The electron-phonon (e-ph) interaction manifests itself in a wide range of phenomena, including the electrical resistivity, superconductivity, ¹ the Kohn effect, ² the Peierls instability, ³ and polaronic transport in organic materials. ⁴ In recent years, considerable attention has been drawn to the e-ph problem by photoemission experiments which revealed sharp signatures of this interaction in the quasiparticle spectra of high-temperature cuprate superconductors.^5,6 In addition, recently, the effect of the e-ph interaction in nanoscale electron transport has attracted considerable interest since there is evidence that phonon-limited carrier lifetimes might hinder the ballistic operation of carbon-based field-effect devices. ⁷
电子-声子 (e-ph) 相互作用表现在多种现象中，包括电阻率、超导性、 ¹ Kohn 效应、 ² Peierls 不稳定性、 ³和有机材料中的极化子输运。 ⁴近年来，光电发射实验引起了人们对 e-ph 问题的极大关注，这些实验揭示了高温铜酸盐超导体准粒子光谱中这种相互作用的鲜明特征。 ^{5, 6}此外，最近，纳米级电子传输中的 e-ph 相互作用的影响引起了人们极大的兴趣，因为有证据表明声子有限的载流子寿命可能会阻碍碳基场效应器件的弹道操作。 ⁷

The e-ph interaction has been the subject of theoretical investigations since the early attempts to explain the temperature dependence of the electrical resistivity of metals. ⁸ Following the pioneering investigations of Fröhlich, ⁹ Holstein, ¹⁰ Bardeen and Pines, ¹¹ and later the BCS theory of superconductivity, ¹ the e-ph interaction has become the prototypical example of a fermion-boson interaction and is now used as a standard benchmark for field-theoretical Green’s functions methods. ^12–15 Despite the continued interest in the e-ph problem, the computational methods developed so far, ranging from frozen-phonon approaches ^16–18 to first-principles linear-response techniques, ^19–22 still remain unpractical. As a consequence, many important aspects, such as the effects of anisotropy within the Eliashberg theory,^23,24 the validity of the Migdal theorem in the normal state, ²⁵ and the range of validity of the Migdal-Eliashberg theory in the superconducting state, ²⁶ remain only partially explored. In some cases, such as the Holstein polaron problem, ¹⁵ a first-principles approach has not even been attempted to the authors’ knowledge. The present situation is equally unsatisfactory from the point of view of applications, since current calculations are still limited to simple systems with a few atoms per unit cell, and only very few attempts have been made to address complex systems such as carbon nanostructures, ²⁷ doped superconductors, ²⁸ or metallic nanowires. ²⁹
自从早期尝试解释金属电阻率的温度依赖性以来，e-ph 相互作用一直是理论研究的主题。 ⁸继 Fröhlich、 ⁹ Holstein、 ¹⁰ Bardeen 和 Pines、 ¹¹以及后来的 BCS 超导理论的开创性研究之后， ¹ e-ph 相互作用已成为费米子-玻色子相互作用的典型例子，现在被用作标准基准用于场论格林函数方法。 ^12–15尽管人们对 e-ph 问题持续感兴趣，但迄今为止计算方法的发展，包括冻结声子方法^16-18线性响应技术的第一原理， ^19-22仍然不切实际。因此，许多重要方面，例如 Eliashberg 理论中各向异性的影响， ^{23、24正常}状态下 Migdal 定理的有效性， ²⁵以及超导状态下 Migdal-Eliashberg 理论的有效性范围， ^{26 个}仍处于部分探索状态。在某些情况下，例如荷斯坦极化子问题， ¹⁵据作者所知，甚至还没有尝试过第一性原理方法。 从应用的角度来看，目前的情况同样不能令人满意，因为当前的计算仍然仅限于每个晶胞只有几个原子的简单系统，并且只有很少的尝试来解决复杂的系统，例如碳纳米结构， ²⁷掺杂超导体， ²⁸或金属纳米线。²⁹

This situation is partly due to the significant computational burden of an e-ph calculation, which often requires a very accurate description of electron and phonon scattering processes in the proximity of the Fermi surface.^30,31
这种情况部分是由于 e-ph 计算的巨大计算负担，这通常需要对费米表面附近的电子和声子散射过程进行非常准确的描述。 ^{30, 31}

