Density-functional calculations are used to identify one-atom-thick metallic In overlayers on the Si(111) surface, which have long been sought in quest of the ultimate two-dimensional (2D) limit of free-electron-like metallic properties. We predict two metastable single-layer In phases, one √ 7 × √ 3 phase with a coverage of 1.4 monolayer (ML; here 1 ML refers to one In atom per top Si atom) and the other √ 7 × √ 7 phase with 1.43 ML, which indeed match well with experimental evidences. Both phases reveal quasi-1D arrangements of protruded In atoms, leading to 2D-metallic but anisotropic band structures and Fermi surfaces. This directional feature contrasts with the free-electron-like In-overlayer properties that are known to persist up to the double-layer thickness, implying that we may have achieved the 2D limit of free-electron-like In overlayers in previous studies of double-layer In phases.How thin can metal films be yet retaining free-electronlike metallic properties [1,2]? It might be one atomic layer that represents the ultimate 2D limit of a crystalline film. This fundamental question is in fact the very motivation underlying extensive experimental studies of the In/Si(111)-( √ 7 × √ 3) surface [3][4][5][6][7][8][9][10][11][12][13][14][15][16], which has long been considered to represent one-atom-thick indium overlayers [3,4]. Fascinating 2D electronic features were reported, including the free-electron-like parabolic bands and circular Fermi surfaces [6], the persistence of superconductivity with a high T c close to the bulk value [8,9], and the intriguing metallic transport behavior [10], all of which have been referred to as revealing the ultimate 2D limit.Unlike the expectations, however, the In/Si(111)-( √ 7 × √ 3) surface was recently verified by densityfunctional theory (DFT) calculations [17,18] to actually represent two-atom-thick In overlayers, either rectangular (hereafter, √ 7-rect) or hexagonal ( √ 7-hex), with 2.4 ML In coverage. So far, there are two single-layer In phases with the coverage verified as 1.0 ML. One is the 4×1 phase, which is metallic but definitely one dimensional with weakly coupled In chains [19][20][21], and the other 2×2 phase is known as an insulating 2D honeycomb lattice [7,22,23]. Thus, the single-layer limit of 2D-metallic In overlayers still remains to be explored.Noteworthy in this regard is that there is another In/Si(111)-( √ 7 × √ 3) surface, differing from the verified √ 7-rect and √ 7-hex double-layer phases. This phase appears intermediately in between the 2×2 and √ 7-rect phases when prepared by room-temperature (RT) In deposition onto the In/Si(111)-( √ 3 × √ 3) surface and is known to transform into the honeycomb-like √ 7 × √ 7 phase during cooling down in the range from 265 to 225 K [7]. This RT √ 7 × √ 3 phase was regarded as the √ 7-hex phase on the basis of similar STM images [7], but a recent DFT study clarified that it should be distinguished from the double-layer √ 7-hex phase [18]. Moreover, in a latest STM study [16], the RT √ 7 × √ 3 phase was ...