Key to our understanding of how electrons behave in crystalline solids is the band structure that connects the energy of electron waves to their wavenumber. Even in phases of matter with only short-range order (liquid or amorphous solid), the coherent part of electron waves still has a band structure. Theoretical models for the band structure of liquid metals were formulated more than five decades ago^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}, but, so far, band-structure renormalization and the pseudogap induced by resonance scattering have remained unobserved. Here we report the observation of the unusual band structure at the interface of a crystalline insulator (black phosphorus) and disordered dopants (alkali metals). We find that a conventional parabolic band structure of free electrons bends back towards zero wavenumber with a pseudogap of 30–240 millielectronvolts from the Fermi level. This is wavenumber renormalization caused by resonance scattering, leading to the formation of quasi-bound states in the scattering potential of alkali-metal ions. The depth of this potential tuned by different kinds of disordered alkali metal (sodium, potassium, rubidium and caesium) allows the classification of the pseudogap of p-wave and d-wave resonance. Our results may provide a clue to the puzzling spectrum of various crystalline insulators doped by disordered dopants^{16,17,18,19,20}, such as the waterfall dispersion observed in copper oxides.

我们理解电子在结晶固体中的行为方式的关键是连接电子波能量与其波数的能带结构。即使在只有短程有序的物质相（液体或无定形固体）中，电子波的相干部分仍然具有能带结构。液态金属能带结构的理论模型是在 5 多年前制定的^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}，但是到目前为止，共振散射引起的能带结构重整化和赝能带仍未被观察到。在这里，我们报告了对晶体绝缘体（黑磷）和无序掺杂剂（碱金属）界面处异常能带结构的观察。我们发现自由电子的传统抛物线带结构向零波数弯曲，与费米能级有 30-240 毫电子伏特的赝隙。这是由共振散射引起的波数重整化，导致碱金属离子的散射势形成准束缚态。由不同种类的无序碱金属（钠、钾、铷和铯）调节的这种电位的深度允许对p波和d的赝能隙进行分类波共振。我们的结果可能为各种晶体绝缘体掺杂无序掺杂剂^{16、17、18、19、20}的令人费解的光谱提供线索，例如在氧化铜中观察到的瀑布色散。