Band effective mass ($m^{\ast }$)

In one dimension the band effective mass is defined by $1/m^{\ast }(k)=E^{\prime \prime }(k)/{\hslash }^2$. Together with $v=E^{\prime }(k)/\hslash$ and $\hslash \dot{k}=F$, this gives $\dot{v}=F/m^{\ast }$. Negative band curvature produces negative effective mass for the electron wave packet; near a filled-band maximum the equivalent missing-electron description uses positively responding holes. In several dimensions the inverse mass is the Hessian ${\partial }_i{\partial }_{j}E/{\hslash }^2$.

The curvature tensor of a nondegenerate energy band, determining the linear relation between force and acceleration in semiclassical band dynamics. From group velocity $v_i={\hslash }^{-1}{\partial }_{k_i}E$ and $\hslash {\dot{k}}_j=F_j$, one obtains ${\dot{v}}_i=(M^{-1}{)}_{ij}{F}_j$. Near a parabolic minimum, $E-E_0={\hslash }^2\delta {\mathbf{k}}^T{M}^{-1}\delta \mathbf{k}/2$. The principal values are reciprocal curvature effective masses; a cyclotron experiment measures a direction-dependent combination rather than every tensor entry separately.

For an isotropic valence-band maximum, the electron curvature is negative. The missing-electron excitation has positive charge and positive mass $m_h^{\ast }={\hslash }^2/|E\prime \prime |$. Its cyclotron angular frequency is $eB/m_h^{\ast }$. An anisotropic orbit requires the corresponding cyclotron mass rather than an arbitrary single directional curvature.

Take the nearest-neighbour hopping $t(a)=-2E_1(1+a/a_1)e^{-a/a_1}$, with $E_1,a_1>0$, in a simple-cubic nearest-neighbour tight-binding band. Substituting into $m^{\ast }=-{\hslash }^2/(2t{a}^2)$ gives the displayed result. With $s=a/a_1$, its logarithmic derivative is $1-2/s-1/(1+s)$, which vanishes only at $s=1+\sqrt{3}>0$. The mass diverges at both positive-domain ends in this formula, so that point is its unique minimum. Its bandwidth has derivative $d\ln W/d\ln a=-s^2/(1+s)$: a small compression increases $W$. The small-$a$ continuation lies beyond reliable localized-orbital tight binding.

For a semiclassical wave packet in an isolated energy band, $v_i={\hslash }^{-1}{\partial }_{k_i}\epsilon$ and $\hslash {\dot{k}}_j=F_j$ imply ${\dot{v}}_i={\sum }_j(m^{\ast -1}{)}_{ij}{F}_j$. The inverse effective-mass tensor is the Hessian matrix of the dispersion divided by ${\hslash }^2$. Along a unit vector $\mathbf{e}$, $1/m_{\mathbf{e}}^{\ast }={\sum }_{ij}{e}_i(m^{\ast -1}{)}_{ij}{e}_j$. The Laplacian is its trace: ${\hslash }^{-2}{\nabla }_k^2\epsilon =\mathrm{tr}(m^{\ast -1})$. Thus in $d$ isotropic dimensions the scalar mass is $d{\hslash }^2/({\nabla }_k^2\epsilon )$, rather than ${\hslash }^2/({\nabla }_k^2\epsilon )$.

In a one-dimensional two-wave nearly-free electron model, set ${\epsilon }_B={\hslash }^2{G}^2/(8m)$ and let the coupling magnitude be $U>0$. Near the Bragg plane,Expansion at $q=0$ gives $E_{\pm }={\epsilon }_B\pm U+[{\hslash }^2/(2m)](1\pm 2{\epsilon }_B/U)q^2+O(q^4)$, henceThe lower edge has negative electronic curvature, and the upper edge positive curvature. The last expression is a leading weak-potential approximation, not the exact curvature of the two-wave dispersion.

Band curvature is the Hessian matrix of an energy band as a function of wavevector. Dividing it by ${\hslash }^2$ gives the inverse band effective mass tensor. At a one-dimensional band minimum the curvature is positive, whereas at a maximum it is negative. Near a weak two-wave avoided crossing with reciprocal wavenumber $G$ and coupling magnitude $U$, the two edge curvatures are $E_{\pm }^{\prime \prime }={\hslash }^2/m\pm {\hslash }^4{G}^2/(4m^{2}U)$. This is a crystal response coefficient, not a change in the bare electron mass.

For $t(a)=-2E_1{e}^{-a/a_1}$ with $E_1,a_1>0$, a nearest-neighbour cubic tight-binding model has zone-centre band effective massIts logarithmic derivative is $1/a_1-2/a$. The unique minimum occurs at $a=2a_1$ and has value ${\hslash }^2\exp (2)/(16E_1{a}_1^2)$. The explicit $a^2$ in band curvature competes with the exponentially weakening hopping integral. Extrapolation to vanishing atomic separation lies outside the localized-orbital approximation.