Intermediate case: parabolic, anisotropic dispersion relation Effective mass (solid-state physics)

constant energy ellipsoids in silicon near 6 conduction band minima. each valley (band minimum), effective masses mℓ = 0.92me ( longitudinal ; along 1 axis) , mt = 0.19me ( transverse ; along 2 axes).

in important semiconductors (notably, silicon) lowest energies of conduction band not symmetrical, constant-energy surfaces ellipsoids, rather spheres in isotropic case. each conduction band minimum can approximated by

e

(

k

)

=

e

0


+



ℏ

2



2

m

x


∗







(

k

x


−

k

0
,
x


)


2


+



ℏ

2



2

m

y


∗







(

k

y


−

k

0
,
y


)


2


+



ℏ

2



2

m

z


∗







(

k

z


−

k

0
,
z


)


2




{\displaystyle e\left(\mathbf {k} \right)=e_{0}+{\frac {\hbar ^{2}}{2m_{x}^{*}}}\left(k_{x}-k_{0,x}\right)^{2}+{\frac {\hbar ^{2}}{2m_{y}^{*}}}\left(k_{y}-k_{0,y}\right)^{2}+{\frac {\hbar ^{2}}{2m_{z}^{*}}}\left(k_{z}-k_{0,z}\right)^{2}}

where x, y, , z axes aligned principal axes of ellipsoids, , mx, , mz inertial effective masses along these different axes. offsets k0,x, k0,y, , k0,z reflect conduction band minimum no longer centered @ 0 wavevector. (these effective masses correspond principal components of inertial effective mass tensor, described later.)

in case, electron motion no longer directly comparable free electron; speed of electron depend on direction, , accelerate different degree depending on direction of force. still, in crystals such silicon overall properties such conductivity appear isotropic. because there multiple valleys (conduction-band minima), each effective masses rearranged along different axes. valleys collectively act give isotropic conductivity. possible average different axes effective masses in way, regain free electron picture. however, averaging method turns out depend on purpose: