Transfer length method (TLM)

In plain words

A measurement that separates the resistance of a device’s contacts from the resistance of the material itself, by comparing devices with channels of different lengths and working out what would be left at zero length – much as weighing a jar with different amounts of jam in it reveals the weight of the empty jar.

Going deeper

Measuring one device gives the sum of the channel and its contacts. Measuring several of different lengths separates them: the slope gives the sheet resistance, and the intercept is twice the contact resistance.

Two unknowns, one straight line

Any two-terminal measurement returns the channel resistance and the added together, and no amount of care with a single device will separate them. The transfer length method makes a set of otherwise identical devices whose channels differ only in length. The channel contributes in proportion to that length; the contacts contribute a constant.

Plotting total resistance against channel length therefore gives a straight line whose slope is the divided by the device width, and whose intercept at zero length is twice the contact resistance – twice, because there are two of them. That intercept is the number quoted as a contact resistance, conventionally normalised by width and given in ohm-micrometres so that devices of different sizes can be compared.

What the transfer length is

Extrapolating the same line the other way, to zero resistance, gives a negative length whose magnitude is twice the transfer length. This is not an artefact: current does not enter the channel at the contact’s edge but spreads along it, and the transfer length is the distance over which most of that transfer happens.

It has a practical meaning. Making a contact much longer than the transfer length adds area without reducing resistance, because the far end of the contact carries almost no current. Knowing it tells a designer how much contact to make, and it sets the limit on how far contacts can be shrunk before contact resistance takes over – a limit that is becoming binding in scaled devices generally, and is already binding in 2D ones.

What makes it lie

The method assumes every device on the ladder is identical apart from its length. On a 2D channel that is a strong assumption: a patch, a bubble, a or a slightly different contact edge under one device shifts a point, and fitting a line through scattered points gives an intercept with an error bar much larger than the fit suggests. Plotting the individual points, not just the line, is the minimum.

The larger caveat is that contact resistance in these devices is not a constant. Because the contacts are usually -like and the is set by a gate, the contact resistance depends strongly on , and often on temperature. A single contact resistance number is therefore meaningless without the carrier density it was measured at, and a transfer-length analysis has to be done at fixed gate voltage, repeated across the range. Where possible, a four-probe measurement on the same channel gives an independent check on the sheet resistance, and a disagreement between the two is a sign that the assumptions have failed.

For specialists

Measuring resistance across channels of different lengths to separate contact resistance from channel resistance.

Where this comes from

Models for contacts to planar devices Berger · Solid-State Electronics 15, 145 (1972) cited by 929

Resistivity and sheet resistance

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.

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 contact resistance drops out.

Full explanation

Layer transfer

Moving an atomically thin layer from where it was made to where it is needed – from the copper foil it grew on to a silicon wafer, or from a flake on tape onto another flake. The layer is too thin to pick up on its own, so it is carried on a polymer film or a soft stamp, put down, and the carrier removed. Each step risks tears, wrinkles, trapped bubbles and a film of leftover polymer, which is why transfer is often the dirtiest step in making a 2D device.

The step that moves a 2D layer from its growth substrate or exfoliation source to a target substrate or stack. Wet transfer coats the layer with a supporting polymer, usually PMMA, detaches it by etching the growth substrate or by electrochemical bubbling, and floats it onto the target before the polymer is dissolved. Dry transfer picks layers up with a viscoelastic PDMS stamp or a polymer film such as polycarbonate, often using the van der Waals adhesion of a top flake, and releases them by heating. The costs are cracks and wrinkles, trapped water, hydrocarbons and polymer residue, strain and doping, and – at wafer scale – yield that compounds with every transfer, which is why growth directly on the target and transfer-free routes are pursued.

Full explanation

Gating

Changing a material by putting a voltage on a nearby electrode – a gate – separated from it by an insulator. The voltage pulls electrons into the material or pushes them out, as in every transistor. Because a 2D material is so thin, the gate reaches all of it, so the number of electrons can be dialled up and down continuously: a single device can be turned from insulator to metal, its magnetism strengthened, or superconductivity switched on. Two gates, above and below, can also apply an electric field across the layer.

Electrostatic control of carrier density and electric field through a gate coupled capacitively across a dielectric: the induced density equals the gate capacitance times the voltage beyond threshold, divided by the electron charge, reaching a few 10¹³ cm⁻² with oxide or hBN gates before breakdown. Dual gating sets density and perpendicular displacement field independently, opening the gap of bilayer graphene and tuning moiré flat bands. Ionic-liquid and solid-electrolyte gating form an electric double layer about a nanometre thick and reach 10¹⁴–10¹⁵ cm⁻², enough to induce superconductivity in MoS₂ or raise the Curie temperature of Fe₃GeTe₂, but can intercalate or react electrochemically and work only while the ions are mobile. Hysteresis from traps, contact effects and quantum capacitance are the usual caveats.

Full explanation