage points. [214] An additional approximate EPC matrix was defined in the same work:

$$G_{mn,ij}^{a\kappa ,\mathrm{AVE}}(𝐤,𝐪)=\left(H_{ij}^{a\kappa ,(1)}(𝐤,𝐪)-\frac{1}{2}({ϵ}_{n𝐤}-{ϵ}_{m𝐤+𝐪})S_{ij}^{a\kappa ,(1)}(𝐤,𝐪)\right).$$

(3.180)

For lattice periodic ($𝐪=0$) perturbations in metallic systems (specifically where fractional occupations are present) the response of the Fermi level must additionally be accounted for:

$${\tilde{g}}_{mm}^{a\kappa }(𝐤,0)=\sum _{ij}{\left(C_{m𝐤}^j\right)}^{\ast }{C}_{m𝐤}^i\left(H_{ij}^{a\kappa ,(1)}(𝐤,0)-{ϵ}_{m𝐤}{S}_{ij}^{a\kappa ,(1)}(𝐤,0)\right)-{ϵ}_{\mathrm{F}}^{a\kappa ,(1)}.$$

(3.181)

This applies only to intraband elements, and thus is only relevant for Allen’s formula (Eq 3.175) where $𝐪=0$ intraband elements contribute. Intraband elements do not contribute to the single delta expression of type Eq 3.174 which tends to $0$ as $\mathrm{\hslash }\omega \to 0$.

The numerical evaluation of the friction tensor involves either two delta functions (low temperature limit, Eq 3.175) or a single delta function (Eq 3.173). These delta functions cannot be reliably evaluated directly for ab initio calculations due to the discrete nature of the density of states represented on a finite $𝐤$-point mesh. Typically the delta function(s) are replaced by a broadening function of finite width, the most common choice is a Gaussian function:

$$\widehat{\delta }\left({ϵ}_i-{ϵ}_j\right)=\frac{1}{\sqrt{2\pi }\sigma }\exp \left\{\frac{-{\left({ϵ}_i-{ϵ}_j\right)}^2}{2{\sigma }^2}\right\},$$

(3.182)

other choices include a Lorentzian broadening function which can be defined as follows:

$${\widehat{\delta }}_{\mathrm{Lorentz}.}\left({ϵ}_i-{ϵ}_j\right)=\frac{1}{\pi }\frac{\frac{1}{2}\sigma }{{\left({ϵ}_i-{ϵ}_j\right)}^2+{\left(\frac{1}{2}\sigma \right)}^2}$$

(3.183)

An additional normalization technique was previously proposed for single delta function calculations when relatively large smearing widths are applied. [214] This was later found to affect the magnitude of electronic friction without increasing stability and so is no longer recommended. [40]

$$\tilde{\delta }\left({ϵ}_i-{ϵ}_j\right)=\frac{\widehat{\delta }\left({ϵ}_i-{ϵ}_j\right)}{{\int }_0^{\mathrm{\infty }}\widehat{\delta }\left[ϵ-\left({ϵ}_i-{ϵ}_j\right)\right]𝑑ϵ}=\frac{\widehat{\delta }\left({ϵ}_i-{ϵ}_j\right)}{\frac{1}{2}\left[1-\mathrm{erf}\left(\frac{{ϵ}_j-{ϵ}_i}{\sqrt{2}\sigma }\right)\right]}.$$

(3.184)

The choice of smearing parameter can have a large effect on the vibrational linewidths and electronic friction tensor [214, 233], by including contributions from a wider range of electron-hole pair excitations. For the calculation of vibrational linewidths, the delta function(s) are typically evaluated within the weak EPC limit, whereby $\sigma$ is chosen to be as small as possible whilst being converged with respect to computationally feasible $𝐤$-meshes. However, MDEF has been most commonly applied to the simulation of atomic/molecular beam scattering experiments [48, 284], where impinging atoms and molecules with high translational energy and possible high vibrational and rotational excitation are fired at metallic surfaces. For this case, the appropriate choice of $\sigma$ is unclear, as the coupling can now be much stronger than when calculating vibrational linewidths. A common choice of $\sigma$ for evaluating electronic friction is within $0.1-0.6$ eV. [300, 214, 284].

The electron-phonon/electronic friction calculation in FHI-aims involves following steps:

1. 1.

   SCF calculation.

2. 2.

   Finite-difference or DFPT calculation of electron-phonon coupling matrix elements. Optionally, these matrix elements can be read from files.

3. 3.

   Calculate the electronic friction tensor from electronic structure data.

The module currently allows to:

• •

  calculate the nonadiabatic relaxation rate tensor in the quasi-static limit using finite-difference matrix elements for cluster and periodic systems. The latter only works for the Gamma point ($𝐪=0$).

• •

  atomwise output of Gamma point ($𝐪=0$) EPC matrix elements

Additionally the keyword empty_states should be set to a large number, or the keyword calculate_all_eigenstates should be used, to make sure the code generates sufficient unoccupied states from the basis set. This number will also be limited automatically by the code to the maximum number that can be generated from the basis set.