Silicon Quantum Wells and the Six Conduction-Band Valleys

Why Is Silicon Different from GaAs?

In the parabolic-band tutorial, we described a quantum well using a conduction-band minimum and a single effective mass. That is a useful starting point for a direct-gap material such as GaAs, whose lowest conduction state is at the centre of the Brillouin zone (the Gamma point). Silicon requires an additional idea: its lowest conduction states are found away from Gamma.

Instead of having just one conduction-band minimum, silicon has six equivalent minima, conventionally called Delta (Δ) valleys. As illustrated in ??, they lie along the positive and negative directions of the three cubic axes: +x, −x, +y, −y, +z and −z. Each valley is a region of the band structure in which electrons can occupy states near a local minimum in energy.

In an unstrained bulk silicon crystal, symmetry makes all six minima equal in energy. When silicon is made into a quantum well, however, its finite thickness singles out one direction. Some valleys then respond differently to confinement, even though they belonged to the same energy group in the bulk material.

What Does a Valley Actually Represent?

A valley is not a physical hollow inside the silicon. It is a minimum on a graph of electron energy against crystal wavevector. If we examine the band structure close to one such minimum, its energy again looks approximately parabolic. But unlike the single isotropic parabola used in our introductory model, the curvature of a silicon valley depends on the direction in which we move through reciprocal space.

We can picture the constant-energy surface around each minimum as an elongated ellipsoid, shown schematically in the figure above. The long direction of this ellipsoid defines the valley axis. The corresponding effective mass is called the longitudinal mass, while the masses in the two perpendicular directions are called transverse masses.

It is important to distinguish the position of a valley in reciprocal space from a real-space electron trajectory. A +z valley does not mean that electrons can only travel upwards through a device; it means that the minimum being described lies along the +z direction of the crystal's band structure. Electron motion around that minimum still has three directional components, with different effective masses.

Longitudinal and Transverse Effective Mass

Near a given valley minimum, the electron energy can be represented using an anisotropic parabolic approximation. For a valley whose axis is along z, we can write:

\[ E(\mathbf{q})=E_{c,\Delta} +\frac{\hbar^2 q_x^2}{2m_t} +\frac{\hbar^2 q_y^2}{2m_t} +\frac{\hbar^2 q_z^2}{2m_l}. \]

Here, \(\mathbf{q}\) measures wavevector relative to the centre of that particular valley, rather than relative to Gamma. The symbol \(E_{c,\Delta}\) is the energy at the valley minimum. Silicon has two characteristic masses:

• Longitudinal mass, \(m_l\): the effective mass along the valley axis, approximately \(0.916m_0\).
• Transverse mass, \(m_t\): the effective mass in either perpendicular direction, approximately \(0.190m_0\).

The longitudinal mass is substantially heavier than the transverse mass. Consequently, the energy increases at different rates depending on the direction of motion. The same expression applies to an x- or y-oriented valley after assigning the longitudinal mass to the appropriate axis.

This distinction is what makes silicon quantum wells interesting: the effective mass relevant to confinement depends on which valley we are looking at.

What Happens When We Make a Silicon Quantum Well?

Consider a silicon layer confined between suitable barriers, such as a Si/SiGe heterostructure. We will take its thin, growth direction to be z, with the semiconductor layers extending across the x–y plane. As in our earlier quantum-well introduction, motion across the layer becomes quantised, while motion parallel to the layer remains free. But now the six valleys do not all have the same mass in the confined direction.

The two valleys whose axes point along +z and −z have the heavy longitudinal mass \(m_l\) in the confinement direction. The other four valleys, pointing along ±x and ±y, have the lighter transverse mass \(m_t\) in that direction. For [001]-oriented growth these two groups are called:

• Δ₂: the two valleys aligned with the growth direction (+z and −z).
• Δ₄: the four valleys whose axes lie parallel to the layer (±x and ±y).

Why does this difference matter? Recall the result for the lowest energy of an infinite quantum well:

\[ E_1=\frac{\pi^2\hbar^2}{2m_z L^2}. \]

Here \(m_z\) is the effective mass across the well and \(L\) is the well width. The equation tells us that a heavier confinement mass gives a smaller confinement energy. All else being equal, the Δ₂ states therefore have lower confinement energies than the Δ₄ states in a [001] quantum well. Confinement alone can separate their subband minima, even before we introduce strain.

Notice that we are not saying a valley is either completely confined or completely free. All six valleys experience confinement along z; they simply experience it with different effective masses. Their energy dispersions parallel to the layers are also different, because each valley has its own assignment of longitudinal and transverse masses.

How Does Strain Change the Valley Energies?

There is another important way to control these valleys. When a thin silicon layer is grown on a material with a different lattice constant, its crystal lattice can be stretched or compressed. This is called strain. Because strain changes the crystal environment, it shifts the conduction-band minima. The shift need not be the same for every valley.

