Making Sense of Effective Mass

Part 1: Why can effective mass be confusing?

This article has been gestating for months but I put off working on it in favor of the paid design and consulting work I'm actually supposed to be doing. Also, I don't imagine the audience clamoring for an accessible discussion of the complexities of effective mass is very large, and that consideration didn't elevate this project's priority. However, I'm finally taking the opportunity to get this downloaded from my head and into a series of articles.

The reason I wrote this article is that when I was learning semiconductor device physics in school, I encountered effective mass in a somewhat disjointed way that appeared inconsistent between sources. In particular, some sources gave a single value each for the electron and hole effective masses of a given semiconductor, whereas other sources tabulated multiple direction-dependent components of the electron effective mass as well as masses for multiple types of hole. Here's an example: the first (i.e. undergraduate-level) device physics book I had as assigned reading for a class tabulated one electron effective mass value each for three technologically important semiconductors (circled in blue in the table below), and made very cursory mention of the theory concealed by those round numbers.

Compare that table to the multiple electron effective mass parameters for the same materials, published in this research paper by S. Richard et al.:

The astute reader will notice that not only are there many more electron effective mass parameters listed per semiconductor in the second table, but that only one entry in the second table, m(Γ) for GaAs, is numerically the same as the electron effective mass found in the first table. It turns out that there is a good reason for this: the tables respectively give values for two different types of effective mass with different definitions. The first table lists density-of-states (DOS) effective masses related to how many electron and hole states there are at a given energy, and the second table lists electron inertial effective mass components related to how electrons accelerate in response to externally-applied forces.

The first table, from 4th-edition Streetman (which may well have been fixed in later editions; the book is now in its 7th edition) is particularly confusing because even the derivation of conduction band DOS in its appendix fails to differentiate between inertial and DOS effective masses, nor does it mention that DOS effective masses are computed from inertial effective masses. But even armed with this knowledge, a second point of confusion persists: why does the number of tabulated effective mass parameters vary depending on material? Moreover, why do some sources provide arcane formulas resembling geometric averages to calculate DOS effective masses from inertial effective masses yet others don't mention the issue at all, using the same mass for both (as in the case of the GaAs electron effective mass)?

At the time I was introduced to effective mass I did not have the benefit of a comprehensive overview, so these apparent inconsistencies were quite puzzling. My aim with this article is to provide the primer that I lacked. Because this is LinkedIn, and not a specialized publication, I'm going to go light on the formal math and concentrate on the physical concepts as much as possible. My objective is to clarify what's going on -- not to focus on walking through derivations with full mathematical rigor. However, it remains to be seen if my idea of what constitutes "just enough math to aid understanding" is widely shared.

Before I get into the meat of the topic in subsequent articles, I will summarize where I'm going. When I eventually untangled all this for myself, I realized that my earlier confusion about effective mass and the seeming inconsistency between different sources comes from the following root causes:

