Effective Mass

The core idea

Put a free electron in vacuum and push it with a force F. It accelerates exactly as Newton says: a = F/m₀, with m₀ = 9.109×10⁻³¹ kg. Now put that same electron inside a crystal and apply the same force. It still accelerates — but by a different amount, because at every instant the periodic electric field of the ion cores is also pushing and pulling on it. Tracking that lattice force explicitly is hopeless. The elegant trick of solid-state physics is to absorb all of it into a single rescaled number, the effective mass m*, so that the electron once again obeys a = F_external/m*. The crystal's complicated internal forces vanish into one parameter.

The deep point is that an electron in a crystal is not a billiard ball — it is a Bloch wave, a wavefunction spread over the periodic lattice and labelled by a crystal momentum ħk. What matters for its motion is not its bare charge alone but how its energy E depends on k. That dependence, E(k), is the band structure, and the curvature of E(k) is what effective mass measures.

Effective mass = band curvature

A wave packet built from Bloch states moves with the group velocity v = (1/ħ)·dE/dk. Apply a force and the crystal momentum changes as ħ·dk/dt = F. Differentiate the velocity to get the acceleration, and you find that the proportionality between force and acceleration is the second derivative of the band:

v_group = (1/ħ) · dE/dk
a = dv/dt = (1/ħ) · (d²E/dk²) · dk/dt = (1/ħ²) · (d²E/dk²) · F

⇒   1/m* = (1/ħ²) · d²E/dk²        (the effective mass)

So effective mass is literally the inverse curvature of the energy band:

• Sharply curved band (large d²E/dk²) → small m* → light, fast, highly mobile carriers.
• Flat band (small d²E/dk²) → large m* → heavy, sluggish carriers, as in many d-band metals and flat-band materials.
• Negative curvature (near a band maximum) → negative m* → the carrier accelerates opposite to the applied force, which we re-describe as a positive-mass hole.

For a simple parabolic band near its minimum, E(k) = E₀ + ħ²k²/(2m*), the curvature is constant and m* is a single number. Real bands are parabolic only near their extrema, which is why m* is most meaningful for carriers close to a band edge — exactly where the thermally populated carriers actually sit.

It is generally a tensor

In an anisotropic crystal the band curves differently along different directions, so effective mass is really a tensor:

(1/m*)_ij = (1/ħ²) · ∂²E / (∂k_i ∂k_j)

The textbook example is silicon. Its conduction-band minima are not at the zone centre but in six equivalent valleys along the <100> directions, and each valley is a cigar-shaped (prolate) ellipsoid. An electron in such a valley is heavy along the valley axis and light across it:

Material / band	Effective mass (in units of m₀)	Comment
Free electron (vacuum)	1.000	The reference
GaAs conduction band (Γ valley)	0.067	Isotropic, very light → fast electrons
InSb conduction band	0.014	Extremely light; huge mobility
Silicon conduction (longitudinal m_l)	0.98	Heavy along valley axis
Silicon conduction (transverse m_t)	0.19	Light across valley axis
Silicon light hole	0.16	Sharply curved valence sub-band
Silicon heavy hole	0.49	Flatter valence sub-band

Because of this anisotropy you cannot speak of "the" effective mass without saying which average you mean. For counting states you use the density-of-states effective mass; for transport you use the conductivity effective mass:

m_DOS = (N² · m_l · m_t²)^(1/3)        (N = number of equivalent valleys)
1/m_cond = (1/3) · (1/m_l + 2/m_t)     (harmonic average over axes)

For silicon with N = 6 valleys this gives m_DOS ≈ 1.08 m₀ and m_cond ≈ 0.26 m₀ — note how different they are, and how neither equals either principal mass.

