Crossover from lattice to plasmonic polarons of a spin-polarised electron gas in ferromagnetic EuO
铁磁 Eu2O 中自旋极化电子气从晶格到等离子体极化子的交叉

A pronounced electron−phonon coupling in solids is known to mediate the formation of polarons—composite quasiparticles of an electron dressed with a phonon cloud¹. Polarons exhibit significantly enhanced quasiparticle masses that modify charge carrier transport and are proposed to play a key role in unconventional superconducting and colossal magnetoresistive states in compounds including high-T_c cuprates^2,3, SrTiO ₃ -based electron gases^4,5,6,7,8, manganites^9,10 and superconducting monolayer FeSe¹¹. Developing control over polaronic states in solids therefore holds exciting potential for manipulating the collective states of quantum materials. Phonons, however, are typically only weakly modified for experimentally accessible tuning parameters. In contrast, we demonstrate in this work that polarons which are formed via a coupling to collective plasmon excitations of an electron gas provide a highly tuneable system.
众所周知，固体中明显的电子-声子耦合会介导极化子的形成，极化子是由声子云包裹的电子的复合准粒子¹ 。极化子表现出显着增强的准粒子质量，可改变载流子传输，并被认为在包括高温_铜酸盐^{2 , 3} , SrTiO ₃基电子气^{4 , 5 , 6在内的化合物中的非常规超导和巨磁阻态中发挥关键作用。 7、8 ，}亚锰酸盐^9、10和超导单层FeSe ¹¹ 。因此，开发对固体中极化子态的控制对于操纵量子材料的集体态具有令人兴奋的潜力。然而，声子通常仅针对实验上可访问的调谐参数进行微弱修改。相比之下，我们在这项工作中证明，通过与电子气的集体等离子体激发耦合形成的极化子提供了高度可调的系统。

We investigate this in the doped ferromagnet EuO. Stoichiometric EuO is insulating, with a Curie temperature of T_c = 69 K. In an ionic picture, Eu²⁺ has seven electrons half-filling the Eu 4f shell. These unpaired electrons align according to Hund’s rule, producing a large energetic splitting between occupied and unoccupied Eu 4f states via strong onsite Coulomb interactions, U (Fig. 1a). The temperature-dependent magnetisation follows an almost perfect Brillouin function¹², reaching a saturation magnetisation of 7 μ_B/Eu. EuO is thus often described as an almost ideal manifestation of a Heisenberg ferromagnet¹³. Oxygen reduction or atomic substitution of trivalent ions, for example Gd³⁺, for divalent Eu²⁺ dopes electrons into an Eu-derived 5d conduction band. This has a dramatic effect on its physical properties, increasing T_c¹⁴, stabilising a giant temperature-dependent metal−insulator transition with up to 13 orders of magnitude change in conductivity¹⁵, inducing giant magnetoresistance¹⁶, and realising a half-metallic phase, enabling almost 100% spin injection into Si and GaN¹⁷.
我们在掺杂铁磁体 EuO 中对此进行了研究。化学计量的EuO 是绝缘的，居里温度为T _c = 69 K。在离子图中，Eu ²⁺有七个电子半填充Eu 4 f壳层。这些不成对的电子根据洪德法则排列，通过强的现场库仑相互作用U ，在占据和未占据的Eu 4 f态之间产生大的能量分裂（图1a ）。与温度相关的磁化强度遵循几乎完美的布里渊函数¹² ，达到 7 μB / _Eu的饱和磁化强度。因此，EuO 通常被描述为海森堡铁磁体¹³的近乎理想的表现形式。氧还原或三价离子（例如Gd ³⁺ ）的原子取代二价Eu ²⁺将电子掺杂到Eu衍生的5d导带中。这对其物理性质产生了巨大影响，增加了T _c ¹⁴ ，稳定了巨大的温度依赖性金属-绝缘体转变，电导率变化高达 13 个数量级¹⁵ ，诱导巨磁阻¹⁶ ，并实现了半金属相，实现几乎 100% 自旋注入 Si 和 GaN ¹⁷ 。

Electronic structure of the ferromagnetic semiconductor EuO. a Schematic energy level diagram of EuO, with a half-filled Eu 4f band split by strong Coulomb interactions yielding a band gap between Eu 4f (5d) valence (conduction) bands. b DFT calculations of the electronic structure reproduce these general features and indicate the conduction band minimum (CBM) is located at the Brillouin zone face, X, point. c ARPES measurements ($h\nu$ = 48 eV) from a lightly Gd-doped sample (Eu_1−xGd _x O, x = 0.023) qualitatively match the DFT valence band dispersions. d The charge carrier doping additionally populates the spin-majority Eu 5d conduction-band state at the X-point (region shown by black box in (c))
铁磁半导体EuO的电子结构。 a EuO 的能级示意图，其中半填充的 Eu 4 f带被强库仑相互作用分裂，在 Eu 4 f (5 d ) 价带（导带）之间产生带隙。 b电子结构的 DFT 计算重现了这些一般特征，并表明导带最小值 (CBM) 位于布里渊区面 X 点。 c ARPES 测量（ $h\nu$ = 48 eV）来自轻掺杂 Gd 的样品 (Eu _{1− x} Gd _x O, x = 0.023) 定性地匹配 DFT 价带色散。 d电荷载流子掺杂另外填充 X 点处的自旋多数 Eu 5 d导带态（( c ) 中黑框所示的区域）

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We synthesise epitaxial films of Eu_1−xGd _x O via oxide molecular-beam epitaxy (MBE), allowing fine control over its charge carrier concentration, and transfer these in situ to a synchrotron-based angle-resolved photoemission spectroscopy (ARPES) system (see Methods). This provides a powerful opportunity to probe the electronic structure of this three-dimensional and air-sensitive compound. Our corresponding ARPES measurements, together with ab initio many-body calculations^18,19, reveal how charge carrier doping fundamentally changes the nature of the underlying electronic liquid in EuO. In particular, we directly image the emergence of well-defined satellites in the spectral function whose energy separation, unusually, grows as $\sqrt{n}$, where n is the free carrier density. This points to an intriguing polaronic state, arising due to the coupling of the induced charge carriers to the conduction-electron plasmon excitations.
我们通过氧化物分子束外延 (MBE) 合成了 Eu _{1− x} Gd _x O 外延薄膜，可以对其载流子浓度进行精细控制，并将这些薄膜原位转移到基于同步加速器的角分辨光电子能谱 (ARPES) 系统（参见方法）。这为探测这种三维空气敏感化合物的电子结构提供了绝佳的机会。我们相应的 ARPES 测量以及从头开始的多体计算^{18 、 19}揭示了电荷载流子掺杂如何从根本上改变 EuO 中基础电子液体的性质。特别是，我们直接对光谱函数中明确定义的卫星的出现进行成像，其能量分离异常地增长为 $\sqrt{n}$ ，其中n是自由载流子密度。这表明了一种有趣的极化状态，这是由于感应电荷载流子与传导电子等离子体激发的耦合而产生的。