1. Multiple dynamical or "inertial" effective mass parameters apply to any given semiconductor, each specific to a particular locus of momentum states within a particular electron or hole band of that material. The loci of interest vary somewhat among materials, based on which momentum states are likely to be occupied by carriers, so the same parameters are not always tabulated for different semiconductors. Inertial effective mass describes how electrons (or holes) accelerate, on average, in response to an externally-applied force. Inertial effective mass is a real physical property of carriers, in the sense that there actually are carriers which accelerate inside a semiconductor as though they have this mass.
2. There are also two "density-of-states" (DOS) effective masses defined for every semiconductor. In this context, DOS means the differential change of the number of electron or hole states as a function of state energy, per unit volume of semiconductor material. Combined with the Fermi-Dirac distribution of state occupancy, the DOS determines the concentration of carriers in a semiconductor. The DOS effective masses respectively apply to all the electron or hole states in a semiconductor that are likely to be populated near thermal equilibrium, which can include loci of states centered at multiple values of momentum within a band, or even loci in different bands. DOS effective mass values are defined in such a way as to give the correct DOS when plugged into a minimum-complexity model that is based on an energy-vs-momentum dispersion relationship that often doesn't correspond to the states that are actually present in the semiconductor. As such, DOS effective mass is less a physical property of carriers and more like a lumped parameter in a semi-physical model, in the sense that there may not actually be any carriers that accelerate inside a semiconductor as though they have this mass.
3. Finally, "reduced" or "joint DOS" effective masses are defined that apply to those pairs of electron and hole states at particular loci of their respective bands that can make transitions without violating conservation of energy or momentum (factoring in the energy and momentum of a third particle that mediates the transition, like a photon). A reduced effective mass is used to compute a reduced/joint DOS that gives the differential change with respect to transition energy of the number of state pairs per unit volume that are separated by the transition energy and are of compatible momentums for a transition to occur. This quantity is needed to calculate transition rates, because the more ways there are for a transition to occur (the more compatible state pairs), the faster it happens. The reduced DOS calculation makes use of the same minimum-complexity dispersion relationship as the DOS calculation, and similar to the DOS effective mass, the reduced effective mass is a model parameter rather than a measure of how any particular particle accelerates in response to applied forces.
4. Beyond the multiple types of effective mass, some confusion arises because inertial effective mass is in general a direction-dependent tensor quantity, but "sphericalized" (directionally-averaged) scalar values are often tabulated in contexts where the directional nature can be ignored. The electron inertial effective mass for the locus of states near zero momentum actually is an isotropic scalar value, and those states are occupied by electrons in direct-bandgap materials like GaAs. However, the electrons in indirect-gap materials like silicon mostly occupy momentum states that have tensor effective mass, as do most of the holes in both types of material. The directionality of inertial effective mass is important in nanostructures like quantum wells, superlattices, and quantum dots, where there is quantum confinement along particular directions of motion. It also factors into interactions with directionally-oriented phenomena like polarized light. On the other hand, repeated scattering tends to average-out effects caused by the directionality of inertial effective mass for carrier diffusion in bulk materials, and DOS effective masses are specifically constructed to be scalar. This leads to sphericalized inertial effective masses being tabulated in some publications, while other publications provide tensor components of electron masses for the same materials, or give the valence band parameters needed to compute hole masses along particular directions.
5. Another source of potential confusion is the practical desire to lump together carriers traveling in different hole bands, so that one may make a single calculation for "holes" rather than separate calculations for "heavy holes", "light holes", and "split-off" holes. Like sphericalized inertial effective mass, this works in some contexts but not others. For example, when calculating aggregate hole concentration it's feasible to combine masses for multiple hole bands into a single hole DOS effective mass. However, anything like transition rates that depend on the details of the energy-vs-momentum dispersion relations require hole bands to be treated separately. Again, some publications tabulate a single hole mass, others provide sphericalized heavy and light hole masses plus a split-off hole mass, and still others publish either Luttinger parameters or equivalent momentum matrix element parameters that relate to the Dresselhaus model of tensor hole mass.
6. Lastly, some apparent inconsistency in the literature has to do with material-specific rules for constructing DOS effective masses from inertial effective masses. Some sources are either silicon-centric or compound semiconductor-centric, and tend to provide formulas that work only for the band structure of one specific class of semiconductor rather than providing general rules. The essential issues are the multiplicity of electron states that have equivalent energy-vs-momentum dispersion relations but differ with respect to the spatial orientation of the momentum, the directional dependences of the electron dispersion relations, and how remote the split-off hole band is in energy from the light and heavy hole bands. These are all ways in which carrier dispersion relations can differ among different semiconductors and carrier types, and because the DOS model is based on a simple dispersion relation with a mathematical form that often doesn't match reality, different calculational contortions are required to cast inertial effective masses for different materials and carrier types into a form that gives the right DOS when plugged into the DOS model.

If a literature source is unclear about the type of effective mass tabulated in a publication there is ample opportunity to conflate one with another. This leads to modeling states that don't exist (e.g. in transition rate or energy level calculations), or unrealistic DOS estimates. And without the knowledge to make use of electron effective mass tensor components or valence band momentum matrix element parameters, someone simply trying to get to the DOS from a literature search about a new material will be at a loss if they find a band theory paper that only presents these more detailed quantities. My aim is to clarify the important distinctions, and to consolidate in a set of articles the knowledge needed to interpret and use effective mass in all its forms.