Motivated by these considerations, we have developed a technique which makes use of Wannier functions to dramatically reduce the computational cost of an e-ph calculation. The basic idea is to exploit the localization of both electronic and lattice Wannier functions in order to compute only a limited set of electronic and vibrational states and e-ph matrix elements from first principles, and then using these results to obtain the corresponding quantities for arbitrary electron and phonon wave vectors by a generalized Fourier interpolation. In this way, it becomes possible to sample accurately the Brillouin zone at the computational cost of a standard phonon dispersion calculation. ³² Besides the significant computational advantage, the Wannier representation proves to be an ideal analytical tool for studying the e-ph interaction in terms of simplified tight-binding models (for the electrons) and force-constant models (for the phonons) while preserving the accuracy of a full first-principles calculation.
出于这些考虑，我们开发了一种技术，利用 Wannier 函数来显着降低 e-ph 计算的计算成本。基本思想是利用电子和晶格 Wannier 函数的局域性，以便根据第一原理仅计算一组有限的电子和振动状态以及 e-ph 矩阵元素，然后使用这些结果获得任意值的相应量通过广义傅里叶插值计算电子和声子波矢量。这样，就可以以标准声子色散计算的计算成本对布里渊区进行精确采样。 ³²除了显着的计算优势外，Wannier 表示法被证明是一种理想的分析工具，可用于研究简化的紧束缚模型（对于电子）和力常数模型（对于声子）的 e-ph 相互作用，同时保留完整第一性原理计算的准确性。

In order to illustrate our method, we present an application to boron-doped diamond. Superconductivity above liquid He temperature has recently been observed in B-doped diamond, ³³ and investigations are ongoing to explore the possibility of increasing 𝑇𝑐 beyond 10K by tuning sample preparation and doping treatments. ³⁴ Previous theoretical works showed that superconductivity in diamond is crucially linked to the presence of the B atoms.^28,35 Boron provides both the hole carriers participating in the supercurrent and the localized vibrations of the B⁢C4 tetrahedra responsible for the pairing field. To keep the focus on the methodology, in the present work, we prefer to adopt a simplified point of view, and we describe B-doped diamond by a virtual crystal model.^36,37
为了说明我们的方法，我们提出了掺硼金刚石的应​​用。最近在 B 掺杂金刚石中观察到高于液态 He 温度的超导性， ³³并且正在进行研究以探索增加超导性的可能性 𝑇𝑐 超过 10K 通过调整样品制备和掺杂处理。 ³⁴先前的理论工作表明，金刚石的超导性与 B 原子的存在密切相关。 ^{28, 35}硼提供参与超流的空穴载流子和局部振动 B⁢C4 四面体负责配对场。为了将注意力集中在方法上，在目前的工作中，我们倾向于采用简化的观点，并通过虚拟晶体模型来描述 B 掺杂金刚石。 ^{36, 37}

The present paper is organized as follows. In Sec. II, we review the current techniques to compute the electron and phonon self-energies arising from the e-ph interaction. In Sec. III, we introduce the electron and phonon Wannier functions and derive the e-ph matrix element in the Wannier representation. Section IV describes the generalized Wannier-Fourier interpolation of the e-ph matrix element and its practical implementation within a density-functional framework. In Sec. VI, we illustrate the theory by calculating the electron and phonon linewidths, the Eliashberg function, and the electron-phonon mass enhancement parameter of boron-doped diamond.
本文的结构如下。在秒。 II ，我们回顾了当前计算 e-ph 相互作用产生的电子和声子自能的技术。在秒。第三，我们引入了电子和声子Wannier函数并推导了Wannier表示中的e-ph矩阵元素。第四节描述了 e-ph 矩阵元素的广义 Wannier-Fourier 插值及其在密度泛函框架内的实际实现。在秒。第六，我们通过计算掺硼金刚石的电子和声子线宽、Eliashberg函数和电子-声子质量增强参数来说明该理论。