In the silicon Δ-valley model, the energy shift is written:

\[ \boxed{ \Delta E_{c,i}= \Xi_d\,\operatorname{Tr}(\varepsilon) +\Xi_u\,\varepsilon_{ii} } \]

Although the notation looks involved, this equation contains just two physical effects. The strain tensor \(\varepsilon\) describes how the crystal is deformed, and its trace is the sum of the three normal strain components:

\[ \operatorname{Tr}(\varepsilon) =\varepsilon_{xx}+\varepsilon_{yy}+\varepsilon_{zz}. \]

• \(\Xi_d\), the dilatational deformation potential: multiplies the total volume-related strain and shifts all six valleys by the same amount.
• \(\Xi_u\), the uniaxial deformation potential: multiplies the strain along each valley's own axis, allowing different valley orientations to shift by different amounts.

In the second term, the index \(i\) identifies the valley axis: an x-oriented valley uses \(\varepsilon_{xx}\), a y-oriented valley uses \(\varepsilon_{yy}\), and a z-oriented valley uses \(\varepsilon_{zz}\). The + and − valleys on the same axis receive the same shift in this model.

Why Do We Obtain Δ₂ and Δ₄ Splitting?

The most common example is a [001] silicon layer under biaxial strain. The strain is the same in both directions parallel to the layers, so \(\varepsilon_{xx}=\varepsilon_{yy}=\varepsilon_{\parallel}\). The strain perpendicular to them, \(\varepsilon_{zz}\), is generally different because the crystal relaxes elastically in that direction.

Our strain equation therefore produces one shift for the two Δ₂ valleys and another for the four Δ₄ valleys:

\[ \begin{aligned} \Delta E_{\Delta_2} &=\Xi_d\operatorname{Tr}(\varepsilon) +\Xi_u\varepsilon_{zz},\\[5pt] \Delta E_{\Delta_4} &=\Xi_d\operatorname{Tr}(\varepsilon) +\Xi_u\varepsilon_{\parallel}. \end{aligned} \]

Subtracting these two expressions makes the reason for the splitting particularly clear: the common dilatational contribution cancels, leaving

\[ \boxed{ \Delta E_{\Delta_2}-\Delta E_{\Delta_4} =\Xi_u\big(\varepsilon_{zz}-\varepsilon_{\parallel}\big) }. \]

For the usual case of tensile in-plane strain in silicon, the perpendicular strain is compressive. With the conventional positive value of \(\Xi_u\), the Δ₂ valleys move down relative to Δ₄, as illustrated in ??. This changes how electrons are distributed between the available valleys.

We can now see that a silicon quantum well is affected by two distinct mechanisms: the orientation-dependent confinement energy discussed above, and the orientation-dependent shift caused by strain. Both contribute to the final positions of the Δ₂ and Δ₄ subbands.

How Does OghmaNano Calculate Silicon Quantum Wells?

The six-valley description might initially sound like a large multiband problem, but each valley can be treated using the same scalar effective-mass idea introduced on our parabolic-band page. The key difference is that each valley has a different directional mass and potentially a different strain-shifted band edge.

For a given valley, OghmaNano solves the one-dimensional envelope-function equation across the quantum well:

\[ -\frac{\hbar^2}{2}\frac{d}{dz} \left[\frac{1}{m_z(z)}\frac{d\psi(z)}{dz}\right] +V_{\Delta}(z)\psi(z)=E\psi(z). \]

The effective mass \(m_z(z)\) is selected according to the valley's orientation and local semiconductor material, while \(V_{\Delta}(z)\) includes the relevant band-edge profile and strain shift. In-plane motion is then added with the appropriate x- and y-direction masses:

\[ E_n(k_x,k_y)=E_n(0,0) +\frac{\hbar^2 k_x^2}{2m_x} +\frac{\hbar^2 k_y^2}{2m_y}. \]

In a [001] structure, the +z and −z valleys have the same mass assignment in this independent-valley approximation, and likewise the ±x and ±y pairs can be grouped by orientation. The calculation therefore produces the confinement energies and in-plane dispersions associated with the six silicon valleys, without requiring a single large coupled Hamiltonian. A complete Si/SiGe example using this model is given in the silicon Δ-valley quantum-well tutorial.

There is, however, an important limitation. In this model, the two opposite valleys on the same axis are treated as equivalent independent states. Any additional splitting caused by atomically sharp interfaces coupling the +z and −z valleys is not described by the simple mass-and-strain equation above. That requires an explicit valley-coupling treatment.

Why Is Valley Splitting Useful?

Moving the Δ₂ and Δ₄ groups relative to each other changes which electron states are occupied at a given temperature and carrier density. Because the valley groups have different transport masses, changing their relative populations can alter transport parallel to the semiconductor layers. This is one of the important physical ideas behind strained-silicon and Si/SiGe heterostructures.

Silicon also illustrates a wider lesson in quantum-well modelling: knowing the thickness and band offset of a layer is not always enough. We must also know where the band minima occur in reciprocal space, how the effective mass depends on direction, and how the crystal changes under strain.