Holes: making sense of negative mass

Near the top of the valence band the energy curves downward: d²E/dk² < 0, so the electrons there have negative effective mass. A negative-mass particle accelerates against the applied force — bookkeeping nightmare in a band that is almost completely full. The physicist's escape is to stop tracking ~10²³ electrons and instead track the few missing ones. An empty state in a sea of electrons behaves exactly like a particle with positive charge +e and positive effective mass equal to the magnitude of the missing electron's negative mass. That fictitious particle is a hole. Holes are the majority carriers in p-type semiconductors, and the entire device physics of pn junctions, bipolar transistors, and LEDs is written in terms of electrons and holes drifting and recombining.

Effective mass sets mobility and the density of states

Effective mass is not an abstraction — it controls two of the most measurable properties of a material.

Mobility. Carriers accelerate in a field, scatter, and reach a drift velocity. The Drude result gives

μ = q·τ / m*          (mobility)
σ = n·q·μ = n·q²·τ / m*   (conductivity)

where τ is the mean free time between collisions and n the carrier density. A lighter m* gives a higher mobility for the same scattering, which is why GaAs and InSb make superb high-frequency and infrared devices while silicon, with its heavier electrons, dominates digital logic for reasons of oxide quality rather than speed.

Material	Electron m*/m₀	Electron mobility (cm²/V·s, 300 K)
Silicon	0.26 (conductivity)	≈ 1400
Germanium	0.12	≈ 3900
GaAs	0.067	≈ 8500
InSb	0.014	≈ 77000

Density of states. The number of available quantum states per unit energy in a 3D parabolic band scales as g(E) ∝ (m*)^(3/2)·√E. A heavier effective mass packs more states near the band edge, which raises the effective carrier density a band can supply at a given temperature. This is why the intrinsic carrier concentration and the effective conduction- and valence-band densities of states N_C, N_V depend directly on the density-of-states effective mass.

How effective mass is measured

• Cyclotron resonance. Put the crystal in a magnetic field B; carriers orbit at the cyclotron frequency ω_c = qB/m*. Sweep a microwave field and find the absorption peak; ω_c gives m* directly. This is the classic technique that mapped silicon's anisotropic valleys.
• Shubnikov–de Haas and de Haas–van Alphen oscillations. The temperature damping of quantum oscillations in resistivity or magnetisation yields the cyclotron effective mass at the Fermi surface.
• Optical and ARPES probes. Angle-resolved photoemission maps E(k) directly, and its curvature is read off as m*. Infrared reflectivity (the plasma edge) also encodes m* through the plasma frequency ω_p² = n·q²/(ε₀·m*).

Beyond the band picture: heavy fermions

So far m* came purely from the single-electron band curvature. In strongly correlated materials, electron–electron interactions renormalise it further. In heavy-fermion compounds such as CeCu₆ or UPt₃, the quasiparticle effective mass can reach hundreds — even 1000 — times the free-electron mass. The electrons are so dressed by interactions that they crawl, yet they still form a coherent Fermi liquid. At the opposite extreme, in graphene the low-energy bands are linear (Dirac cones), so d²E/dk² is ill-defined and the usual parabolic effective mass concept fails entirely — the carriers behave as massless relativistic particles with a velocity-like parameter instead.

Common mistakes

• Thinking m* is a property of the electron. It is a property of the band at the carrier's location in k-space. The same material has different m* in its conduction and valence bands, and at different points in the zone.
• Using one scalar for an anisotropic band. Silicon's m_l and m_t differ by a factor of five. Collapsing them prematurely gives wrong mobilities and wrong state counts.
• Confusing the density-of-states mass with the conductivity mass. They are different averages of the same tensor and answer different questions (how many states vs. how fast carriers move).
• Forgetting the parabolic approximation breaks down. Far from a band edge, or in narrow-gap materials, non-parabolicity makes m* energy-dependent; a single number is only valid near the extremum.
• Treating negative effective mass as unphysical. It is perfectly real near a band maximum; we simply relabel those carriers as positive-mass holes for convenience.
• Equating low m* with high conductivity outright. Conductivity is n·q²·τ/m*; a very light band with few carriers or short τ can still conduct poorly.