The six-valley silicon conduction-band model

Learning objectives. After reading this page you should be able to:

Explain why silicon has six equivalent conduction valleys.
Understand the anisotropic longitudinal and transverse valley masses.
State the strain shift \(\Delta E_c = \Xi_d\mathrm{Tr}(\varepsilon)+\Xi_u\varepsilon_{\text{valley}}\).
Explain why [001] strain splits the valleys into \(\Delta_2\) and \(\Delta_4\) groups.
Describe how OghmaNano solves each valley as a scalar quantum well.

Prerequisites. The effective-mass equation and the strain page.

Estimated reading time. 25 minutes.

Silicon is an indirect semiconductor: its conduction-band minima lie not at the zone centre but near the six equivalent \(X\) points along the \(\langle 100\rangle\) axes. Each minimum, or valley, is an anisotropic ellipsoid with one heavy and two light masses. Modelling a silicon or SiGe quantum well therefore means solving six single-band problems, one per valley, and combining them. OghmaNano does exactly this.

1. Six valleys

The conduction minima sit along \(+x, -x, +y, -y, +z, -z\), giving six equivalent valleys related by the cubic symmetry of the crystal. Because each is a partial ellipsoid folded into the zone, they are conventionally counted as six half-valleys or, equivalently, as three pairs. In the bulk all six are degenerate; confinement and strain break this degeneracy, which is the key to band-structure engineering in Si/SiGe.

The six anisotropic constant-energy ellipsoids of the silicon conduction band along the three cubic axes. — Figure 1. Silicon's six conduction valleys are prolate ellipsoids aligned with the cubic axes, each with a longitudinal mass along its axis and a transverse mass across it.

2. Anisotropic valley masses

Each valley is an ellipsoid: heavy along its own axis (the longitudinal mass \(m_l\)) and light in the two perpendicular directions (the transverse mass \(m_t\)), with \(m_l \approx 0.92\,m_0\) and \(m_t \approx 0.19\,m_0\) in silicon. OghmaNano assigns the masses according to the valley orientation relative to the growth direction \(z\). For a valley whose axis is along \(z\) (a \(\Delta_2\) valley in [001] growth), the growth-direction mass is \(m_l\) and the in-plane masses are \(m_t\); for a valley whose axis is in the plane (a \(\Delta_4\) valley), the growth mass is \(m_t\) and one in-plane direction carries \(m_l\). This assignment is what makes the two valley groups behave differently under confinement, even before strain is applied: the heavier growth mass of the \(\Delta_2\) valleys pushes their confinement energy lower.

3. Strain shift of the valleys

Strain shifts each valley by an amount that depends on its orientation, through two deformation potentials: the dilatational \(\Xi_d\), which responds to the volume change, and the uniaxial \(\Xi_u\), which responds to the strain resolved along the valley axis. OghmaNano implements

\[ \Delta E_c = \Xi_d\,\mathrm{Tr}(\varepsilon) + \Xi_u\,\varepsilon_{\text{valley}}, \]

where \(\varepsilon_{\text{valley}}\) is the strain component along the valley axis (\(\varepsilon_{xx}\) for an \(x\)-valley, \(\varepsilon_{yy}\) for a \(y\)-valley, \(\varepsilon_{zz}\) for a \(z\)-valley). The first term shifts all six valleys together; the second term shifts them differently according to their orientation, which is what breaks the degeneracy.

4. Why [001] strain splits \(\Delta_2\) and \(\Delta_4\)

Under biaxial [001] strain the two \(z\)-valleys see \(\varepsilon_{\text{valley}}= \varepsilon_{zz}\) while the four in-plane valleys see \(\varepsilon_{\text{valley}}= \varepsilon_{xx}=\varepsilon_{yy}\). Since \(\varepsilon_{zz}\) and \(\varepsilon_{xx}\) have opposite signs, the uniaxial term shifts the two groups in opposite directions, splitting the six-fold degeneracy into a two-fold \(\Delta_2\) group (axis along growth) and a four-fold \(\Delta_4\) group (axis in plane). For tensile in-plane strain the \(\Delta_2\) valleys drop below the \(\Delta_4\) valleys, concentrating electrons in the two valleys with the light in-plane transport mass — the mechanism behind strained-silicon mobility enhancement.

Strain splitting the six degenerate silicon valleys into a two-fold Delta2 group and a four-fold Delta4 group. — Figure 2. Biaxial [001] strain lifts the six-fold valley degeneracy into the \(\Delta_2\) (growth-axis) and \(\Delta_4\) (in-plane) groups through the uniaxial deformation potential \(\Xi_u\).

5. The scalar valley solver

Within a single valley the conduction band is a single, anisotropic parabola, so the quantum-well problem is scalar: one envelope per node, giving a tridiagonal matrix exactly like the parabolic solver. The growth-direction kinetic term uses the valley's growth mass with reciprocal-mass averaging at the interfaces (averaging \(1/m(z)\) rather than \(m(z)\), the natural quantity for this operator), and the in-plane term \( \hbar^2(k_x^2/m_x + k_y^2/m_y)/2 \) uses the anisotropic in-plane masses. OghmaNano solves this scalar problem once for each of the three distinct valley orientations, applies the appropriate strain shift, and assembles the six-valley conduction structure. The material data also carry intervalley coupling constants for the \(g\)- and \(f\)-type processes, used by the scattering routines.

Key points

Silicon has six equivalent conduction valleys along the cubic axes, each an anisotropic ellipsoid.
Valley masses are assigned as longitudinal along the valley axis and transverse across it.
Strain shifts each valley by \(\Xi_d\mathrm{Tr}\varepsilon+\Xi_u\varepsilon_{\text{valley}}\).
[001] strain splits the valleys into \(\Delta_2\) and \(\Delta_4\) groups; each valley is solved as a scalar quantum well.