Electronic structure of lightly doped EuO
轻掺杂EuO的电子结构

Figure 1 summarises the electronic structure of our Gd-doped EuO films. After incorporating an onsite Coulomb repulsion, density-functional theory calculations (DFT, see Methods) reproduce the generic features of the electronic structure described above, including its ferromagnetic nature. A majority-spin Eu 4f level is fully occupied, sitting above an O 2p-derived valence band that inherits a much weaker exchange splitting. Our ARPES measurements of the valence band dispersions from a lightly doped sample (x = 0.023, Fig. 1c) are in good general agreement with these DFT calculations, as well as with prior ARPES studies of the valence band electronic structure of EuO^20,21. For a weakly interacting semiconductor, charge carrier doping would simply induce a rigid shift of the Fermi level into the majority-spin Eu 5d conduction band, whose minimum is located at the X-point of the Brillouin zone (see also Supplementary Fig. 1). Consistent with previous experiment²¹, we indeed find that conduction-band states become populated at X with electron doping (Fig. 1d). Our measured spectra (evident in Fig. 1d and shown magnified in Fig. 2a), however, are not consistent with a simple rigid-band filling.
图1总结了我们的 Gd 掺杂 EuO 薄膜的电子结构。结合现场库仑斥力后，密度泛函理论计算（DFT，参见方法）再现了上述电子结构的一般特征，包括其铁磁性质。大多数自旋Eu 4 f能级被完全占据，位于O 2 p衍生的价带上方，继承了更弱的交换分裂。我们对轻掺杂样品（ x = 0.023，图1c ）的价带色散进行的 ARPES 测量与这些 DFT 计算以及之前对 EuO ^{20 、 21}价带电子结构的 ARPES 研究基本一致。 。对于弱相互作用半导体，电荷载流子掺杂只会引起费米能级刚性移动到多数自旋 Eu 5 d导带中，其最小值位于布里渊区的 X 点（另见补充图1 ） ）。与之前的实验²¹一致，我们确实发现通过电子掺杂，导带态在 X 处填充（图1d ）。然而，我们测量的光谱（图1d中明显并在图2a中放大显示）与简单的刚性带填充不一致。

Instead, we observe a series of replica bands offset in energy from the main quasiparticle band which intersects the Fermi level. This is a characteristic spectroscopic signature of Fröhlich polaron formation^22,23, whereby strong electron−phonon interactions give rise to shake-off excitations involving small q scattering processes. These yield satellite features in the spectral function, shifted to successively higher binding energy and evenly spaced by the corresponding phonon mode frequency. From our measured energy-distribution curves (EDCs, Fig. 2c), we observe three such distinct replica bands which, from fitted peak positions, are separated by a constant value of $\mathrm{\Delta }E\approx (56\pm 3)\mathrm{m}\mathrm{e}\mathrm{V}$. This agrees well with the longitudinal optical phonon mode frequency measured from EuO single crystals²⁴ as well as with the optical phonon branch obtained from our ab initio calculations (Supplementary Fig. 2). We thus attribute these observed spectral features in very lightly doped EuO as polaronic satellites arising from a strong electron−phonon coupling.
相反，我们观察到一系列复制能带的能量偏离与费米能级相交的主准粒子能带。这是 Fröhlich 极化子形成^{22 、 23}的特征光谱特征，由此强的电子-声子相互作用产生涉及小q散射过程的抖落激发。这些在谱函数中产生卫星特征，转移到连续更高的结合能，并由相应的声子模式频率均匀间隔。从我们测量的能量分布曲线（EDC，图2c ）中，我们观察到三个不同的复制带，它们从拟合的峰值位置开始被恒定值分开 $\mathrm{\Delta }E\approx (56\pm 3)\mathrm{m}\mathrm{e}\mathrm{V}$ 。这与从 EuO 单晶²⁴测量的纵向光学声子模式频率以及从我们从头计算获得的光学声子分支（补充图2 ）非常吻合。因此，我们将这些在极轻掺杂的EuO中观察到的光谱特征归因于强电子声子耦合产生的极化子卫星。

Spectroscopic observation of lattice polarons in dilutely doped EuO. a Measured and b calculated occupied part of the single-particle spectral function of dilutely doped Eu_1−xGd _x O (x = 0.023, n = 9.3×10¹⁷ cm⁻³, see Methods). Replica satellite bands below the main quasiparticle band that crosses the Fermi level are evident. c These are visible up to third order in an EDC taken at k = k_X, visible as distinct peaks (green shading) separated from the main quasiparticle peak (orange shading) by integer multiples of the LO phonon energy. d Similar features are evident in our ab initio many-body calculations which explicitly treat electron−phonon coupling from first principles
稀掺杂 EuO 中晶格极化子的光谱观察。 a测量值和b计算值占据了稀掺杂Eu _{1− x} Gd _x O（ x =0.023， n =9.3×10 ¹⁷ cm ^-3 ，参见方法）的单粒子光谱函数的一部分。穿过费米能级的主准粒子带下方的复制卫星带是明显的。 c这些在k = k _X处获取的 EDC 中高达三阶可见，可见不同的峰（绿色阴影）与主准粒子峰（橙色阴影）隔开 LO 声子能量的整数倍。 d类似的特征在我们的从头算多体计算中很明显，该计算明确地从第一原理处理电子声子耦合

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To confirm that this is an intrinsic property of the spectral function, we perform many-body ab initio calculations within the cumulant expansion method^25,26, thereby including the effects of electron−phonon coupling from first principles (see Methods). Apart from a small overall energetic shift, our calculations, performed for the same carrier doping as in our experiments, yield a spectral function in excellent agreement with the one measured by ARPES (Fig. 2c, d), including the spacing and approximate spectral weights of the replica features. Indeed, this level of agreement is remarkable given that the calculations are performed fully ab initio and there are no tuning parameters employed. They reveal a pronounced quasiparticle mass renormalisation, m^*/m₀ = 2.1, where m₀ is the bare band mass, pointing to a strong electron−phonon coupling, and supporting that dilutely doped EuO is in the polaronic limit. We note that similar spectral features and electron−phonon coupling strengths have been observed recently in other lightly doped oxides including TiO₂, Sr_2−xLa _x TiO₄, as well as ZnO- and SrTiO₃-based two-dimensional electron gases^{4,5,6,8,27,28,29}. Their observation here, within the markedly different system of the bulk-doped three-dimensional and spin-polarised electron pocket of EuO, suggests that polaron formation is likely universal to lightly doped polar oxides.
为了确认这是谱函数的固有属性，我们在累积量展开方法中进行了多体从头计算^{25 、 26} ，从而包括了第一原理中电子声子耦合的影响（参见方法）。除了小的总体能量偏移之外，我们对与实验中相同的载流子掺杂进行的计算产生了与 ARPES 测量的光谱函数非常一致的光谱函数（图2c，d ），包括间距和近似光谱权重的复制品功能。事实上，考虑到计算完全从头开始并且没有使用调整参数，这种一致性水平是显着的。他们揭示了明显的准粒子质量重整化， m ^* / m ₀ = 2.1，其中m ₀是裸带质量，表明存在强电子声子耦合，并支持稀掺杂的EuO处于极化极限。我们注意到，最近在其他轻掺杂氧化物中观察到了类似的光谱特征和电子声子耦合强度，包括 TiO ₂ 、Sr _{2− x} La _x TiO ₄以及基于 ZnO 和 SrTiO ₃的二维电子气^{4 、 5 、 6 、 8 、 27 、 28 、 29} 。他们在这里观察到，在EuO的体掺杂三维和自旋极化电子袋的明显不同的系统中，表明极化子的形成可能对于轻掺杂的极性氧化物是普遍存在的。

