How heavy an electron seems as it moves through a crystal. Pushed by an electric field, it speeds up as if its mass were different from a free electron’s – lighter in some materials, heavier in others, and in graphene as if it had no mass at all. Light carriers generally make for faster devices.
Going deeper
Left: near a band edge the energy rises like a parabola, and its curvature sets the effective mass – a sharply curved band gives light carriers, a gently curved one heavy carriers. Right: graphene’s bands are straight lines meeting at the Dirac point, so they have no curvature at all; its carriers are called massless and all move at about 106 m/s.
Mass from a band’s curvature
A free electron’s energy grows with the square of its momentum, as a ball’s kinetic energy does. Near the bottom of a , or the top of a valence band, a crystal’s band is also close to a parabola, but with a different curvature, so a carrier there responds to an electric or magnetic field as if its mass were different: small when the band curves sharply, large when it is nearly flat. Nothing about the electron itself has changed. The mass summarises how the crystal’s periodic potential helps or hinders it – strongly overlapping orbitals make wide, steep bands and light carriers, weakly overlapping ones narrow bands and heavy carriers.
Masses are quoted in units of the free-electron mass, m0. Silicon’s electrons have about 0.19 m0 across each of their conduction-band and 0.98 m0 along them; MoS2’s are around half the free mass, and InSe’s about 0.14 m0. A hole at the top of a valence band has an effective mass too, usually heavier than the electrons’.
What the mass controls
is the charge times the average time between collisions, divided by the mass, so light carriers pick up more speed from the same field before they scatter. The of a is proportional to the mass, so heavy carriers pack more electrons into each slice of energy – more charge per volt on the gate, and stronger interactions relative to their kinetic energy, which is one reason host . Confinement energies scale inversely with the mass, so light carriers feel thinning most: InSe’s light electrons are part of why its gap widens so strongly as it is thinned. And tunnelling falls off faster for heavy carriers, so light ones leak more easily through thin barriers.
Those effects pull in different directions in a . Light carriers give a high injection velocity and fast switching, but at gate lengths of a few nanometres they also tunnel straight from source to drain; heavy ones hold the leakage down but carry less current. That trade-off is why device studies for the shortest channels favour moderate masses rather than the lightest possible. In crystals such as black phosphorus the mass depends on direction, and so do mobility and conductivity: current flows more easily along the armchair direction than along the zigzag one.
Massless electrons, and how masses are measured
In graphene the bands near the are cones: energy rises in proportion to momentum, the curvature is zero, and a mass defined by curvature loses its meaning. Every carrier moves at about 106 m/s whatever its energy, the way particles with no rest mass do – hence Dirac fermions. Graphene’s carriers still take time to turn in a magnetic field, though, and that cyclotron mass, the Fermi energy divided by the square of the Fermi velocity, is a few hundredths of m0 at ordinary densities and grows with the square root of the density. Novoselov and colleagues measured exactly that dependence in 2005, one of the signatures that established the Dirac picture.
In other materials masses are measured by cyclotron resonance, the absorption of microwaves or infrared light at the frequency with which carriers circle in a magnetic field; by the way quantum-oscillation amplitudes die away as the temperature rises; and from the curvature of bands seen in . Optical spectra give the reduced mass of an pair. Calculated give masses too, and their curvatures are generally more trustworthy than their band gaps.
For specialists
The mass m* with which a carrier near a band edge responds to forces as if it were free, set by the band curvature: 1/m* = (1/ħ2) d2E/dk2. It enters the mobility (μ = eτ/m*), the density of states, confinement energies and tunnelling rates, and in anisotropic crystals it is a tensor – in black phosphorus the armchair and zigzag masses differ several-fold. Graphene’s linear bands have no curvature, and its carriers are described as massless Dirac fermions; their cyclotron mass, the Fermi energy divided by the square of the Fermi velocity, grows with the square root of the .