Metal-induced gap states

Also called MIGS

In plain words

Electron states that leak from a metal contact into the it touches. They make the contact behave the same whichever metal is used, which is a big reason why contacts to 2D semiconductors are hard to improve.

Going deeper

At energies inside a semiconductor’s gap there are no propagating states, but there are decaying ones. Where a metal touches the semiconductor, its electrons occupy those decaying solutions, filling the gap with a continuum that fixes where the Fermi level can sit.

States that only exist because the crystal ends

Solving a semiconductor’s for energies inside the gap gives no real wavevector, but it does give complex ones – solutions that oscillate and decay. In an infinite crystal those are discarded as unphysical. At a surface or an interface they are not, because a decaying solution is perfectly acceptable if it decays away from the boundary.

When a metal is placed against the semiconductor, its electrons tail into exactly those states. The result is a continuum of states at energies throughout the gap, localised within roughly a nanometre of the interface, whose character changes across the gap: -like near the top of the valence band, conduction-band-like near the bottom of the conduction band. The energy where the character crosses over is the charge neutrality level, or branch point.

What the picture predicts

Because those states can hold charge, they act as a reservoir. If the sits above the charge neutrality level they fill and acquire net negative charge; below it, they empty and charge positively. Either way the resulting dipole pushes the Fermi level back, so it ends up pinned near the neutrality level whatever metal was used, and the is set by the semiconductor’s band structure rather than by the metal’s work function.

Tersoff made this quantitative: taking the neutrality level from the bulk band structure alone gives canonical barrier heights that match measurements across many semiconductor–metal pairs. The degree of pinning is expressed as a slope – how much the barrier moves per electronvolt of work function – which depends on the density of gap states and on how well the interface screens.

Why a monolayer is a harder case

The decay length of these states is comparable to a ’s thickness, which means there is no part of a 2D channel that is far from the metal. In a bulk semiconductor the pinned region is a thin skin on a thick crystal; in a monolayer it is the whole channel. Measured slopes on monolayer MoS₂ of around 0.11 – where 1 would mean no pinning at all – reflect that.

There is a second, non-intrinsic contributor that is easy to confuse with this one. Evaporating a metal onto a monolayer damages it, and the resulting and reacted regions add their own gap states. The cleanest discriminator is to remove the intrinsic mechanism: inserting a of a few ångström, either by transferring a peeled metal film or by a thin buffer, cuts the evanescent tails enough that barrier heights begin to follow work functions again. That contacts made this way behave so differently is the strongest evidence that the leakage, not only the damage, is real.

For specialists

Evanescent metal wavefunctions penetrating a semiconductor’s gap at a contact; a major cause of Fermi-level pinning.

Where this comes from

Theory of surface states Heine · Physical Review 138, A1689 (1965) cited by 1,336
Schottky barrier heights and the continuum of gap states Tersoff · Physical Review Letters 52, 465 (1984) cited by 1,389

Band structure

The map of the energies an electron may have in a crystal. The allowed energies come in ranges called bands, with forbidden gaps between them – like the decks of a multi-storey car park, where a car can stand on a deck but never between two. Whether a material conducts, gives off light or lets its electrons move easily can all be read from it, and in a layered material the map changes with the number of layers.

The electron energies E(k) of a periodic crystal as functions of crystal momentum, one branch per band, usually plotted along high-symmetry lines of the Brillouin zone. Band edges, slopes and curvatures give the gap, group velocities and effective masses; crossings and inversions give the topology. In layered crystals interlayer hybridisation makes it depend on thickness: the valence-band maximum of MoS₂ moves from Γ in the bulk to K in the monolayer, turning an indirect gap into a direct one.

Full explanation

Point defect

A flaw in a crystal that involves a single atomic site: an atom missing (a vacancy), an atom of the wrong kind in a site (a substitution, or an antisite when the crystal’s own two kinds of atom swap places), or an extra atom squeezed in between (an interstitial). Every real crystal has some. In a sheet a few atoms thick each one sits at the surface, so a handful of them can change how the material conducts, glows or reacts far more than in a thick crystal.

A defect confined to one or a few lattice sites – vacancy, interstitial, substitutional impurity, antisite or adatom – or a small complex of them. What it does is set by the levels it introduces: shallow levels dope, deep levels trap carriers and act as non-radiative recombination centres, and an isolated deep level in a wide gap can emit single photons. Formation energies and charge states depend on the chemical conditions of growth and on the Fermi level. In 2D materials every defect is a surface defect, less screened and more exposed to the environment, and densities of 10¹² to 10¹³ per cm² are typical of TMDC monolayers.

Full explanation

Metal, semimetal and insulator

The three basic ways a solid treats electricity, set by how its electrons fill the energies open to them. In a metal they fill a band of energies only partly, so the smallest push sets them moving, and it conducts at any temperature. In an insulator they fill a band completely and a wide gap separates it from the next, so nothing can move. A semimetal sits in between: two bands just touch or overlap a little, leaving a small number of mobile electrons and holes. NbSe₂ is a metal, graphene a semimetal, hBN an insulator.

A classification by where the Fermi level falls in the band structure. In a metal it crosses a partly filled band, leaving a Fermi surface, a carrier density of order one per atom (about 10²² cm⁻³) and a resistivity that falls on cooling. In a band insulator it lies in a gap of several electronvolts; a semiconductor is an insulator whose gap is small enough to be doped or gated. A semimetal has a slight overlap between conduction and valence bands (graphite, bismuth, bulk WTe₂) or bands that touch at isolated points (graphene), giving low and nearly equal densities of electrons and holes. Interactions can open gaps that band theory does not predict: Mott insulators from half-filled bands, excitonic insulators from semimetals.

Full explanation