Doping-dependent spectral function
掺杂相关的光谱函数

We show in Fig. 3 how the spectral function evolves with increasing carrier doping. By increasing the density of the Gd³⁺ dopants, the band filling can be controllably increased, as evidenced by the increased quasiparticle bandwidth as well as the larger Fermi surface volume, shown inset in Fig. 3a–d. With even a small increase in carrier density, however, the pronounced multi-peak satellite structure observed in the lowest-doped sample is no longer apparent. This points to a rapid reduction in the electron−phonon coupling strength, a phenomenon which we return to below. Nonetheless, a broadened satellite peak is still observed below the quasiparticle band (Fig. 3b, c), evident as a hump in EDCs (Fig. 3e) which persists over at least two orders of magnitude increase in carrier density. To demonstrate this more clearly, we show in Fig. 4a the residual of the measured EDC intensity after subtraction of a background function accounting for the quasiparticle peak intensity (see also Supplementary Fig. 3).
我们在图3中展示了光谱函数如何随着载流子掺杂的增加而演变。通过增加 Gd ³⁺掺杂剂的密度，可以可控地增加能带填充，如图3a-d中插图所示，准粒子带宽的增加以及费米表面积的增加证明了这一点。然而，即使载流子密度略有增加，在最低掺杂样品中观察到的明显的多峰卫星结构也不再明显。这表明电子声子耦合强度迅速降低，我们将在下面讨论这一现象。尽管如此，在准粒子带下方仍然观察到加宽的卫星峰（图3b，c ），这在EDC中明显表现为驼峰（图3e ），其在载流子密度增加至少两个数量级的情况下持续存在。为了更清楚地证明这一点，我们在图4a中显示了扣除准粒子峰值强度的背景函数后测得的 EDC 强度的残差（另见补充图3 ）。

Doping-dependent plasmonic polarons. a–d Evolution of the measured spectral function of Eu_1−xGd _x O with increasing charge carrier doping, showing not only a strong increase in band filling of the quasiparticle band, but also a substantial evolution of the satellite peak structure. The insets show Fermi surface contours (hν = 137 eV), indicating the increasing doping. While replica bands can still be observed to high doping, as clearly evident as peak-dip-hump structures in measured EDCs (e), these show a strong broadening and blue-shift relative to the quasiparticle peak with increasing doping. f–i Our ab initio calculations reproduce this general trend when both electron−phonon and electron−plasmon interactions are considered, identifying the hump feature in the higher-density samples as arising from plasmonic polarons
掺杂依赖性等离子体极化子。 a – d随着电荷载流子掺杂的增加，Eu _{1− x} Gd _x O 的测量光谱函数的演变，不仅显示出准粒子带的带填充的强烈增加，而且还显示了卫星峰结构的实质性演变。插图显示费米表面轮廓（ hν = 137 eV），表明掺杂不断增加。虽然在高掺杂下仍然可以观察到复制带，与测量的 EDC 中的峰-倾-驼峰结构一样清晰可见（ e ），但随着掺杂的增加，这些带相对于准粒子峰显示出强烈的展宽和蓝移。 f – i当同时考虑电子-声子和电子-等离子体激元相互作用时，我们的从头计算重现了这种总体趋势，识别出由等离子体极化子引起的较高密度样品中的驼峰特征

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Tuning and disentangling the interplay of electron−phonon and electron−plasmon coupling. a Normalised residual intensity plot of EDCs at the centre of the electron pocket (see Supplementary Fig. 3), revealing a clear satellite structure that shifts to higher binding energy with increasing charge carrier doping. b Fits to the measured raw EDCs reveal that the separation of this satellite from the quasiparticle band follows the functional form of a plasmon mode (see Methods), while an additional weak satellite feature is found to remain at a constant energy for the lower-doped samples, which we attribute to a phonon-induced replica band. Error bars reflect the uncertainty in extracting the Luttinger area and satellite binding energies from the experimental measurements, and incorporate statistical errors in peak fitting as well as systematic experimental uncertainties. c Decomposition of the coupling strength to phonon and plasmon modes from the ab initio calculations reveals a rich carrier-density-driven crossover in the underlying nature of dominant many-body interactions in this system
调整和解开电子-声子和电子-等离子体耦合的相互作用。电子袋中心 EDC 的归一化残余强度图（参见补充图3 ），揭示了清晰的卫星结构，随着载流子掺杂的增加，该结构转变为更高的结合能。 b与测量的原始 EDC 的拟合表明，该卫星与准粒子带的分离遵循等离激元模式的函数形式（参见方法），同时发现一个额外的弱卫星特征对于低掺杂的粒子保持恒定能量样本，我们将其归因于声子引起的复制带。误差线反映了从实验测量中提取路廷格面积和卫星结合能的不确定性，并纳入了峰拟合中的统计误差以及系统实验的不确定性。 c从头算计算对声子和等离激元模式的耦合强度的分解揭示了该系统中占主导地位的多体相互作用的潜在性质中丰富的载流子密度驱动的交叉

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The satellite peak broadens with increasing charge carrier doping, but remains clearly resolved up to a carrier density n ≈ 10²⁰ cm⁻³. At the same time, the satellite exhibits a pronounced shift to higher binding energy with increasing doping. The shift is much faster than the increase in filling of the conduction band. Indeed, from fits to the measured data, we find that the separation of this hump feature from the band bottom of the quasiparticle band grows with a $\sqrt{n}$ dependence, where n is the three-dimensional electron density (Fig. 4b). This indicates electron−boson coupling to a mode which hardens with increasing carrier density. This is in striking contrast to the expectations for a phonon mode, which should be nearly carrier density independent. Instead, it agrees well with the functional form of the mode energy expected for a plasmon (red line in Fig. 4b).
卫星峰随着电荷载流子掺杂的增加而变宽，但在载流子密度n ≈ 10 ²⁰ cm ^-3之前仍然清晰地解析。与此同时，随着掺杂的增加，卫星表现出向更高结合能的明显转变。这种转变比导带填充的增加快得多。事实上，根据对测量数据的拟合，我们发现该驼峰特征与准粒子带带底的分离随着 $\sqrt{n}$ 依赖性，其中n是三维电子密度（图4b ）。这表明电子-玻色子耦合到一种随着载流子密度增加而硬化的模式。这与声子模式的期望形成鲜明对比，声子模式应该几乎与载流子密度无关。相反，它与等离子体激元预期的模式能量的函数形式非常吻合（图4b中的红线）。

