Effective mass

The effective mass $m^{\ast }$ is the mass an electron or hole appears to have when it is accelerated inside a crystal, $m^{\ast }={\mathrm{\hslash }}^2\mathrm{/}(d^{2}E\mathrm{/}d{k}^2)$, set by the curvature of its energy band near the band edge. A strongly curved band gives a light carrier: conduction electrons in GaAs have $m^{\ast }=0.067m_0$, where $m_0$ is the free-electron mass, while in silicon the electron mass is $0.19m_0$ across its constant-energy ellipsoids and $0.98m_0$ along them. With this one number per band, electrons and holes in the periodic potential of the crystal can be treated as free particles, which is how mobility, the density of states, carrier statistics and the optical gain of lasers are calculated.

Definition

For a parabolic band near an extremum at $\mathbf{k}={\mathbf{k}}_0$,

$$E(\mathbf{k})=E_0+\frac{{\mathrm{\hslash }}^2(\mathbf{k}-{\mathbf{k}}_0{)}^2}{2m^{\ast }},$$

and in general the inverse effective mass is a tensor,

$${\left(\frac{1}{m^{\ast }}\right)}_{ij}=\frac{1}{{\mathrm{\hslash }}^2}\frac{{\mathrm{\partial }}^{2}E}{\mathrm{\partial }{k}_i\mathrm{\partial }{k}_j}\mathrm{.}$$

For an isotropic band, such as the conduction band of GaAs, $m^{\ast }$ is a scalar; for anisotropic bands, such as the conduction bands of Si, Ge, GaP and AlAs, it has a longitudinal and a transverse component. The parabolic form holds near the band edge; higher in the band the curvature falls and the mass grows, which is called non-parabolicity.

Values for common semiconductors

In the direct-gap III-V materials holes are much heavier than electrons, and in all of these materials the valence band consists of several subbands: heavy holes, light holes and a split-off band. The hole masses depend on direction and differ between compilations by 10 to 20%, so for band calculations the Luttinger parameters of a consistent set (Vurgaftman et al.) are preferred to single masses. From the GaAs parameters, ${\gamma }_1=6.98$, ${\gamma }_2=2.06$ and ${\gamma }_3=2.93$, the heavy-hole mass is $m_0\mathrm{/}({\gamma }_1-2{\gamma }_2)=0.35m_0$ along [100] and $m_0\mathrm{/}({\gamma }_1-2{\gamma }_3)=0.89m_0$ along [111]; single values such as 0.51 are averages over direction.

Light electrons in narrow-gap semiconductors

In a direct-gap semiconductor the conduction band curves because it interacts with the valence band across the gap, and $k\cdot p$ theory gives an electron mass roughly proportional to the bandgap: 0.067 in GaAs (1.42 eV), 0.041 in InGaAs lattice matched to InP (0.75 eV), 0.023 in InAs (0.35 eV) and 0.014 in InSb (0.17 eV). The light electrons of InGaAs give it:

• High electron mobility, about 12,000 cm²/V·s at 300 K against 1,450 in silicon
• A small effective density of states, $N_c\approx 2\times {10}^{17}$ cm⁻³, so the electrons become degenerate at low densities
• Strong band filling at high density, which shifts the absorption edge to higher energy (the Burstein-Moss shift)

High-speed transistors (HEMTs) and photoreceivers use InGaAs for these reasons.

Density of states and mobility

The mass sets the effective density of states of a band, which for the conduction band is

$$N_c=2{\left(\frac{2\pi m_e^{\ast }{k}_{B}T}{h^2}\right)}^{3\mathrm{/}2}\mathrm{.}$$

A mass ten times heavier gives an effective density of states ${10}^{1.5}\approx 32$ times larger. For silicon, whose six conduction-band valleys give a density-of-states mass of $1.08m_0$ (below), the formula gives $N_c=2.8\times {10}^{19}$ cm⁻³ at 300 K; for GaAs, with $0.067m_0$, it gives $4.4\times {10}^{17}$ cm⁻³, 65 times less. Tabulated values, such as $4.7\times {10}^{17}$ cm⁻³ for GaAs, differ slightly with the masses used.

In the Drude model, $\mu =q\tau \mathrm{/}{m}^{\ast }$, where $\tau$ is the mean time between scattering events. A light mass gives a high mobility for a given scattering time, which is the main reason electron mobilities in the narrow-gap III-V materials far exceed their hole mobilities and the mobilities of silicon:

These are values for lightly doped material at 300 K; compilations differ by 10 to 20%, and doping lowers them.

Optical transitions and excitons

The joint density of states for band-to-band transitions, and with it the absorption coefficient and the gain, depends on the reduced mass

$$m_r^{\ast }=\frac{m_e^{\ast }{m}_h^{\ast }}{m_e^{\ast }+m_h^{\ast }}\mathrm{.}$$

For GaAs, with $m_e^{\ast }=0.067m_0$ and $m_{hh}^{\ast }=0.51m_0$, $m_r^{\ast }=0.059m_0$, close to the electron mass because the heavy hole is much heavier. The same reduced mass sets the binding energy of the exciton, the hydrogen-like bound state of an electron and a hole:

$$E_b=\frac{m_r^{\ast }}{m_0{ϵ}_r^2}\times 13.6\text{ eV},$$

where ${ϵ}_r$ is the dielectric constant. For GaAs, with $m_r^{\ast }=0.059m_0$ and ${ϵ}_r=12.9$, this hydrogenic estimate gives 4.8 meV, against a measured 4.2 meV; for ZnO the binding energy is about 60 meV, larger than $k_{B}T$ at room temperature. Materials whose binding energy exceeds $k_{B}T$ show clear excitonic peaks in absorption at room temperature, while in GaAs they appear clearly only at low temperature or in quantum wells, where confinement raises the binding energy.