The present work extends and improves upon the method proposed in Ref. 35. In particular, in the present work, the electron and phonon coordinates are treated on the same footing, leading to a joint electron-phonon Wannier representation and a simultaneous electron-phonon Fourier interpolation. In the Appendix, we establish the connection with the procedure outlined in Ref. 35, and we discuss the relative merits of the two strategies.
目前的工作对参考文献中提出的方法进行了扩展和改进。 35 .特别是，在目前的工作中，电子和声子坐标被同等对待，从而产生联合电子-声子 Wannier 表示和同时电子-声子傅里叶插值。在附录中，我们建立了与参考文献中概述的程序的联系。 35 ，我们讨论了这两种策略的相对优点。

The formalism for addressing the e-ph interaction has been set by the seminal contributions of Fröhlich, ⁹ Bardeen and Pines, ¹¹ and Engelsberg and Schrieffer. ³⁸ The e-ph Hamiltonian derived in these studies is conveniently dealt with by standard Green’s functions techniques.^1,14 The interacting electron and phonon propagators are, in principle, fully determined through Dyson’s equation once the corresponding electron (𝛴) and phonon (𝛱) self-energy operators associated with the mutual interactions are known. Instead of reviewing the possible approximations to the self-energy operators, we focus here on the simplest one, which consists in replacing the dressed e-ph vertex 𝛤 by its bare counterpart 𝑔 (Fig. 1). ¹ This approximation is connected with the Born-Oppenheimer adiabatic theorem and is generally referred to as the Migdal approximation.^25,26 In this work, we replace the dressed electron Green’s function by the corresponding free propagator, thereby avoiding complications associated with self-consistency. On the other hand, the fully renormalized phonon frequencies, as obtained from density-functional calculations, will be adopted in the phonon propagator. This is generally considered to be a sensible approximation. ¹⁴
解决 e-ph 相互作用的形式主义是由 Fröhlich、 ⁹ Bardeen 和 Pines、 ¹¹以及 Engelsberg 和 Schrieffer 的开创性贡献确定的。 ³⁸这些研究中导出的 e-ph 哈密顿量可以通过标准格林函数技术方便地处理。 ^{1, 14}一旦相应的电子和声子传播体相互作用，原则上就可以通过戴森方程完全确定 (𝛴) 和声子 (𝛱) 与相互作用相关的自能算子是已知的。我们不回顾自能算子的可能近似，而是关注最简单的一种，其中包括替换修饰的 e-ph 顶点 𝛤 通过其裸露的对应物 𝑔 （图1 ）。 ¹该近似与玻恩-奥本海默绝热定理相关，通常称为 Migdal 近似。 ^{25, 26}在这项工作中，我们用相应的自由传播子代替了修饰电子格林函数，从而避免了与自洽相关的复杂性。另一方面，声子传播器将采用从密度泛函计算获得的完全重整化的声子频率。这通常被认为是一个合理的近似值。 ¹⁴

FIG. 1 如图。 1

The electron and phonon self-energies arising from the e-ph interaction (Fig. 1) read ¹²
e-ph 相互作用产生的电子和声子自能（图1 ）为¹²

where 𝐺0 and 𝐷 are the bare electron and the dressed phonon Green’s functions, respectively, 1=(𝐤,𝜔) the quadrimomentum in compressed notation ( 𝐤 being the wave vector and 𝜔 the energy), and 𝑔⁡(1,2) the electron-phonon matrix element. ³⁹ Equations (1) and (2) were originally derived for the electron gas and need to be rewritten within the reduced-zone scheme for a practical calculation.
在哪里 𝐺0 和 𝐷 分别是裸电子和修饰声子格林函数， 1=(𝐤,𝜔) 压缩符号中的四动量 ( 𝐤 是波矢量并且 𝜔 能量），以及 𝑔⁡(1,2) 电子声子矩阵元素。 ³⁹方程（1）和（2）最初是针对电子气推导出来的，需要在缩减区域方案中重写以进行实际计算。