The satellite features we observe here for our higher-density samples therefore point to the formation of plasmonic polarons, where the conduction electrons become dressed by charge-density fluctuations of their own electron gas³⁰. This interpretation is confirmed by our ab initio calculations, where we are able to treat electron−phonon and electron−plasmon coupling on an equal footing. Our obtained spectral functions (Fig. 3f–i) reproduce the general trends observed experimentally, also yielding plasmonic polaron satellites shifted below the quasiparticle band by the conduction electron plasmon energy. Given that EuO is a half-metal for the levels of doping investigated here, these plasmon−polarons must necessarily also be spin-polarised. Indeed, the spin-polarised conduction band of EuO has led to significant interest in using this material for spin-injection in spintronics applications¹⁷. The polaronic nature of the spin-polarised charge carriers in EuO, and consequent limited intrinsic carrier mobilities that would be expected, should be carefully considered for such applications. More generally, the excellent agreement that we find between our experimental and ab initio spectral functions for a real, complex, multi-orbital and magnetic system such as EuO suggests opportunities to exploit such advanced calculation schemes for not only understanding, but increasingly predicting, the interacting electronic states and properties of functional materials.
因此，我们在这里观察到的高密度样品的卫星特征表明等离激元极化子的形成，其中传导电子通过其自身电子气的电荷密度波动进行修饰³⁰ 。我们的从头计算证实了这种解释，我们能够在平等的基础上对待电子-声子和电子-等离子体激元耦合。我们获得的光谱函数（图3f-i ）再现了实验观察到的总体趋势，还产生了通过传导电子等离子体能量移动到准粒子带以下的等离子体极化子卫星。鉴于这里研究的掺杂水平，EuO 是半金属，这些等离激元极化子也必然是自旋极化的。事实上，EuO 的自旋极化导带引起了人们对使用这种材料在自旋电子学应用中进行自旋注入的极大兴趣¹⁷ 。对于此类应用，应仔细考虑 Eu2O 中自旋极化电荷载流子的极化性质，以及由此产生的预期的有限本征载流子迁移率。更一般地说，我们在真实、复杂、多轨道和磁系统（例如EuO）的实验谱函数和从头算谱函数之间发现了极好的一致性，这表明有机会利用这种先进的计算方案，不仅可以理解，而且可以越来越多地预测，相互作用的电子态和功能材料的性质。

Tuneable plasmon polarons
可调谐等离激元极化子

We focus below on the origin, and unique properties, of the plasmon−polarons discovered here. For a three-dimensional electron gas, the plasmon dispersion, ω(q), remains gapped in the long-wavelength limit (ω(|q|→0) = ω_p, where ω_p is the plasma frequency), while the electron−plasmon coupling strength goes as 1/|q|. Our direct observation of satellite structures spaced by the plasma energy here indicates that this is sufficient to generate well-defined replica bands, similar to those generated by the Fröhlich electron−phonon interaction. We note that this is different to the occurrence of sharp plasmon-mediated features in the spectral function of graphene, which rely upon pseudospin conservation and phase-space restrictions from matching the group velocity of plasmon and band dispersions³¹ which would not be expected in the current system. Instead, the plasmon–polarons observed here can be expected as a generic feature in the low- to medium-doping limit of a doped three-dimensional semiconductor. Moreover, we note that the polarons observed here have markedly different characteristics to plasmonic polaron band structures that have been predicted to occur via excitation of high-energy valence plasmons^30,32,33, with experimental signatures recently observed in silicon³³ and graphite³⁴. In such systems, the plasmon energy scales are ~3 orders of magnitude larger than the other excitations in the system. In contrast, the plasmons considered here have comparable energy to the Fermi energy, and so can be expected to have a much more dramatic influence on the low-energy properties of the system, such as enhancing the quasiparticle mass and limiting charge carrier mobilities.
下面我们重点讨论这里发现的等离子体激元-极化子的起源和独特性质。对于三维电子气，等离子体激元色散ω ( q ) 在长波长极限内保持有间隙（ ω (| q |→0) = ω _p ，其中ω _p是等离子体频率），而电子-等离子体耦合强度为1/| q |。我们对由等离子体能量间隔的卫星结构的直接观察表明，这足以产生明确的复制带，类似于弗罗利希电子声子相互作用产生的带。我们注意到，这与石墨烯光谱函数中尖锐等离子体介导特征的出现不同，后者依赖于赝自旋守恒和相空间限制来匹配等离子体激元和带色散的群速度^31，这在当前系统。相反，这里观察到的等离子体激元-极化子可以预期为掺杂三维半导体的低至中掺杂极限的一般特征。此外，我们注意到这里观察到的极化子与等离子体极化子能带结构具有明显不同的特征，这些结构预计是通过高能价等离子体激^{元30、32、33的}激发而发生的，最近在硅³³和石墨³⁴中观察到了实验特征。在此类系统中，等离子体激元能量尺度比系统中的其他激发大约 3 个数量级。 相比之下，这里考虑的等离子体激元具有与费米能量相当的能量，因此预计会对系统的低能特性产生更显着的影响，例如增强准粒子质量和限制载流子迁移率。

Moreover, the conduction-electron plasmons here are highly tuneable via charge carrier doping, with a characteristic mode energy that can be driven into resonance with, for example, phonon modes of the system as shown in Fig. 4. When close in energy, the two bosonic modes will in general couple to each other, leading to hybrid phonon−plasmon polaritons. Such a mode coupling is not explicitly considered in our calculations, although would be consistent with our experimentally determined satellite structure (Supplementary Fig. 4). Our study thus motivates the development of theoretical approaches and targeted experiments to investigate the polaronic signatures that might be expected in this intriguing regime.
此外，这里的传导电子等离子体激元可以通过电荷载流子掺杂进行高度可调，其特征模式能量可以被驱动与系统的声子模式共振，如图4所示。当能量接近时，两种玻色子模式通常会相互耦合，从而产生混合声子-等离子体激元。尽管这种模式耦合与我们通过实验确定的卫星结构一致（补充图4 ），但我们的计算中并未明确考虑这种模式耦合。因此，我们的研究促进了理论方法和有针对性的实验的发展，以研究在这个有趣的体系中可能预期的极化子特征。

