# Effective mass (solid-state physics)

**Effective mass** is the mass that a charge carrier, an electron or a hole, appears to have when it moves through a crystal, defined by the curvature of its energy band: in one dimension, \( \hbar^{2}/m^{*} = d^{2}E/dk^{2} \), so a strongly curved band gives a light carrier and a flat band gives a heavy one.<sup>[1](https://www.tf.uni-kiel.de/matwis/amat/td_kin_ii/kap_1/backbone/r_se2.html)</sup> The concept lets the complicated band structure \( E(\vec{k}) \) of a periodic crystal be replaced, for nearly all analytical work, by quasi-free particles whose only adjustable parameter is this mass.<sup>[1](https://www.tf.uni-kiel.de/matwis/amat/td_kin_ii/kap_1/backbone/r_se2.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( \hbar^{2}/m^{*}_{e}(k) = d^{2}E(k)/dk^{2} \); in 3D the inverse mass tensor is the Hessian of \( E(\vec{k}) \)<sup>[1](https://www.tf.uni-kiel.de/matwis/amat/td_kin_ii/kap_1/backbone/r_se2.html)</sup> |
| Holes | At the top of the valence band the electron effective mass is negative; the hole mass is defined as its negative, \( m^{*}_{h} = -m^{*}_{e} \), and is positive<sup>[2](https://www.phy.cuhk.edu.hk/~pmhui/course/phys0003/download/4/Ch%20IX%20Semiconductor%20Physics%20Part%201%20-%20Gaps%20and%20Lattice%20Constants%2C%20Concept%20of%20Holes.pdf)</sup> |
| GaAs values | Electron mass 0.063–0.067 \( m_{0} \); heavy hole 0.51 \( m_{0} \), light hole 0.082 \( m_{0} \)<sup>[3](https://www.cleanroom.byu.edu/Semiconductor-Constants)</sup> |
| Silicon values | Longitudinal 0.92–0.98 \( m_{0} \), transverse 0.19 \( m_{0} \), reflecting anisotropic conduction-band minima<sup>[4](https://courses.cit.cornell.edu/mse5470/handout16.pdf)</sup><sup> • </sup><sup>[5](https://archive.nptel.ac.in/content/storage2/courses/117102061/LMB2A/6a.htm)</sup> |
| Cyclotron mass | \( m^{*}_{c} = (\hbar^{2}/2\pi)\,\partial A/\partial E \), where \( A \) is the area of the constant-energy orbit; it equals \( m^{*} \) for a spherical surface<sup>[6](https://web.mit.edu/course/6/6.732/www/6.732-pt1.pdf)</sup> |
| Conductivity mass | For silicon's six valleys the conductivity effective mass is \( 3/(1/m_{l} + 2/m_{t}) \), giving an isotropic tensor<sup>[4](https://courses.cit.cornell.edu/mse5470/handout16.pdf)</sup> |
| Practical consequence | GaAs electron mobility at room temperature (8500 cm² V⁻¹ s⁻¹) far exceeds silicon's (1250 cm² V⁻¹ s⁻¹), attributed to the lower electron effective mass<sup>[5](https://archive.nptel.ac.in/content/storage2/courses/117102061/LMB2A/6a.htm)</sup> |

## Definition and tensor form

The effective mass is, up to a scaling factor, the inverse of the curvature of the band structure.<sup>[1](https://www.tf.uni-kiel.de/matwis/amat/td_kin_ii/kap_1/backbone/r_se2.html)</sup> In three dimensions the inverse effective mass tensor is defined by the Hesse matrix of the band energy,

\[ \hbar^{2}\,(m^{-1}_{e})_{ij} = \frac{\partial^{2} E(\vec{k})}{\partial k_{i}\,\partial k_{j}} \,, \]

and the diagonal elements of this tensor, after a unitary transformation if needed, are often used for a one-dimensional representation.<sup>[1](https://www.tf.uni-kiel.de/matwis/amat/td_kin_ii/kap_1/backbone/r_se2.html)</sup> In practice the mass is obtained by fitting the actual \( E \)-\( k \) diagram near the conduction-band minimum or the valence-band maximum with a parabola.<sup>[7](https://apachepersonal.miun.se/~gorthu/halvledare/Effective%20mass%20in%20semiconductors.htm)</sup>

