Resistivity and sheet resistance

In plain words

Two ways to put a number on how strongly a material resists a current. Resistivity belongs to the material itself, whatever the size of the piece: copper’s is tiny, glass’s enormous. For a sheet so thin that its thickness is hard to pin down, the sheet resistance is used instead: the resistance of a square of it, measured from one edge to the opposite one, which comes out the same for a square of any size. It is quoted in ohms per square.

Going deeper

Resistivity belongs to the material; the resistance of a piece follows from its length and cross-section. For a thin film the resistance of any square is the same, so a film is described by its sheet resistance in ohms per square. Both are measured with four contacts: current through the outer pair, voltage across an inner pair that carries no current.

Why ohms per square

A strip of film twice as long has twice the resistance; twice as wide, half. So a strip exactly as long as it is wide has the same resistance whether it is a micrometre or a metre across, and that value describes the film. A device’s resistance is then the sheet resistance times its number of squares, the length divided by the width.

For a the sheet resistance is the meaningful number. Turning it into a resistivity needs a thickness, and the thickness of a single layer is a matter of convention – usually the layer spacing of the bulk crystal – so resistivities quoted for monolayers carry that assumption. Combined with the it gives the : graphene with 10¹² carriers per cm² and a sheet resistance of 1 kΩ per square has a mobility of about 6,000 cm²/Vs.

Four probes, not two

Measured with two contacts, the resistance includes the contacts themselves, which in 2D devices can exceed that of the channel. Four contacts separate them: a current is driven through the outer pair, and the voltage is read by an inner pair that carries no current, so their own resistance does not matter. The arrangement goes back to Wenner’s measurements of the earth’s resistivity in 1915 and was taken up for in the 1950s; in thin it takes the form of a , whose side contacts also give the Hall voltage.

For samples of irregular shape, van der Pauw showed in 1958 that four small contacts anywhere on the edge of a uniform sheet without holes are enough: two measurements with the contacts swapped round give the sheet resistance exactly. The conditions matter – small contacts, at the edge, on a uniform film – and flakes with cracks, bubbles or uneven doping break them without any visible sign.

What the numbers mean

Copper has a resistivity of about 1.7 µΩ·cm and glass one at least 10¹⁸ times higher; semiconductors span the range in between according to their doping. Transparent electrodes are judged by sheet resistance at a given transparency: indium tin oxide reaches some tens of ohms per square at about 90 percent transmission, and doped graphene has reached about 30 ohms per square at a similar transparency, though not yet cheaply and stably over large areas. Reduced graphene oxide stays between hundreds and tens of thousands of ohms per square.

The change with temperature says more than any single value. Resistivity that falls on cooling and then levels off marks a metal, and the ratio between room temperature and the lowest temperature measures how clean it is; resistivity that rises marks an or semiconductor; kinks and jumps mark , such as the onset of a or of .

For specialists

Resistivity ρ relates electric field to current density, in Ω·m or Ω·cm, so that a bar of length L and cross-section A has resistance ρL/A. For a film of thickness t the sheet resistance is ρ/t, quoted in Ω per square, and a strip L long and W wide has the sheet resistance times L/W. For an atomically thin layer it is the natural quantity, since the thickness is a convention, and it equals 1/neμ for sheet density n and mobility μ. Both are measured with four probes – a Hall bar, a collinear four-point probe or the van der Pauw method for arbitrary shapes – so that the voltage is sensed away from the current contacts and the drops out.

Where this comes from

A method of measuring specific resistivity and Hall effect of discs of arbitrary shape van der Pauw · Philips Research Reports 13, 1 (1958)
The 100th anniversary of the four-point probe technique: the role of probe geometries in isotropic and anisotropic systems Miccoli et al. · Journal of Physics: Condensed Matter 27, 223201 (2015)
Roll-to-roll production of 30-inch graphene films for transparent electrodes Bae et al. · Nature Nanotechnology 5, 574 (2010)

Hall effect

Send a current along a strip in a magnetic field and the moving charges are pushed towards one edge, so a small voltage appears across the strip. Its size tells how many charge carriers there are, and its sign whether they are electrons or holes, which makes it the standard way to count them. In a magnetic material a sideways voltage appears even without an outside field – the anomalous Hall effect, a handy way to see whether a tiny flake is magnetic.

The transverse voltage that the Lorentz force produces when a current flows in a perpendicular magnetic field. For one type of carrier in a 2D sheet the Hall resistance is B/ne, independent of thickness, so its slope gives the sheet density n and its sign the carrier type; combined with the sheet resistance it gives the Hall mobility. Two carrier types make the Hall resistance non-linear in B and call for a two-band fit. In magnetic conductors an extra term that follows the magnetisation – the anomalous Hall effect – comes from the Berry curvature of the bands (intrinsic) or from skew and side-jump scattering (extrinsic); quantised, it becomes the quantum anomalous Hall effect.

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

Phase transition

A sudden change in the state of a material when temperature, pressure or another control is varied – ice melting into water, a magnet losing its magnetism when heated, a metal becoming a superconductor when cooled. Each happens at a definite transition temperature. In quantum materials, transitions mark where new kinds of order appear: charge density waves, magnetism, superconductivity, ferroelectricity. A map of which state appears where, against temperature and a second knob such as doping or pressure, is a phase diagram.

A non-analytic change of thermodynamic state at a transition point, described by an order parameter that vanishes in the disordered phase and is finite in the ordered one – magnetisation, polarisation, charge-density-wave amplitude, superconducting gap. First-order transitions show latent heat, phase coexistence and hysteresis; continuous ones show a diverging correlation length and susceptibility and universal critical exponents set by dimension and symmetry (Landau theory, the renormalisation group). In two dimensions fluctuations are stronger: the Mermin–Wagner theorem forbids breaking a continuous symmetry at finite temperature, leaving anisotropic, Ising-like order or Berezinskii–Kosterlitz–Thouless quasi-order. Transition temperatures change with layer number, gating, strain and twist, which makes 2D materials tunable platforms for phase diagrams, including quantum phase transitions driven at zero temperature by non-thermal parameters.

Full explanation