It's a bit of a shame that there's no feedback from the OP. My answer would be that it is not true that there are no metallic thin film systems with 2D electrons.

And what qualifies as 2D is maybe also a question of what criterion(s) one selects or considers (-> posts by Artur Braun & Ioannis Samaras). And then, one may also ponder the question of "what qualifies as a metallic thin film (for the OP)?"

When do you consider electrons or a material to be "truly" 2D? Layered crystals were mentioned by Alexander N Titov.

• Consider the family of TMDCs (e.g. MoS2, MoSe2; semiconductors, though), said to be van der Waals crystals. Very anisotropic conductivities (=2D?) but nonetheless non-vanishing band dispersion along the hexagonal axis. [By the same token: even rare gas solids have sizeable band dispersions. So -- is this a criterion?].
• Consider 2D-silicides (formed e.g. after deposition of 1ML of RE metal onto Si(111) and careful annealing). These are truly 2D metals. Would they qualify as "metallic film" in the perception of the OP?
• Finally, even in metal-on-metal systems, there have been numerous studies of so-called quantum well states [e.g.

  Article d-band quantum well states in ultrathin silver films on V(100)

  , to mention only one].
• However, the specific states investigated in the previous example reside well below the Fermi level. So maybe you don't find them relevant for what interests you in 2D physics...

One general reason why low-dimensionality appears more prominently in semiconductors (SCs) as compared to many metals can be roughly and qualitatively 'understood' from looking at their (bulk) band structures. When the relevant band edge (determining transport) is at the Γ (i.e. the BZ center), then the wave vector relevant to transport is small, i.e. the corresponding wavelength is long. Such electronic states are bound to strongly feel the presence of surfaces and/or interfaces, even if they're not atomically sharp. This is different (at a quantitative level) in most (or "typical") metals, where wave vectors at the Fermi surface(s) amount to sizeable fractions of a reciprocal lattice vector and therefore the wavelengths just to few lattice constants. They therefore acquire bulk-like, "3D" character much more quickly with growing film thickness.

The structure of graphene prevents electrons from traveling perpendicular to the layers, so they can only move in two dimensions - it is like an extended net of connected benzene rings, so that electrons can move in the pi orbitals from ring to ring in the plane but cannot jump to another plane. In a thin metal film, however, even if it is only 1 nm thick, the material is a 3-D conductor so the electrons are not completely confined to the plane of the film, they have some freedom of movement in the perpendicular direction. I hope this is helpful.

In metal film electrons may be at discrete transverse level, Catherine. Then, they are also two-dimensional.

What about if metal film is at atomic lèvel ??

Thin metal films act as dielectrics because they are not continuous. Electrons can only travel through percolation. The critical thickness at which they become conducting depends on the grain size and type of metal. Graphene's linear band structure makes it metal-like at all energies, except at the Dirac point where density of states drops to zero.

The 2D honeycomb network represented by graphene formed from one or two atomic layers, has also received substantial research interest because of its intriguing structure and fascinating properties. The high mechanical strength, flexibility and optical transparency.

Interesting and also following

In general, in thin films, the metal must be two-dimensional. It is possible that it is difficult to obtain such a film. In layered crystals, the bond between the layers is weak and therefore the outer layer can be regarded as almost insulated from the substrate. With metal, it does not work out that way. It will interact with the substrate. A film, which is free of substrate, 1 nm thick, is probably not obtained yet.

But there are no principal reasons for not being 2D electrons in a thin metal film.

Does "two-dimensional electron" refer to:

Dear Michael Lusk,

I meant the first. With a two-dimensional density of states.

A 2D-DOS (density of states) occurs if the (high mobility) carriers are confined to follow a 2D-topology (structure), either in a (2D case of same[1] material's) layer (few layer graphene, or HOPG, as a virtual 2D-metal), or at an interface, between two (2D-film) layers, case of different materials[2]. The vertical confinement of the e-gas means that the energy spacing (of all the energetically accessible, low or zero mobility particles-in-a-box) in the z-direction[3] is, usually, greater than the kBT. So, the carriers[4] can not, easily, move (classically) in the z-direction, and the (high mobility) transport is confined in a 2D -DOS[5] topology.

1. Two-dimensional electron and hole gases at the surface of graphite http://faculty.washington.edu/cobden/papers/morozov05.pdf ; Article Growth and properties of few-layer graphene prepared by chem...

2. Say, the interface between a thin Si and SiO2 insulating layer or the interface between GaAs and AlGaAs; https://en.wikipedia.org/wiki/Two-dimensional_electron_gas

3. The DOS in the vertical z-direction is discrete, and the z-component of the wave function has a standing wave form (CDW).

4. Either common intrinsic electronic carriers or even other extrinsic (and intercalants) carriers.

5. at quite low kBT.

It's a bit of a shame that there's no feedback from the OP. My answer would be that it is not true that there are no metallic thin film systems with 2D electrons.

And what qualifies as 2D is maybe also a question of what criterion(s) one selects or considers (-> posts by Artur Braun & Ioannis Samaras). And then, one may also ponder the question of "what qualifies as a metallic thin film (for the OP)?"

When do you consider electrons or a material to be "truly" 2D? Layered crystals were mentioned by Alexander N Titov.

• Consider the family of TMDCs (e.g. MoS2, MoSe2; semiconductors, though), said to be van der Waals crystals. Very anisotropic conductivities (=2D?) but nonetheless non-vanishing band dispersion along the hexagonal axis. [By the same token: even rare gas solids have sizeable band dispersions. So -- is this a criterion?].
• Consider 2D-silicides (formed e.g. after deposition of 1ML of RE metal onto Si(111) and careful annealing). These are truly 2D metals. Would they qualify as "metallic film" in the perception of the OP?
• Finally, even in metal-on-metal systems, there have been numerous studies of so-called quantum well states [e.g.

  Article d-band quantum well states in ultrathin silver films on V(100)

  , to mention only one].
• However, the specific states investigated in the previous example reside well below the Fermi level. So maybe you don't find them relevant for what interests you in 2D physics...

One general reason why low-dimensionality appears more prominently in semiconductors (SCs) as compared to many metals can be roughly and qualitatively 'understood' from looking at their (bulk) band structures. When the relevant band edge (determining transport) is at the Γ (i.e. the BZ center), then the wave vector relevant to transport is small, i.e. the corresponding wavelength is long. Such electronic states are bound to strongly feel the presence of surfaces and/or interfaces, even if they're not atomically sharp. This is different (at a quantitative level) in most (or "typical") metals, where wave vectors at the Fermi surface(s) amount to sizeable fractions of a reciprocal lattice vector and therefore the wavelengths just to few lattice constants. They therefore acquire bulk-like, "3D" character much more quickly with growing film thickness.

Interesting discussion. Following with intent.

Regards