The concept of effective mass follows from physicists' love for simple relations such as Ohm's law (current density = conductivity x electric field intensity) or Newton's second law of motion (acceleration = force / mass). For a free electron, the mass in the latter is just the electron mass. But, if one wants to write a similar relation for a charge carrier in a crystal lattice, the situation changes. Going through the math (see any solid state physics textbook) allows you to write F = m x a, where the force is charge x electric field, but the mass is no longer the mass of the electron, but reflects the curvature of the conduction band bottom (for electrons) or valence band top (for holes), as it is inversely proportional to the second derivative of the energy as a function of k. This is the so-called effective mass. Again, physicists like simple things, so one often expresses the effective mass as a constant x electron mass, although this has very little (or nothing) to do with the actual physics of the situation. Hence: it is just a practical mathematical construction aimed at simplifying equations (in a similar manner as the reciprocal lattice, for example). The hole itself is also a mathematical construction helping us to avoid using negative values for mass in the simple equations (the effective mass for conduction occurring through the unoccupied electron states in the valence band would be negative, if we didn't invert the charge).

In the majority of cases, the top of the valence band is clearly "flatter" than the bottom of the conduction band. From this follows that the hole effective mass is often larger than the electron effective mass. The top of the valence band tends to be flatter due to the asymmetry of the situation: you are talking about the highest occupied states for electrons. The bottom of the conduction band is formed by the lowest unoccupied states.

The concept of effective mass comes from the influence of periodic potential on the mass, and therefore, the movement of the holes to be slower, and mobility holes are less than the mobility of electrons.

The concept of effective mass follows from physicists' love for simple relations such as Ohm's law (current density = conductivity x electric field intensity) or Newton's second law of motion (acceleration = force / mass). For a free electron, the mass in the latter is just the electron mass. But, if one wants to write a similar relation for a charge carrier in a crystal lattice, the situation changes. Going through the math (see any solid state physics textbook) allows you to write F = m x a, where the force is charge x electric field, but the mass is no longer the mass of the electron, but reflects the curvature of the conduction band bottom (for electrons) or valence band top (for holes), as it is inversely proportional to the second derivative of the energy as a function of k. This is the so-called effective mass. Again, physicists like simple things, so one often expresses the effective mass as a constant x electron mass, although this has very little (or nothing) to do with the actual physics of the situation. Hence: it is just a practical mathematical construction aimed at simplifying equations (in a similar manner as the reciprocal lattice, for example). The hole itself is also a mathematical construction helping us to avoid using negative values for mass in the simple equations (the effective mass for conduction occurring through the unoccupied electron states in the valence band would be negative, if we didn't invert the charge).

In the majority of cases, the top of the valence band is clearly "flatter" than the bottom of the conduction band. From this follows that the hole effective mass is often larger than the electron effective mass. The top of the valence band tends to be flatter due to the asymmetry of the situation: you are talking about the highest occupied states for electrons. The bottom of the conduction band is formed by the lowest unoccupied states.

From the quantum mechanical point of view the mass is no more an inertial concept as it is generally associated in classic mechanic domain. It is referred as effective mass and it is a mathematical operator that is related to the inverse of the second derivative of energy as a function of the wave number k. For hole carriers we are speaking of the top of the valence band; for free electrons we are speaking of the bottom of the conduction band. In semiconductor materials the bands may be considered ellipsoids with curvatures W(k) more “open” or flat for holes. Therefore, the hole mass is greater than the electron mass. In terms of the conduction or drift transport, this fact leads to electron mobilities greater than the hole mobilities.

Filip Tuomisto has elaborated it very well. And so also Carlos Fernandes. If you do not want an extensive answer like that, mass is measured by the difficulty posed for accelerating or de-accelerating an object.. Some mass and charge in in an environment where other fields exist, the accelerating force (e.g. an electric field) might make it easy or difficult to accelerate. This results in effective mass. The difference between effective mass of a hole and that of an electron results from second derivative of energy vs k curve and is a bit technical but is available in text books.

For real crystals, there isn't really an intuitive answer for why the effective mass of a hole in the valence band is generally larger than for an electron in the conduction band. You just have to find the band structure of your material and compute the second derivative of energy w.r.t. k-vector. As Filip Tuomisto mentioned, the valence band edge is usually "flatter" than the conduction band edge.

But you might find some intuition in the nearly free electron model of energy bands (see e.g. Kittel or Ashcroft & Mermin). In this model, you start by assumming that electrons are nearly free, so their energy is proportional to momentum squared, and the band diagram in reciprocal space is a parabola. Then you add periodic boundary conditions, so you get another parabola at every multiple of 2 pi / a, where "a" is the lattice parameter. Energy gaps open where the parabolas intersect (k = +-pi / a) due to electron wave interference. If you replace the intersection of two positive parabolas with a small band gap, you'll have a valence band maximum and a conduction band minimum in reciprocal space. But parabolas get steeper as they go up, so the conduction band will have to curve a little bit more than the valence band to smoothly connect the parabolas around the band edge. Thus, the curvature of the conduction band is larger in this model, and the effective mass for an electron is lower.