Multi-valley semiconductors

Silicon has six equivalent conduction-band valleys and germanium four, each an ellipsoid with a longitudinal mass $m_l^{\ast }$ and a transverse mass $m_t^{\ast }$. Carrier densities use the density-of-states mass

$$m_{DOS}^{\ast }=M_c^{2\mathrm{/}3}(m_l^{\ast }{m}_t^{\ast 2}{)}^{1\mathrm{/}3},$$

where $M_c$ is the number of valleys; for Si it is $1.08m_0$, larger than either principal mass because the six valleys add their states.

Transport uses the conductivity mass, an average of the inverse masses over the three directions:

$$\frac{1}{m_c^{\ast }}=\frac{1}{3}(\frac{1}{m_l^{\ast }}+\frac{2}{m_t^{\ast }})\mathrm{.}$$

For Si it is $0.26m_0$.

Quantum wells, strain and polarons

Confinement in a quantum well lifts the electron above the band edge, where non-parabolicity makes the band less curved: in a 10 nm GaAs well the first electron level lies a few tens of meV up, and the mass there is a few percent heavier than at the band edge. For holes the change is larger, because confinement separates the heavy-hole and light-hole bands and each has different masses along the growth axis and in the plane of the well. Near the zone center, with the GaAs Luttinger parameters above, the heavy hole has $m_0\mathrm{/}({\gamma }_1-2{\gamma }_2)=0.35m_0$ along the growth axis but $m_0\mathrm{/}({\gamma }_1+{\gamma }_2)=0.11m_0$ in the plane, and the light hole the reverse, $0.090m_0$ and $0.20m_0$; away from the zone center the mixing of the two bands makes the in-plane dispersion strongly non-parabolic.

Biaxial strain shifts and warps the valence bands further. Compressive strain, as in InGaAs wells on GaAs, raises the heavy-hole band further above the light-hole band and lowers the in-plane mass of the top valence band toward that of the electrons, which reduces the carrier density needed for transparency and the threshold current of a laser. This was proposed in 1986 by Adams and, independently, by Yablonovitch and Kane, and strained wells are now standard in telecom lasers and 980 nm pump lasers.

In polar crystals a moving carrier drags a lattice polarization with it, forming a polaron, and its mass rises by a factor of about $1+\alpha \mathrm{/}6$ for weak coupling, where $\alpha$ is the Fröhlich coupling constant. For GaAs, with $\alpha$ of about 0.07, the increase is about 1%, and measured masses such as 0.067 already include it; in strongly polar crystals such as the alkali halides the coupling is strong and the correction large.

Common questions

What is effective mass?

The mass that an electron or hole appears to have when a force accelerates it inside a crystal, $m^{\ast }={\mathrm{\hslash }}^2\mathrm{/}(d^{2}E\mathrm{/}d{k}^2)$. It differs from the free-electron mass because the carrier also feels the periodic potential of the lattice; the electron effective mass is $0.067m_0$ in GaAs and $0.041m_0$ in InGaAs.

Why is the effective mass different from the real mass of the electron?

The electron's mass does not change. The effective mass describes how its energy depends on its crystal momentum, and it absorbs the forces of the lattice's periodic potential, so that the carrier responds to an applied field as a free particle of mass $m^{\ast }$ would.

Can the effective mass be negative?

Yes. Near the top of a band the curvature $d^{2}E\mathrm{/}d{k}^2$ is negative, and an electron there accelerates opposite to the applied force. The empty states at the top of the valence band are therefore described as holes, with positive charge and positive mass.

What is the effective mass of electrons in silicon?

Silicon's conduction band has six equivalent valleys, each with a longitudinal mass of $0.98m_0$ and a transverse mass of $0.19m_0$. Together they give a density-of-states mass of $1.08m_0$, used for carrier densities, and a conductivity mass of $0.26m_0$, used for mobility.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 16; Yu & Cardona, Fundamentals of Semiconductors (4th ed., 2010), Ch. 2; Coldren, Corzine & Mašanović, Diode Lasers and PICs (2nd ed., 2012), Ch. 4; S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (Wiley, 2007); I. Vurgaftman, J. R. Meyer and L. R. Ram-Mohan, "Band parameters for III-V compound semiconductors and their alloys," Journal of Applied Physics 89, 5815 (2001); O. Madelung, Semiconductors: Data Handbook, 3rd ed. (Springer, 2004); A. R. Adams, "Band-structure engineering for low-threshold high-efficiency semiconductor lasers," Electronics Letters 22, 249 (1986); E. Yablonovitch and E. O. Kane, "Reduction of lasing threshold current density by the lowering of valence band effective mass," Journal of Lightwave Technology 4, 504 (1986).