Even without considering such mode hybridisation, our ab initio calculations shown in Fig. 4c already reveal a rich doping-dependent interplay of the coupling strength of charge carriers to different bosonic modes in the system. For the lowest carrier density investigated, a large electron−phonon coupling of λ_e−ph > 1 is obtained, which is the dominant coupling in the system. Given this, and that the plasma frequency is very small for this level of charge carrier doping, the multi-peak satellite structure observed experimentally thus predominantly arises due to electron−phonon interactions (cf. Figs. 2b and 3f). In this regime (plasma energy much smaller than the phonon mode energy), the system hosts Fröhlich polarons³⁵.
即使不考虑这种模式杂化，图4c中所示的从头计算也已经揭示了电荷载流子与系统中不同玻色子模式的耦合强度之间丰富的掺杂相关相互作用。对于研究的最低载流子密度，获得了λ _e−ph > 1 的大电子声子耦合，这是系统中的主要耦合。鉴于此，并且等离子体频率对于该电荷载流子掺杂水平非常小，因此实验观察到的多峰卫星结构主要是由于电子-声子相互作用而产生的（参见图2b和3f ）。在这种状态下（等离子体能量远小于声子模式能量），系统拥有Fröhlich极化子³⁵ 。

With increasing carrier density, the plasma frequency becomes larger than the phonon frequency (Fig. 4b). The electron−phonon interaction therefore becomes efficiently screened²³, and so the electron−phonon coupling strength shows a rapid drop-off with increasing charge carrier doping. Nonetheless, a satellite peak remains visible in both our calculations (Fig. 3g) and in fits to our measured experimental data (Fig. 4b) until the Fermi energy becomes comparable to the phonon mode frequency (approximately at the dashed line in Fig. 4b). Beyond this point, the system moves into a Fermi liquid regime²³, and the electron−phonon interaction instead leads to a more conventional kink in the band dispersion near to the Fermi level. Weak signatures of this are visible in our experimental data (Fig. 3c, d), although they are somewhat obscured by broadening due to a poor k _z resolution resulting from the inherent surface sensitivity of photoemission.
随着载流子密度的增加，等离子体频率变得大于声子频率（图4b ）。因此，电子-声子相互作用被有效屏蔽²³ ，因此电子-声子耦合强度随着载流子掺杂的增加而快速下降。尽管如此，卫星峰值在我们的计算（图3g ）和与我们测量的实验数据（图4b ）的拟合中仍然可见，直到费米能量变得与声子模式频率相当（大约在图4b中的虚线处） ）。超过这一点，系统进入费米液相态²³ ，并且电子声子相互作用反而导致费米能级附近的能带色散出现更传统的扭结。在我们的实验数据中可以看到这种微弱的特征（图3c，d ），尽管由于光电子固有的表面敏感性导致的k _z分辨率较差，它们在某种程度上被展宽所掩盖。

Despite these qualitative changes in the nature of the electron−phonon interactions, the electron−plasmon coupling strength evolves more smoothly with increasing charge carrier density (Fig. 4c). It thus plays a more dominant role at somewhat higher carrier densities, where the electron−phonon interaction is more efficiently screened. It still, however, displays a pronounced dependence on charge carrier doping in the system. To investigate this over a wider carrier density range, we show in Fig. 5a the effective electron−plasmon coupling constant, α, and the plasmonic polaron radius derived from the self-energy for a homogeneous electron gas with the same effective mass and dielectric permittivity as EuO (see Methods). The increase of α with decreasing carrier density suggests that a strong coupling regime between electrons and plasmons may be approached at low doping concentrations. In practice, the critical doping density which marks the onset of a Mott metal−insulator transition, poses a strict limit to the highest coupling that will be achievable in practice, since below this value the system becomes insulating and plasmons cannot be excited. The critical density in EuO, for example, is ~10¹⁷ cm⁻³ and it is marked by the vertical dashed line in Fig. 5a. At this doping, we find α ≃ 2.9 and a polaron radius of 53 Å. These values and their dependence on carrier concentrations are compatible with the results obtained from our first-principles calculations at the experimental doping densities (marked by dotted lines in Fig. 5a) reported in Supplementary Table 1.
尽管电子-声子相互作用的性质发生了这些质的变化，但电子-等离子体耦合强度随着载流子密度的增加而演变得更加平稳（图4c ）。因此，它在较高的载流子密度下发挥着更主导的作用，其中电子-声子相互作用被更有效地屏蔽。然而，它仍然表现出对系统中电荷载流子掺杂的明显依赖性。为了在更广泛的载流子密度范围内研究这一点，我们在图5a中显示了有效电子-等离子体激元耦合常数α以及从具有相同有效质量和介电常数的均匀电子气的自能导出的等离子体极化子半径作为EuO（参见方法）。 α随着载流子密度的降低而增加，表明电子和等离子体激元之间可以在低掺杂浓度下实现强耦合机制。在实践中，标志着莫特金属-绝缘体转变开始的临界掺杂密度，对实践中可实现的最高耦合提出了严格的限制，因为低于该值，系统变得绝缘并且等离子体激元无法被激发。例如，EuO中的临界密度约为10 ¹⁷ cm ^{-3 ，}并且在图5a中由垂直虚线标记。在这种掺杂下，我们发现α ≃ 2.9 且极化子半径为 53 Å。这些值及其对载流子浓度的依赖性与我们在补充表1中报告的实验掺杂密度（由图5a中的虚线标记）下的第一原理计算获得的结果一致。

Plasmonic polaron structure. a Evolution of the electron−plasmon coupling constant α (red symbols) and of the plasmonic polaron radius (black symbols) with carrier density. The black-dashed lines indicate the experimental doping levels from Fig. 3, while the blue line marks the critical density at the metal-insulator transition. b, c Square modulus of the polaron wavefunction for n = 9.3×10¹⁷ cm⁻³ (b) and n = 1.7×10²⁰ cm⁻³ (c). For clarity the wavefunctions are represented only in the xz plane, with 20 replicas of the unit cell shown along each direction
等离激元极化子结构。 a电子-等离子体耦合常数α （红色符号）和等离子体极化子半径（黑色符号）随载流子密度的演变。黑色虚线表示图3中的实验掺杂水平，而蓝线表示金属-绝缘体转变处的临界密度。 b , c n = 9.3×10 ¹⁷ cm ^-3 ( b ) 和n = 1.7×10 ²⁰ cm ^-3 ( c ) 的极化子波函数的平方模量。为了清楚起见，波函数仅在xz平面中表示，沿每个方向显示 20 个晶胞的副本