With that preamble and summary of the confounding issues, let's circle back to the beginning. What is effective mass?

Inertial effective mass is a physical property of carriers, whereas DOS effective mass and reduced effective mass are more like lumped parameters in a semi-physical model, so it makes sense to talk about inertial effective mass first. Simply put, inertial effective mass is a way of accounting for the fact a carrier is inside a crystal, while otherwise ignoring the crystal, when determining the carrier's motion in response to an externally-applied force.

A free particle accelerates in line with the net force applied to it, and is as easy to accelerate in one direction as any other, in keeping with this common statement of Newton's second law,

F = m a.

To better represent force as the cause and acceleration as the effect, Newton's second law can also be written as:

a = m?1 F.

In these equations, boldface type indicates that net force F and acceleration a are both vector quantities, and normal type indicates particle mass m as well as its inverse, m?1, are scalar quantities. Since multiplying a vector by a scalar doesn't change where the vector points, the net force and resulting acceleration are aligned in these equations. Moreover, the magnitude of m (or m?1) doesn't depend on the direction of F; it's just m (or m?1). That means the mass is isotropic -- the same in all directions. This is the familiar, intuitive state of affairs: objects accelerate in the same direction you push them, and are as easy to shove one way as another.

An electron inside a crystal is not free: it is bound in the merged potential wells of the atoms that make up the crystal, which are arranged in a regular lattice. For the sake of simplicity, we prefer to treat semiconductors and other materials as though they are homogeneous media, but at a microscopic scale, the periodic potential of a semiconductor crystal subjects carriers to a repeating pattern of internal forces. Thus, although a = m?1 F still applies inside a crystal, it only applies if the net force is written as the sum of any externally-applied force and whatever local internal forces the crystal imposes.

Because the internal forces imposed by a crystal potential vary on the atomic scale, it is not tractable to add an external force to the complicated internal forces and then directly calculate a carrier's motion using a = m?1 F (or the equivalent quantum mechanical formulation appropriate to the atomic scale). Fortunately, the periodicity of the crystal potential makes a simpler treatment possible. After a good deal of math, one finds that

<a> = M??1 ? F??.

In this new equation, F?? is the externally-applied force, and the inverse of the inertial effective mass M??1 is a 2nd-rank tensor (basically a 3×3 matrix) which multiplies the force vector; the acceleration a is enclosed in angular brackets to indicate this is now an average quantity. This equation has the very desirable feature of giving the carrier's average acceleration <a> in terms of just the externally-applied force F??. However, because M??1 is no longer a scalar quantity, the average acceleration doesn't necessarily point in the same direction as the externally-applied force, and its magnitude isn't necessarily the same for forces applied along different directions.

What does it mean that a carrier's average acceleration doesn't line up with the externally-applied force? The following illustration is a physical example of how internal forces can deflect a system's response to an external force. The rubber band, under tension and anchored at either end, represents internal forces acting on a particle bound inside a material. The 'particle' is simply a point along the rubber band's length to which a length of string is attached. Pulling on the string applies an external force to the particle in the direction of the string (green arrow), but due to the internal forces associated with the rubber band and its anchor points, the resulting displacement of the particle (red arrow) does not align with the externally-applied force. This is a qualitative illustration only. The case of effective mass is a little more complicated, in that the direction a carrier accelerates in and how much acceleration results from a given force depends on the carrier's momentum in addition to the direction of the force.

My aim is to avoid heavy-duty math in these articles, but phrases like "after a good deal of math" are not very edifying, so in Part 2, I will talk about where <a> = M??1 ? F?? comes from and why M??1 can cause average accelerations that don't align with applied forces. A somewhat detailed understanding of the energy states occupied by electrons and holes inside a semiconductor, and their associated momentums, is a necessary starting point. So buckle up!