**Anisotropy.** When the constant-energy surfaces are ellipsoids rather than spheres, the mass tensor has distinct principal values. Silicon's conduction-band minima lie along the six ⟨100⟩ directions, so each valley has one longitudinal mass \( m_{l} \) and two transverse masses \( m_{t} \); the Cornell lecture notes give \( m_{l} = 0.92\,m_{0} \) and \( m_{t} = 0.19\,m_{0} \),<sup>[4](https://courses.cit.cornell.edu/mse5470/handout16.pdf)</sup> while the NPTEL course and other references list \( m_{l} = 0.98\,m_{0} \) with the same transverse value.<sup>[5](https://archive.nptel.ac.in/content/storage2/courses/117102061/LMB2A/6a.htm)</sup> Germanium's constant-energy surfaces are likewise ellipsoids of revolution about the L points, and an average value \( m^{*}/m \approx 0.12 \) is commonly used for estimates.<sup>[6](https://web.mit.edu/course/6/6.732/www/6.732-pt1.pdf)</sup>

## Holes and negative mass

At the top of the valence band the curvature of \( E(k) \) is negative, so the electron effective mass there is negative.<sup>[1](https://www.tf.uni-kiel.de/matwis/amat/td_kin_ii/kap_1/backbone/r_se2.html)</sup> The hole picture removes this awkwardness: the hole effective mass tensor is defined as the negative of the electron tensor for the empty state, \( (1/m^{*}_{h})_{\alpha\beta} = -(1/m^{*}_{e})_{\alpha\beta} \), so \( m^{*}_{h} = -m^{*}_{e} \) is positive, and the hole dispersion is \( E_{h}(\vec{k}_{h}) = \hbar^{2}k_{h}^{2}/2m^{*}_{h} \).<sup>[2](https://www.phy.cuhk.edu.hk/~pmhui/course/phys0003/download/4/Ch%20IX%20Semiconductor%20Physics%20Part%201%20-%20Gaps%20and%20Lattice%20Constants%2C%20Concept%20of%20Holes.pdf)</sup> A hole is thus treated as a positively charged quasiparticle with positive mass, and both carrier types in a semiconductor then have positive effective masses.<sup>[8](https://learnqm.gatech.edu/Semiconductor-Physics-Visualization/ch7/index.html)</sup>

**Why a full band carries no current.** In a completely filled band, for every electron with negative mass there is an electron with positive mass, and their contributions cancel.<sup>[8](https://learnqm.gatech.edu/Semiconductor-Physics-Visualization/ch7/index.html)</sup> Consistently, the effective mass of electrons in a completely full band evaluates to infinity, meaning no net acceleration and no conduction.<sup>[9](https://homepages.iitb.ac.in/~alsangle/MM765_Knowledge_Check_4.html)</sup> Conduction appears when a band is partially filled, either by electrons in the conduction band or, in semiconductors, by vacancies near the top of the valence band, where the hole quasiparticle description applies.<sup>[14](https://openstax.org/books/university-physics-volume-3/pages/9-5-band-theory-of-solids)</sup>

In semiconductors the valence band is degenerate at \( \vec{k} = 0 \) in unstrained material, giving separate heavy-hole, light-hole, and split-off masses \( m^{*}_{hh} \), \( m^{*}_{lh} \), and \( m^{*}_{so} \), all cited as positive quantities.<sup>[2](https://www.phy.cuhk.edu.hk/~pmhui/course/phys0003/download/4/Ch%20IX%20Semiconductor%20Physics%20Part%201%20-%20Gaps%20and%20Lattice%20Constants%2C%20Concept%20of%20Holes.pdf)</sup> For GaAs these are 0.51 \( m_{0} \) and 0.082 \( m_{0} \) for the heavy and light holes respectively.<sup>[3](https://www.cleanroom.byu.edu/Semiconductor-Constants)</sup> The heavy-hole band is not spherically symmetric, so the density-of-states mass used in carrier-concentration calculations differs from the conductivity mass used in transport.<sup>[7](https://apachepersonal.miun.se/~gorthu/halvledare/Effective%20mass%20in%20semiconductors.htm)</sup>