Full size image

Interestingly, we show in Fig. 5a that the polaron radius decreases with increasing carrier density. In fact, over the doping range considered in our experiments, the plasmonic polaron radius approximately doubles, as illustrated in Fig. 5b, c. This is in striking contrast to the expected behaviour known from phononic polarons, where the polaron radius decreases with increasing coupling strength (i.e., increases with increasing carrier density)³⁶. This unconventional behaviour results from the strong dependence of the plasmon energy on carrier concentration, and may lead to unconventional trends in doping-dependent mobilities. Moreover, this further points to the highly tuneable nature of plasmonic polaron states, whereby a broad spectrum of electron−boson coupling regimes can be explored by tuning the carrier concentrations.
有趣的是，我们在图5a中表明，极化子半径随着载流子密度的增加而减小。事实上，在我们实验中考虑的掺杂范围内，等离子体极化子半径大约加倍，如图5b、c所示。这与从声子极化子已知的预期行为形成鲜明对比，其中极化子半径随着耦合强度的增加而减小（即，随着载流子密度的增加而增加） ³⁶ 。这种非常规行为是由于等离激元能量对载流子浓度的强烈依赖性造成的，并且可能导致掺杂依赖性迁移率的非常规趋势。此外，这进一步指出了等离子体极化子态的高度可调性质，因此可以通过调整载流子浓度来探索广谱的电子-玻色子耦合机制。

We stress that our findings should not be specific to EuO, but rather a general feature of band insulators where conductivity can be induced by dilute charge carrier doping. They suggest substantial opportunity to engineer the relative importance of different bosonic modes, and may allow triggering or controlling instabilities of the collective system via electron−plasmon as well as electron−phonon interactions. Indeed, superconductivity in SrTiO₃/LaAlO₃ interface 2D electron gases has recently been argued to emerge from a phonon polaron liquid, with its superconducting dome linked to a doping-dependent strength of electron−phonon coupling⁵. A similar superconducting instability could generically be expected to occur for the plasmon−polarons introduced here.
我们强调，我们的发现不应该特定于EuO，而是带状绝缘体的一般特征，其中导电性可以通过稀载流子掺杂来诱导。他们提出了设计不同玻色子模式的相对重要性的大量机会，并且可能允许通过电子-等离子体以及电子-声子相互作用来触发或控制集体系统的不稳定性。事实上，SrTiO ₃ /LaAlO ₃界面二维电子气中的超导性最近被认为是从声子极化子液体中产生的，其超导圆顶与电子-声子耦合的掺杂依赖性强度有关⁵ 。对于这里引入的等离子体激元-极化子，通常可以预期会发生类似的超导不稳定性。

As shown above, the coupling strength for both phonon and plasmon polarons decreases with increasing doping. In the phononic case, the mode energy is fixed, and so this decrease in coupling strength must lead to a decrease in superconducting transition temperature as the doping is increased. In contrast, for the plasmonic case, while the coupling strength still decreases with increasing carrier doping, the influence on T_c should be partially offset by a hardening of the relevant mode energy. This could even lead to a doping-dependent crossover from phonon- to plasmon-mediated superconductivity with increasing charge carrier doping. While such considerations would not be relevant for EuO as studied here, due to its ferromagnetic nature, our results point to the intriguing possibility to stabilise unusual doping-dependent superconducting instabilities in, for example, lightly doped oxide semiconductors. Furthermore, they highlight the complexity of charge-carrier doping in oxides even in the absence of strong electronic correlations, opening routes to the targeted design of their materials properties.
如上所示，声子和等离激元极化子的耦合强度随着掺杂的增加而降低。在声子情况下，模式能量是固定的，因此随着掺杂的增加，耦合强度的降低必然导致超导转变温度的降低。相反，对于等离激元情况，虽然耦合强度仍然随着载流子掺杂的增加而降低，但对T _c的影响应该通过相关模式能量的硬化来部分抵消。随着载流子掺杂的增加，这甚至可能导致从声子介导的超导性到等离子体介导的超导性的掺杂依赖性交叉。虽然这些考虑因素与本文研究的EuO无关，但由于其铁磁性质，我们的结果指出了稳定轻掺杂氧化物半导体等异常掺杂相关超导不稳定性的有趣可能性。此外，他们强调了即使在没有强电子相关性的情况下，氧化物中电荷载流子掺杂的复杂性，也为材料性能的目标设计开辟了道路。

Molecular-beam epitaxy 分子束外延

The EuO thin films were grown by MBE utilising a Createc miniMBE system³⁷ installed on the I05 beamline at Diamond Light Source, UK. The films were grown on YAlO₃ substrates in an absorption-controlled, or distillation³⁸, growth mode at a temperature of 425 °C, using an Eu partial pressure of p_Eu ≈ 2.3 × 10⁻⁷ mbar and a molecular oxygen partial pressure of $p_{{\mathrm{O}}_2}\approx 2.0\times {10}^{-8}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$, as measured by a beam flux monitor. Gd dopants were introduced by exposing the films to a Gd partial pressure of p_Gd ≈ 6.3 × 10⁻⁹ mbar during growth, and shuttering the Gd source (for four equal length periods within each monolayer of EuO growth) to further reduce the incorporated Gd concentration. The films were monitored in situ using reflection high energy electron diffraction (see Supplementary Fig. 5), from which the inverse growth rate was determined to be ≈120 s per monolayer. The total thickness of the grown films is ≈20 nm, thick enough to ensure that their electronic structure is bulk-like³⁹. Following growth, the films were transferred under ultra-high vacuum to the HR-ARPES end-station (see below). After the ARPES measurements, they were further characterised by low-energy electron diffraction (see Supplementary Fig. 5) and x-ray photoelectron spectroscopy (Supplementary Fig. 6), which indicated their high crystalline and chemical quality. We note that our undoped samples are highly insulating, pointing to negligible oxygen vacancy concentrations, while our x-ray photoelectron spectroscopy (XPS) measurements indicate an Eu²+ charge state, indicative of the growth of stoichiometric EuO. The samples were then capped with 5–15 nm of amorphous silicon (p_Si ≈ 2.5×10⁻⁸ mbar), allowing these air-sensitive samples to be removed from the ultra-high vacuum environment. A subset of the films were then further probed by superconducting quantum interference device magnetometry to probe their magnetic properties and x-ray absorption spectroscopy to assess the Gd doping (Supplementary Fig. 7). These measurements confirmed material and magnetic properties of our grown EuO films that are in good agreement with previous studies of this compound. We also show in Supplementary Fig. 8 temperature-dependent ARPES measurements of a moderate carrier density sample. The majority band (occupied at low temperature) can be seen moving up through the Fermi level upon increasing through the Curie temperature, driving a temperature-dependent metal−insulator transition as a result of the loss of exchange splitting. This is fully consistent with the expected presence of spin-polarised exchange-split bands at low temperature, entirely in line with our spin-polarised DFT calculations (Fig. 1b).
EuO 薄膜是利用安装在英国钻石光源 I05 光束线上的 Createc miniMBE 系统³⁷通过 MBE 生长的。这些薄膜在 425 °C 温度下以吸收控制或蒸馏³⁸生长模式在 YAlO ₃基底上生长，使用的 Eu 分压为p _Eu ≈ 2.3 × 10 ^-7 mbar，分子氧分压为 $p_{{\mathrm{O}}_2}\approx 2.0\times {10}^{-8}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$ ，由光束通量监视器测量。通过在生长过程中将薄膜暴露于p _Gd ≈ 6.3 × 10 ^-9 mbar 的 Gd 分压来引入 Gd 掺杂剂，并关闭 Gd 源（在每个单层 EuO 生长中持续四个相等长度的周期）以进一步减少掺入的 Gd专注。使用反射高能电子衍射对薄膜进行原位监测（参见补充图5 ），从中确定逆生长速率为每单层约120秒。生长薄膜的总厚度约为 20 nm，足够厚以确保其电子结构是块状的³⁹ 。生长后，薄膜在超高真空下转移至 HR-ARPES 终端站（见下文）。经过 ARPES 测量后，它们通过低能电子衍射（参见补充图5 ）和 X 射线光电子能谱（补充图6 ）进一步表征，这表明它们具有高结晶和化学质量。 我们注意到，我们的未掺杂样品是高度绝缘的，表明氧空位浓度可以忽略不计，而我们的X射线光电子能谱（XPS）测量表明Eu ² + 电荷状态，表明化学计量的EuO的生长。然后用 5–15 nm 的非晶硅 ( p _Si ≈ 2.5×10 ^-8 mbar) 覆盖样品，使这些对空气敏感的样品能够从超高真空环境中取出。然后通过超导量子干涉装置磁力测量进一步探测薄膜的子集，以探测其磁性，并通过 X 射线吸收光谱来评估 Gd 掺杂（补充图7 ）。这些测量证实了我们生长的 EuO 薄膜的材料和磁性特性，与之前对该化合物的研究非常一致。我们还在补充图8中显示了中等载流子密度样本的温度相关 ARPES 测量结果。可以看到，随着居里温度的升高，大多数能带（在低温下占据）向上移动到费米能级，由于交换分裂的损失而驱动了与温度相关的金属-绝缘体转变。这与低温下自旋极化交换​​分裂带的预期存在完全一致，完全符合我们的自旋极化DFT计算（图1b ）。

