Effective mass

The effective mass is the dynamical mass of an electron (or hole) in a semiconductor crystal, derived from the curvature of the band structure near a band extremum. It allows the use of free-electron-like equations of motion in a periodic crystal — the cornerstone simplification that makes semiconductor device physics tractable.

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

$$E(\mathbf{k}) \;=\; E_0 + \frac{\hbar^2 (\mathbf{k} - \mathbf{k}_0)^2}{2 m^*},$$

where the effective mass tensor is:

$$\left( \frac{1}{m^*} \right)_{ij} \;=\; \frac{1}{\hbar^2} \frac{\partial^2 E}{\partial k_i \partial k_j}.$$

For isotropic bands, $m^*$ is a scalar; for anisotropic bands (Si, Ge, GaP, AlAs), $m^*$ is a tensor with longitudinal and transverse components.

Why effective mass matters. With the effective mass abstraction, electrons in a semiconductor behave as free particles with mass $m^*$ rather than the free-electron mass $m_e$. This drastic simplification:

• Replaces the full Schrödinger equation in a periodic potential with effectively the free-particle equation
• Allows transport equations (mobility, diffusion) and optical equations (absorption, gain) to be derived
• Provides a single number per band per material that captures most of the band-structure complexity

Effective masses of common semiconductors.

Material	$m_e^* / m_0$	$m_{hh}^* / m_0$	$m_{lh}^* / m_0$	$m_{so}^* / m_0$
Si (longitudinal)	0.98	0.49	0.16	0.24
Si (transverse)	0.19
Ge (longitudinal)	1.59	0.33	0.043	0.077
Ge (transverse)	0.082
GaAs	0.067	0.51	0.082	0.15
InP	0.077	0.60	0.12	0.121
In₀.₅₃Ga₀.₄₇As	0.041	0.45	0.052	0.131
InAs	0.023	0.41	0.026	0.140
GaN	0.20	1.0
GaSb	0.041	0.40	0.050	0.14
AlAs	0.15	0.51	0.18	0.28

Holes typically have larger effective mass than electrons in III-V materials, and the valence band consists of multiple subbands (heavy-hole, light-hole, split-off).

Why InGaAs has light electrons. In InGaAs LM to InP, the electron effective mass is exceptionally small ($m_e^* = 0.041 m_0$). This is because the small bandgap (0.74 eV) keeps the conduction band wave functions close to the s-like cation states, producing strong band curvature and light effective mass.

Consequences:

• High electron mobility (~10,000 cm²/V·s at 300 K, vs ~1,400 in Si)
• Small electron DOS ($N_c \sim 2 \times 10^{17}$ cm⁻³; reaches degeneracy at low density)
• Sharp absorption edge (small electron DOS in the integrand)
• Strong band-filling at high density (Burstein-Moss shift large)

This explains why high-speed electronics (HEMTs, photoreceivers) use InGaAs.

Effective mass and density of states. Effective mass enters the DOS through:

$$N_c \;=\; 2 \left( \frac{2 \pi m_e^* k_B T}{h^2} \right)^{3/2}.$$

A 10× heavier effective mass produces a $10^{1.5} \approx 32\times$ larger effective DOS. This is why Si ($m_e^* \cdot$ multivalley factor $\sim 1$) has $N_c = 2.8 \times 10^{19}$ cm⁻³, while GaAs ($m_e^* = 0.067$) has $N_c = 4.7 \times 10^{17}$ cm⁻³ — a factor of 60.

Effective mass and mobility. From the Drude model, $\mu = q \tau / m^*$. Light effective mass directly translates to high mobility. This is why electron mobilities in III-V materials far exceed hole mobilities and silicon mobilities:

Material	$\mu_e$ (cm²/V·s, 300 K)	$\mu_h$ (cm²/V·s, 300 K)
Si	1450	500
Ge	3900	1900
GaAs	8800	400
InGaAs	12000	350
InAs	33000	460
InSb	78000	750
GaN	1000	200

Effective mass in quantum wells. Confinement modifies effective mass slightly through non-parabolicity corrections. For typical GaAs QWs of 10 nm thickness, the modification is small (< 10%). For very narrow wells or for high subbands, it becomes significant.

Reduced effective mass for optical transitions. The joint DOS for optical transitions uses the reduced effective mass:

$$m_r^* \;=\; \frac{m_e^* m_h^*}{m_e^* + m_h^*}.$$

For GaAs ($m_e^* = 0.067$, $m_{hh}^* = 0.51$): $m_r^* = 0.059$ — close to the electron mass because the hole mass is much heavier. The reduced mass appears in absorption coefficient formulas and excitonic binding energy.

Excitonic binding energy.

$$E_b \;=\; \frac{m_r^*}{m_e \, \epsilon_r^2} \cdot 13.6 \text{ eV},$$

where $\epsilon_r$ is the dielectric constant. For GaAs: $E_b \approx 4.2$ meV. For ZnO: $E_b \approx 60$ meV. For organic semiconductors: $E_b \approx 100 - 500$ meV. Materials with high exciton binding energy show clear excitonic features in absorption.

Density-of-states effective mass. For multi-valley semiconductors (Si has 6 equivalent valleys; Ge has 4):

$$m_{DOS}^* \;=\; M_c^{2/3} (m_l^* m_t^{*2})^{1/3},$$

where $M_c$ is the number of valleys and $m_l^*, m_t^*$ are the longitudinal and transverse masses. For Si: $m_{DOS}^* = 1.08 m_0$, much larger than either of the principal-axis masses.

Conductivity effective mass. Different from DOS effective mass; relevant for transport:

$$\frac{1}{m_c^*} \;=\; \frac{1}{3}\left(\frac{1}{m_l^*} + \frac{2}{m_t^*}\right).$$

For Si: $m_c^* = 0.26 m_0$.

Polaron effective mass. Electron-phonon coupling renormalizes the bare band effective mass. In strongly-polar materials (alkali halides, oxides), polaron effects can multiply the effective mass by 2 – 5×. For typical III-Vs, polaron correction is small (< 5%).

Strain-modified effective mass. Strain shifts and warps the bands, modifying the effective mass tensor. Compressive strain in QWs typically:

• Splits the heavy-hole and light-hole bands (heavy-hole moves up)
• Reduces the in-plane heavy-hole mass
• Improves laser gain via the reduced effective mass in the valence band

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 16; Yu & Cardona, Fundamentals of Semiconductors (4th ed., 2010), Ch. 2 — canonical treatment of band-structure effective masses; Coldren, Corzine & Mašanović, Diode Lasers and PICs (2nd ed., 2012), Ch. 4; Properties of Group-IV, III-V and II-VI Semiconductors, INSPEC handbook for compiled effective-mass tables.