## Effective masses in magnetic fields

Several distinct mass quantities matter when a magnetic field is applied. The cyclotron effective mass governs the cyclotron frequency \( \omega_{c} = eB/m^{*}_{c} \) for motion transverse to the field and can be computed either as a line integral around the constant-energy contour or, more conveniently, from the orbit area \( A \):

\[ m^{*}_{c} = \frac{\hbar^{2}}{2\pi}\,\frac{\partial A}{\partial E} \,.
\]

For a spherical constant-energy surface this reduces to \( m^{*}_{c} = m^{*} \), and for an ellipsoidal surface the orbit mass is \( m^{*}_{c} = \sqrt{m_{1}m_{2}} \), the geometric mean of the two principal masses perpendicular to the field.<sup>[6](https://web.mit.edu/course/6/6.732/www/6.732-pt1.pdf)</sup> A separate longitudinal mass \( m^{*}_{B} = \hat{b} \cdot \overleftrightarrow{m^{*}} \cdot \hat{b} \) governs motion along the field, obtained by projecting the mass tensor onto the field direction.<sup>[6](https://web.mit.edu/course/6/6.732/www/6.732-pt1.pdf)</sup> Cyclotron resonance measurements relate \( \omega_{c} \) directly to the band structure and are a standard way to probe \( m^{*} \) experimentally.<sup>[10](https://maglabweb.magnet.fsu.edu/user-facilities/emr/measurement-techniques/cyclotron-resonance/)</sup>

**Conductivity mass.** For transport calculations a further averaging is needed. Summing the current density over silicon's six valleys, each holding one sixth of the conduction electrons, yields an isotropic conductivity tensor described by a conductivity effective mass \( m_{e} = 3/(1/m_{l} + 2/m_{t}) \).<sup>[4](https://courses.cit.cornell.edu/mse5470/handout16.pdf)</sup> The Drude formula \( \sigma = ne^{2}\tau/m^{*} \) is recovered for a semiconductor from a parabolic band with constant relaxation time, and the Hall mobility involves the cyclotron mass.<sup>[6](https://web.mit.edu/course/6/6.732/www/6.732-pt1.pdf)</sup>

## The effective mass theorem and its uses

The effective mass theorem states that once \( E(\vec{k}) \) is known, the effect of perturbations on the ideal crystal can be treated by replacing \( \vec{k} \) with \( -i\nabla \) in the band energy, giving an envelope-function equation \( [E_{n}(-i\nabla) + H_{0}]F(\vec{r},t) = i\hbar\,\partial F/\partial t \) that is far easier to solve than the original [Schrödinger equation](https://www.edgepedia.org/schrodinger-equation).<sup>[6](https://web.mit.edu/course/6/6.732/www/6.732-pt1.pdf)</sup> Its classic application is the donor impurity problem: with an isotropic mass and the dielectric screening of the crystal, the donor electron obeys a hydrogenic Schrödinger equation with the substitutions \( m \to m^{*} \) and \( e^{2} \to e^{2}/\varepsilon \). For germanium, with \( \varepsilon \approx 16 \) and \( m^{*}/m \approx 0.12 \), this gives a donor ionization energy \( E_{c} - E_{d} \approx 0.007 \) eV and an effective [Bohr radius](https://www.edgepedia.org/bohr-radius) \( a^{*}_{0} \approx 70 \) Å.<sup>[6](https://web.mit.edu/course/6/6.732/www/6.732-pt1.pdf)</sup> The same envelope-function framework underlies the analysis of quantum wells, where each two-dimensional subband contributes a density of states proportional to \( m^{*} \).<sup>[6](https://web.mit.edu/course/6/6.732/www/6.732-pt1.pdf)</sup>