Angle-resolved photoemission
角分辨光电发射

In situ ARPES was performed using the High-Resolution ARPES instrument (HR-ARPES) of Diamond Light Source, UK. Measurements were performed at temperatures of ≈20 K or below using p-polarised synchrotron light. A Scienta R4000 hemispherical electron analyser was used, with a vertical entrance slit and the light incident in the horizontal plane. Photon energies of 48 and 137 eV was used. For an inner potential of 15 eV, these correspond to measured dispersions which cut centrally through the conduction band Fermi surface of EuO along k _z , with an in-plane dispersion along the short axis of the elliptical Fermi pocket as shown in Supplementary Fig. 1. To determine the carrier density of the doped films, we extracted the Luttinger volume of their measured Fermi surfaces. This is complicated by the three-dimensional nature of these Fermi pockets, and the inherently poor k _z resolution in ARPES arising from its surface sensitivity. We therefore simulated the measured Fermi surface including the effects of k _z broadening, and compared this to our experimental data to determine the correct carrier density (Supplementary Fig. 9). This carrier density enters into the fit for the plasma frequency of a three-dimensional electron gas, ${\omega }_p=\sqrt{n{e}^2/{\text{ε}}_0{\text{ε}}_{\mathrm{\infty }}{m}^{\ast }},$ where ε₀ is the dielectric permittivity of free space, and ε_∞ is the dielectric constant of EuO (4.5 ²⁴). m^* is the effective mass, which is treated as a fit parameter in our analysis (Fig. 4b), from which we find a value of m^* = 0.2 ± 0.1 m_e within the range of previous estimates of the effective mass of EuO^40,41,42.
使用英国Diamond Light Source公司的高分辨率ARPES仪器(HR-ARPES)进行原位ARPES。使用 p 偏振同步加速器光在约 20 K 或更低的温度下进行测量。使用 Scienta R4000 半球形电子分析仪，具有垂直入口狭缝和水平面入射的光。使用 48 和 137 eV 的光子能量。对于 15 eV 的内部电势，这些对应于测量的色散，该色散沿着k _z中心穿过 EuO2 的导带费米表面，并且沿着椭圆费米袋的短轴具有面内色散，如补充图1所示。 。为了确定掺杂薄膜的载流子密度，我们提取了测量的费米表面的卢廷格体积。这些费米口袋的三维性质以及 ARPES 中由于其表面敏感性而固有的较差的k _z分辨率使情况变得复杂。因此，我们模拟了测量的费米表面，包括k _z展宽的影响，并将其与我们的实验数据进行比较，以确定正确的载流子密度（补充图9 ）。该载流子密度适合三维电子气的等离子体频率， ${\omega }_p=\sqrt{n{e}^2/{\text{ε}}_0{\text{ε}}_{\mathrm{\infty }}{m}^{\ast }},$ 其中ε ₀是自由空间的介电常数， ε _∞是EuO (4.5 ²⁴ )的介电常数。 m ^*是有效质量，在我们的分析中被视为拟合参数（图 1）。4b )，从中我们发现m ^* = 0.2 ± 0.1 m _e的值在先前估计的 EuO ^{40 , 41 , 42}有效质量的范围内。

First-principles calculations
第一性原理计算

Density-functional theory calculations including Hubbard corrections (DFT + U)⁴³ for the low-temperature ferromagnetic phase of EuO were performed using Quantum ESPRESSO⁴⁴. We employed the Perdew, Burke and Ernzerhof (PBE)⁴⁵ exchange-correlation functional, an effective onsite Coulomb parameter U _f  = 6 eV for the Eu 4f states, and U _p  = 3 eV for the O 2p states. We used norm-conserving pseudopotentials, a plane wave kinetic energy cutoff of 150 Ry, and a 8 × 8 × 8 uniform k-point mesh to sample the Brillouin zone. Maximally localised Wannier functions were constructed starting from a 4 × 4 × 4 uniform k grid. The effect of electron doping was included in the rigid-band approximation. The lattice vibrational properties were calculated using the projector augmented wave (PAW) method⁴⁶, and effective Coulomb parameters U _f  = 8.3 eV and U _p  = 4.6 eV which yield the same band gap calculated with norm-conserving pseudopotentials. Convergence was ensured by using a kinetic energy cutoff of 70 Ry. The phonon dispersions were obtained by finite differences in a 6 × 6 × 6 supercell, using atomic displacements of 0.01 Å. The longitudinal optical-transverse optical (LO-TO) splitting was accounted for as in ref. ⁴⁷, using the calculated Born effective charges from Ref. ⁴⁸.
使用 Quantum ESPRESSO ⁴⁴进行密度泛函理论计算，包括针对 EuO 低温铁磁相的哈伯德校正 (DFT + U ) ⁴³ 。我们采用 Perdew、Burke 和 Ernzerhof (PBE) ⁴⁵交换相关函数，Eu 4 f态的有效现场库仑参数U _f = 6 eV，O 2 p态的U _p = 3 eV。我们使用范数守恒赝势、150 Ry 的平面波动能截止值和 8 × 8 × 8 均匀k点网格对布里渊区进行采样。最大局部 Wannier 函数是从 4 × 4 × 4 均匀k网格开始构建的。电子掺杂的影响包含在刚性带近似中。使用投影增强波（PAW）方法计算晶格振动特性⁴⁶ ，有效库仑参数U _f = 8.3 eV 和U _p = 4.6 eV 产生与范数守恒赝势计算相同的带隙。通过使用 70 Ry 的动能截止来确保收敛。声子色散是通过 6 × 6 × 6 超级晶胞中的有限差分，使用 0.01 Å 的原子位移来获得的。纵向光学-横向光学 (LO-TO) 分裂的计算方法如参考文献 1 所示。 ⁴⁷ ，使用参考文献中计算的玻恩有效电荷。 ⁴⁸ .