We make the following approximations: (i) we neglect the changes in the electronic wave functions and phonon eigendisplacements arising from the e-ph interaction, ⁴⁰ (ii) we take the expectation value of the self-energy operators on the noninteracting electron and phonon states, and (iii) we restrict our discussion to the imaginary parts of the electron (𝛴″) and phonon (𝛱″) self-energies, i.e., we only consider the corresponding linewidths. These simplifications are intended to illustrate our methodology by focusing on a few specific cases. The inclusion of the off-diagonal corrections and the calculation of the corresponding real self-energies are both feasible and will be the subject of a future communication. ⁴¹ By expressing the free propagators in terms of the noninteracting electronic energy 𝜖𝑛⁢𝐤 (with 𝑛 the band index and 𝐤 the momentum) and vibrational frequency 𝜔𝐪⁢𝜈 (with 𝜈 the branch index and 𝐪 the momentum), Eqs. (1) and (2) can be integrated analytically to yield ¹²
我们进行以下近似：（i）我们忽略由 e-ph 相互作用引起的电子波函数和声子本征位移的变化， ⁴⁰ （ii）我们取非相互作用电子和声子的自能算子的期望值状态，并且（iii）我们将讨论限制在电子的虚部 (𝛴″) 和声子 (𝛱″) 自能，即我们只考虑相应的线宽。这些简化旨在通过关注一些具体案例来说明我们的方法。包含非对角线修正和相应的真实自能的计算都是可行的，并将成为未来交流的主题。 ⁴¹通过用非相互作用电子能量来表达自由传播子 𝜖𝑛⁢𝐤 （和 𝑛 能带指数和 𝐤 动量）和振动频率 𝜔𝐪⁢𝜈 （和 𝜈 分支索引和 𝐪 动量），方程。 (1)和(2)可以通过分析积分得到¹²

where 𝑓𝑛⁢𝐤 and 𝑛𝐪⁢𝜈 are the Fermi-Dirac and Bose-Einstein occupations, respectively, the factor of 2 accounts for the spin degeneracy, and the integrations extend over the Brillouin zone. The e-ph matrix element (vertex) 𝑔SE𝑚⁢𝑛,𝜈⁡(𝐤,𝐪) appearing in Eqs. (3) and (4) is given by
在哪里 𝑓𝑛⁢𝐤 和 𝑛𝐪⁢𝜈 分别是费米-狄拉克和玻色-爱因斯坦占据，因子 2 解释了自旋简并性，并且积分延伸到布里渊区。 e-ph 矩阵元素（顶点） 𝑔SE𝑚⁢𝑛,𝜈⁡(𝐤,𝐪) 出现在方程式中。 (3)和(4)由下式给出

where 𝑚0 is a convenient reference mass, and
在哪里 𝑚0 是一个方便的参考质量，并且

We adopt the superscript “SE” to distinguish between the matrix element appearing in the self-energy expressions ( 𝑔SE, with the dimensions of an energy) and the one used in the Bloch to Wannier transformation ( 𝑔, with dimensions of an energy divided by a length, cf. Sec. III). In Eq. (6), ∣𝑛⁢𝐤⟩ and ∣𝑚⁢𝐤+𝐪⟩ indicate Bloch eigenstates, while the operator 𝜕𝐪⁢𝜈𝑉 is the derivative of the self-consistent potential with respect to a collective ionic displacement corresponding to a phonon with branch index 𝜈 and momentum 𝐪 (cf. Sec. III). ⁴²
我们采用上标“SE”来区分自能表达式中出现的矩阵元素（ 𝑔SE ，具有能量的维度）以及布洛赫到万尼尔变换中使用的能量（ 𝑔 ，能量的维度除以长度，参见。秒。三）。在等式中。 (6) , ∣𝑛⁢𝐤⟩ 和 ∣𝑚⁢𝐤+𝐪⟩ 指示布洛赫本征态，而算子 𝜕𝐪⁢𝜈𝑉 是自洽势相对于对应于具有分支指数的声子的集体离子位移的导数 𝜈 和势头 𝐪 （参见第III节）。 ⁴²

Within the isotropic approximation to Eliashberg theory, ²⁶ the phonon linewidths [Eq. (4)] constitute a key ingredient for calculating the Eliashberg function 𝛼2⁢𝐹 and the associated mass enhancement parameter 𝜆 (Ref. 43):
在 Eliashberg 理论的各向同性近似中， ²⁶声子线宽 [Eq. (4) ] 构成计算 Eliashberg 函数的关键要素 𝛼2⁢𝐹 以及相关的质量增强参数 𝜆 （参考文献43 ）：