## Values and why they matter

Effective masses vary substantially between materials and between electrons and holes in the same material. For a semiconductor with a direct gap, curvature in the conduction band arises from coupling to the valence band over the gap; according to k · p theory, the electron mass scales approximately with the bandgap: 0.067 in GaAs (1.42 eV), 0.023 in InAs (0.35 eV), and 0.014 in InSb (0.17 eV).<sup>[11](https://www.photonica.io/glossary/effective-mass)</sup> The GaAs Γ-valley value of 0.067 \( m_{0} \) is among the lowest for common semiconductors and is a key reason for GaAs's high electron mobility and its use in high-frequency electronics.<sup>[12](https://technav.ieee.org/topic/effective-mass/)</sup> The NPTEL course attributes the room-temperature mobility gap, 8500 cm² V⁻¹ s⁻¹ in GaAs versus 1250 cm² V⁻¹ s⁻¹ in silicon, directly to this mass difference.<sup>[5](https://archive.nptel.ac.in/content/storage2/courses/117102061/LMB2A/6a.htm)</sup> [Reference](https://www.edgepedia.org/reference) tables list electron and hole masses alongside band gaps and dielectric constants as standard semiconductor parameters; for example, silicon is listed with electron mass 0.33 \( m \) and hole mass 0.5 \( m \) in a Kittel-derived table.<sup>[13](https://www.hyperphysics.gsu.edu/hbase/Tables/Semgap.html)</sup>

## References

1. [Free quantum mechanical particles and band structure, Kiel University](https://www.tf.uni-kiel.de/matwis/amat/td_kin_ii/kap_1/backbone/r_se2.html)
2. [Semiconductor Physics: Gaps and Lattice Constants, Concept of Holes, CUHK lecture notes](https://www.phy.cuhk.edu.hk/~pmhui/course/phys0003/download/4/Ch%20IX%20Semiconductor%20Physics%20Part%201%20-%20Gaps%20and%20Lattice%20Constants%2C%20Concept%20of%20Holes.pdf)
3. [Semiconductor Constants, BYU Cleanroom](https://www.cleanroom.byu.edu/Semiconductor-Constants)
4. [Electrical Conduction in Energy Bands, ECE 407 lecture notes, Cornell University](https://courses.cit.cornell.edu/mse5470/handout16.pdf)
5. [Compound Semiconductor FETs (MESFETs), NPTEL course notes](https://archive.nptel.ac.in/content/storage2/courses/117102061/LMB2A/6a.htm)
6. [Solid State Physics (6.732), MIT lecture notes](https://web.mit.edu/course/6/6.732/www/6.732-pt1.pdf)
7. [Effective mass in semiconductors, Mid Sweden University](https://apachepersonal.miun.se/~gorthu/halvledare/Effective%20mass%20in%20semiconductors.htm)
8. [External Electric Fields and Electrons and Holes, Georgia Tech learnQM](https://learnqm.gatech.edu/Semiconductor-Physics-Visualization/ch7/index.html)
9. [MM 765 Knowledge Assessment 4, IIT Bombay](https://homepages.iitb.ac.in/~alsangle/MM765_Knowledge_Check_4.html)
10. [Cyclotron Resonance, National High Magnetic Field Laboratory](https://maglabweb.magnet.fsu.edu/user-facilities/emr/measurement-techniques/cyclotron-resonance/)
11. [Effective mass in semiconductors, with values, Photonica glossary](https://www.photonica.io/glossary/effective-mass)
12. [Effective mass, IEEE Technology Navigator](https://technav.ieee.org/topic/effective-mass/)
13. [Semiconductor Band Gaps, HyperPhysics](https://www.hyperphysics.gsu.edu/hbase/Tables/Semgap.html)
14. [9.5 Band Theory of Solids - University Physics Volume 3, OpenStax](https://openstax.org/books/university-physics-volume-3/pages/9-5-band-theory-of-solids)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Matter and radiation physics › Condensed matter physics › Electronic and magnetic properties › Band theory and electron transport › Semiconductor materials and carrier physics*

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