In this way, you can think of the lower effective mass of electrons compared to holes as being a consequence of their non-relativistic dispersion (i.e. E is proportional to k^2). Contrast this with graphene, in which E is proportional to k. With graphene's relativistic-like dispersion, effective mass is no longer a meaningful quantity, but there is no difference between electrons and holes (as long as the system is undoped, etc.). But of course, real materials are much more complicated, and generally you have to look at the band structure in all three dimensions to get a detailed view of how electrons and holes will respond to an electric field.

Thanks Filip Tuomisto for your elaborate answer.

Because of hole being the collective exitation of many electrons, at least three. Nevetheless there exist light holes,e.g. in p-Si.Me=1.08me, Mh=0.56me :)

In a simple two bands model, effective masses are respectively given by the following relations

1/me = 1/m +R and 1/mh = 1/m - R

with R a posituve quantity. Consequently me < mh

Qualitatively, as has been mentioned, the effective mass in a simple s-band tight binding model is m*~1/t (both at band bottom and band top), t being the hopping parameter i.e. related to the band width. Thus low-energy bands, which are comparatively narrower, will generally have larger masses.

Dear sirs,

Thanks for your kind explanation. really the discussions are interesting and thought provoking.

Based on the relation E = mc^2 , Energy of hole is greater than that of electron, thus holes have mass greater than that of electrons.

I do not understand explanations above.They explain that the effective mass is a consequence of energy  dependence on momentum. Sure! But, why these relations are different for electrons and holes?

The textbook explanation is below.

Let us start with the free electron model, where electrons and holes have the same mass. The only difference between electrons and holes is in their energy and, therefore, in velocity.

The effective mass is a result of electron interaction with lattice, i.e. with phonons. As the hole velocity is smaller,  a hole spends more time in the interaction region, i.e. holes strongly interact with phonons. This leads to larger effective mass.

In scientific words,  phonon renormalization of the effective mass of holes is larger than that for electrons. Effect is described by the real part of the electron(hole)-phonon self-energy. It may be calculated in perturbative and non-perturbative ways. 

Actually, effective mass comes from band structure of material. Effective mass is the mass of electrons or holes in the crystal to fit the Newton's Law for them. We know, in crystals various forces act on electrons or holes. So the effective mass of electrons and holes come from including all these force. Holes are modeled as the empty place of electrons which is positively charged. So effective mass of hole depends on the forces acting on it and the degree of those forces. If we find the band structure including these effects then we can find the effective mass for both the electrons and holes. And it is not generally  true that holes are always heavier than electrons. It depends on the forces that acts within the crystal. :) 

All best for research :) 

The mentioned by Andrei renormalisation of the effective masses via the interaction of electrons and holes  with phonons is important  when the processes with rates comparable (or less) than the phonon frequencies in the crystal are considered. Those effective masses are called "polaron" masses and are usually larger (~10%) than those calculated (or obtained) without consideration interaction with phonons - so called  "bare" effective masses. 

Yes, it is so. For example, in Cu2O crystals electrons are heavier then holes.

Let me start by explaining what is meant by "effective mass."  It's a construct designed to make the math simpler, allowing you to treat electron motion through a solid as a particle with definite position and momentum ("semiclassical" physics), under the assumption that all the moving charge carriers are close to the band edge.

Taking an approximation, rather than directly taking the E-k relationship we want to define a term "m*" or "effective mass" that gets us to something where a semiclassical approach is close enough.

see to the figure below

The approximation is the usual Taylor's series expansion.

E(k) ~ E(0) + k dE(k0)/dk + k^2/2 d^2 E(k0)/dk^2 + ...

At the band edge, dE/dk = 0.

This gives us E(k) ~ E(0) + k^2/2 d^2 E(k0)/dk^2.

Hmm... that looks an awful lot like the classical formula for the relationship between momentum and energy:

E = p^2 / 2m

and we already know by definition that p = hbar * k.

Directly substituting the terms and rearranging them, it leaves us with

m* = hbar^2 / (d^2 E(k0)/dk^2).

So effective mass comes out entirely from the band structure.

Looking at some tables of electron and hole effective mass, I find that it isn't always the case that electrons are so much lighter than holes.  Germanium, in particular, has extraordinarily light holes -- m* = 0.04 for holes, versus m* = 1.64 for electrons.  Many other semiconductors instead have heavy holes and light electrons.

For nano particles or quantum dots there are lot of localized energy levels are formed between CB and VB hence electrons faces more interaction with these localize states however sometimes same same situation is observed for impurity or doped semicondutors. In such cases effective mass of Holes found lesser than effective mass of electrons..

Naveed: actually, Ge has m=0.85 at the L valley, CB minimum. 1.4 is for the X valley (where Si has the minimum). And, yes, in the valence the light hole has 0.045, but the heavy hole has 0.28, and the Gamma electrons also have 0.04. It's a mixed bag, even in 'simple' systems.