The first-principles spectral functions were obtained from the cumulant expansion method^23,49,50 using the electron−phonon and electron−plasmon self-energy as implemented in the EPW code^51,52,53 as a seed:
第一原理谱函数是通过使用电子声子和电子等离子体自能的累积展开法^{23 、 49 、 50}获得的，如 EPW 代码^{51 、 52 、 53}中实现的那样作为种子：

Here, η is a positive infinitesimal, f_{mk + q} and n _{q ν} are Fermi−Dirac and Bose−Einstein occupations, respectively, ε_{mk + q} is the electron energy, and $\hslash {\omega }_{\mathbf{q}\nu }$ is the energy of a plasmon/phonon with wavevector q. The coupling matrix elements due to electron−plasmon and electron−phonon coupling were computed as in ref. ⁵² and ref. ⁵⁴, respectively. For the electron−phonon coupling, dynamical screening arising from the added carriers in the conduction band was taken into account by using nonadiabatic matrix elements²³: $g_{mnv}^{\mathrm{N}\mathrm{A}}\left(\mathbf{k},\mathbf{q}\right)=g_{mnv}\left(\mathbf{k},\mathbf{q}\right)/\text{ε}\left(\mathbf{q},{\omega }_{\mathbf{q}v}+i/{\tau }_{n\mathbf{k}}\right)$. Here ε(q,ω) is the Lindhard dielectric function for a spin-polarised homogeneous electron gas with effective mass m^* and dielectric permittivity ε_∞ of EuO, and $\hslash /{\tau }_{n\mathbf{k}}$ is the electron lifetime near the band edge, taken to be 50 meV. Finite resolution effects were accounted for by applying two Gaussian masks of widths 20 meV and 0.015 Å⁻¹, and by integrating the spectral function along the out-of-plane direction k _z . The temperature broadening at the Fermi level was included via a Fermi–Dirac distribution at T = 20 K. The electron−plasmon and electron−phonon coupling strengths λ were extracted from the self-energy via $\lambda =-{\hslash }^{-1}{\mathrm{\partial }\mathrm{R}\mathrm{e}\text{Σ}({\mathbf{k}}_{\mathrm{F}},\omega )/\mathrm{\partial }\omega |}_{{\text{ε}}_{\mathrm{F}}}$¹⁹. The effective electron−plasmon coupling constants α were obtained from the mass renormalisation 1 + λ_e−pl²², whereas the plasmonic polaron radius was estimated following ref. ⁵⁵: $r_p\simeq {\left(3/0.44\alpha \right)}^{\frac{1}{2}}(2m{\omega }_{\mathrm{p}\mathrm{l}}/\hslash {)}^{-\frac{1}{2}}$. The polaron wavefunction was calculated as the product of the lattice-periodic component of the Kohn−Sham eigenstate at the conduction-band bottom and a Gaussian with isotropic width σ corresponding to the polaron radius.
这里， η是正无穷小， f _{m k + q}和n _{q ν}分别是费米−狄拉克和玻色−爱因斯坦占据， ε _{m k + q}是电子能量，并且 $\hslash {\omega }_{\mathbf{q}\nu }$ 是波矢q的等离子体激元/声子的能量。由电子-等离子体和电子-声子耦合引起的耦合矩阵元素的计算如参考文献中所述。 ⁵²和参考文献。分别为⁵⁴ 。对于电子声子耦合，通过使用非绝热矩阵元素²³考虑了导带中添加的载流子引起的动态屏蔽： $g_{mnv}^{\mathrm{N}\mathrm{A}}\left(\mathbf{k},\mathbf{q}\right)=g_{mnv}\left(\mathbf{k},\mathbf{q}\right)/\text{ε}\left(\mathbf{q},{\omega }_{\mathbf{q}v}+i/{\tau }_{n\mathbf{k}}\right)$ 。这里ε ( q , ω ) 是具有有效质量m ^*和 EuO 介电常数ε _∞的自旋极化均质电子气的 Lindhard 介电函数，并且 $\hslash /{\tau }_{n\mathbf{k}}$ 是能带边缘附近的电子寿命，取 50 meV。通过应用宽度为 20 meV 和 0.015 Å ^-1的两个高斯掩模，并通过沿面外方向k _z积分光谱函数来解释有限分辨率效应。费米能级的温度展宽包含在T = 20 K 时的费米-狄拉克分布中。 电子-等离子体和电子-声子耦合强度λ通过以下方式从自能中提取 $\lambda =-{\hslash }^{-1}{\mathrm{\partial }\mathrm{R}\mathrm{e}\text{Σ}({\mathbf{k}}_{\mathrm{F}},\omega )/\mathrm{\partial }\omega |}_{{\text{ε}}_{\mathrm{F}}}$ ¹⁹ .有效电子-等离子体耦合常数α是从质量重正化 1 + λ _e-pl ²²获得的，而等离子体极化子半径是根据参考文献估计的。 ⁵⁵ ： $r_p\simeq {\left(3/0.44\alpha \right)}^{\frac{1}{2}}(2m{\omega }_{\mathrm{p}\mathrm{l}}/\hslash {)}^{-\frac{1}{2}}$ 。极化子波函数计算为导带底部 Kohn−Sham 本征态的晶格周期分量与对应于极化子半径的各向同性宽度σ的高斯函数的乘积。

Code availability 代码可用性

The calculations were performed using the open-source software projects Quantum ESPRESSO, EPW, and Wannier90, which can be downloaded free of charge from www.quantum-espresso.org, epw.org.uk, and www.wannier.org, respectively. Input files and calculation workflows can be downloaded from the GitHub repository https://github.com/mmdg-oxford/papers.
计算是使用开源软件项目 Quantum ESPRESSO、EPW 和 Wannier90 进行的，这些项目可以分别从www.quantum-espresso.org 、epw.org.uk 和www.wannier.org免费下载。输入文件和计算工作流程可以从 GitHub 存储库https://github.com/mmdg-oxford/papers下载。

Data availability 数据可用性

The data that underpins the findings of this study are available at https://doi.org/10.17630/4e82a731-57c6-4cf5-b8c2-841486b8dbde.
支持本研究结果的数据可参见https://doi.org/10.17630/4e82a731-57c6-4cf5-b8c2-841486b8dbde 。