𝑁𝐹 being the density of electronic states at the Fermi level. The mass enhancement parameter 𝜆 is also referred to as the electron-phonon coupling strength and is widely used to estimate the transition temperature of conventional superconductors by means of the semiempirical McMillan formula. ⁴⁴
𝑁𝐹 是费米能级的电子态密度。质量增强参数 𝜆 也称为电子-声子耦合强度，广泛用于通过半经验麦克米伦公式来估计传统超导体的转变温度。 ⁴⁴

Inspection of Eq. (4) reveals that the calculation of the phonon linewidths requires a summation over electronic transitions where both the initial state ∣𝑛⁢𝐤⟩ and the final state ∣𝑚⁢𝐤+𝐪⟩ are pinned close to the Fermi level by the selection rule 𝜖𝑚⁢𝐤+𝐪=𝜖𝑛⁢𝐤+𝜔𝐪⁢𝜈. As a consequence, the calculation of vibrational linewidths requires a determination of the initial and final electronic states, the phonon states of interest, and the associated e-ph matrix elements with fine energy and momentum resolutions. Whereas total-energy calculations for metals usually require at most a few tens of 𝐤 points in the irreducible wedge of the Brillouin zone, in the present case, a much denser sampling is required to achieve numerical convergence, even up to several millions of 𝐤 points.^30,45,46
检查方程。 (4)表明声子线宽的计算需要对电子跃迁求和，其中初始态 ∣𝑛⁢𝐤⟩ 和最终状态 ∣𝑚⁢𝐤+𝐪⟩ 通过选择规则固定在费米能级附近 𝜖𝑚⁢𝐤+𝐪=𝜖𝑛⁢𝐤+𝜔𝐪⁢𝜈 。因此，振动线宽的计算需要确定初始和最终电子态、感兴趣的声子态以及具有精细能量和动量分辨率的相关 e-ph 矩阵元素。而金属的总能量计算通常最多需要几十 𝐤 布里渊区不可约楔中的点，在目前的情况下，需要更密集的采样才能实现数值收敛，甚至达到数百万个 𝐤 点。 ^{30、45、46​​}

Similar considerations apply to the calculations of the electron self-energy [Eq. (3)], to the Eliashberg function [Eq. (7)], and to the mass enhancement parameter [Eq. (8)]. In particular, the difficulty in the determination of the mass enhancement parameter translates into a large uncertainty in the calculated superconducting transition temperature through the McMillan equation. ³⁵ Even within more sophisticated approaches where superconducting properties are determined directly from first principles, ⁴⁷ the practical implementations suffer from a strong sensitivity to the sampling of e-ph scattering processes near the Fermi surface. ³¹ In the following sections, we describe how it is possible to circumvent such difficulties by reformulating the e-ph vertex in the Wannier representation.
类似的考虑也适用于电子自能的计算[方程1]。 (3) ], 到 Eliashberg 函数 [Eq. (7) ], 以及质量增强参数 [Eq. (8) ]。特别是，质量增强参数确定的困难转化为通过麦克米伦方程计算的超导转变温度存在很大的不确定性。 ³⁵即使在超导特性直接根据第一原理确定的更复杂的方法中， ⁴⁷实际实现也会受到对费米表面附近 e-ph 散射过程采样的强烈敏感性的影响。 ³¹在以下各节中，我们将描述如何通过在 Wannier 表示中重新表述 e-ph 顶点来规避此类困难。

In this section, we introduce the Wannier representation of the e-ph vertex. We first describe the electronic Wannier functions and the phonon perturbation potential in the Wannier representation. Then, we derive the relation between the e-ph matrix elements in the Wannier representation and those in the Bloch representation.
在本节中，我们介绍 e-ph 顶点的 Wannier 表示。我们首先描述电子 Wannier 函数和 Wannier 表示中的声子微扰势。然后，我们推导了Wannier表示中的e-ph矩阵元素与Bloch表示中的e-ph矩阵元素之间的关系。