Effective Mass

DEFINITION

Effective mass is defined by analogy with Newton's Second Law

ec{F}=mec{a}

. Using Quantum Mechanics it can be shown that for an electron in an external electric field ''E'':

:

a =  \cdot  qE

becomes:

} = \hbar^2 \cdot \left[ ight]^{-1}

For a free particle, the dispersion relation is a Quadratic , and so the effective mass would be constant (and equal to the real mass). In a crystal, the situation is far more complex. The dispersion relation is not even approximately quadratic, in the large scale. However, wherever a minimum occurs in the dispersion relation, the minimum can be approximated by a quadratic curve in the small region around that minimum. Hence, for electrons which have energy close to a minimum, effective mass is a useful concept.

In energy regions far away from a minimum, effective mass can be negative or even approach Infinity . Effective mass, being generally dependent on direction (with respect to the Crystal Axes ), is a Tensor . However, for most calculations the various directions can be averaged out.

Effective mass should not be confused with Reduced Mass , which is a concept from Newtonian Mechanics . Effective mass can only be understood with quantum mechanics.

EFFECTIVE MASS FOR SOME COMMON SEMICONDUCTORS (FOR DENSITY OF STATES CALCULATIONS)

Material	Electron effective mass	Hole effective mass
Group IV
Si	0.36 ''m_e''	0.81 ''m_e''
Ge	0.55 ''m_e''	0.37 ''m_e''
III-V
GaAs	0.067 ''m_e''	0.45 ''m_e''
InSb	0.013 ''m_e''	0.6 ''m_e''
II-VI
ZnO	0.19 ''m_e''	1.21 ''m_e''
ZnSe	0.17''m_e''	1.44 ''m_e''

Sources:

S.Z. Sze, ''Physics of Semiconductor Devices,'' ISBN 0-47-105661-8.

W.A. Harrison, ''Electronic Structure and the Properties of Solids,'' ISBN 0-48-666021-4.

EXPERIMENTAL DETERMINATION

c}. In recent years effective masses have more commonly been determined through measurement of Band Structure s using techniques such as angle-resolved Photoemission (ARPES). Effective masses can also be estimated using the coefficient γ of the linear term in the low-temperature electronic Specific Heat at constant volume $c_v$ . The specific heat depends on the effective mass through the density of states at the Fermi level and as such is a measure of degeneracy as well as band curvature. Very large estimates of carrier mass from specific heat measurements have given rise to the concept of Heavy Fermion materials. Since carrier Mobility depends on the ratio of carrier collision lifetime $au$ to effective mass, masses can in principle be determined from transport measurements, but this method is not practical since carrier collision probabilities are typically not known a priori.

SIGNIFICANCE

As the table shows, III-V compounds based on GaAs and InSb have far smaller effective masses than tetrahedral group IV materials like Si and Ge. In the simplest . The ultimate speed of Integrated Circuit s depends on the carrier velocity, so the low effective mass is the fundamental reason that GaAs and its derivatives are used instead of Si in high- Bandwidth applications like Cellular Telephony .

EXTERNAL LINK

NSM archive

¹ This book contains an exhaustive but accessible discussion of the topic with extensive comparison between calculations and experiment.

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Effective Mass