Why is the parabolic shape of holes broader than that of electrons?

[bluejay27]

Why is the parabolic shape for the hole broader than that of the electron?

 

[DrDu]

For whivh substance?

 

[Dr Transport]

generally the effective mass of a hole is different than that of the electron...

 

[learn.steadfast]

In general, you need a model to determine what the band-shape is. A typical model, to understand band theory, is the Kronig Penny model. A decent Graph is here:
http://ecee.colorado.edu/~bart/book/book/chapter2/ch2_3.htm

The model is used to figure out what causes bands in semiconductors, and is very simplified. However, there are some general trends that can be related to very basic atomic physics which ought to apply to most real semiconductors; eg: That there is a lattice of atoms that are approximately equally spaced, and there is a periodic potential associated with that lattice which is causing electrons to be "reflected" with a certain probability at each atom; eg: the atoms are attempting to trap electrons in a potential energy well.

A free electron, with no crystal lattice, has a nearly pure parabolic shape for the relationship between energy and momentum. In the graphs, the green line is a free electron parabola. That's how an electron would behave if no crystal was present at all. The black lines are how the electrons would behave with a small periodic potential. eg: an atomic lattice with atoms having a weak attraction to the electron.

There are a few things to note about the graphed solution:

1) Shrodinger's equation solutions for the Kronig-Penny model are based on sinusoidal functions which repeat periodically; therefore each solution (or "band") actually repeats itself periodically every 2 pi radians as either an "even" or "odd" function depending on whether the momentum at "0" represents either 0 or pi radians.

In all three graphs (see above), all three diagrams are equivalent; they are just "aliased" graphs of functions where the "bands" are drawn over a different period; either -pi to pi, or pi to 2pi, 2pi to 3pi, etc.

2) Each band alternates in whether it starts with it's maximum or minimum at the k=0 point of the diagram. (This is easiest to see in the rightmost graph). Therefore, the bottom of each band alternates in whether or not it acts as a "hole" or an "electron" at zero momentum.

Now, if you look closely at the first graph; you'll notice that each band is taller and narrower than the last; The width of each band is the same (2 pi radians total split up between pi negative radians,and pi positive radians); Notice also that the black line touches the green line at the end of every period: Look at "1 pi radians" and the black line touches the green parabola at ~0.375eV, at "2 pi radians" the black line touches the green line at ~1.5eV, and at "3 pi radians" the black line touches the green parabola at ~3.15 eV.

From basic physics, there is a problem called the "infinite" well, and the Schrödinger equation solves to only have discrete energy levels; the levels are spaced in proportion to n squared. The same is true in the Kronig-Penny model, which can reduce to the infinite square well problem if the periodic potential is set to "infinite." The energy levels where the free electron parabola touches the "bands" is exactly where the square law energy levels are in the infinite well problem.

If you take just a few moments to look at the graph, qualitatively, you'll notice that the bandgaps are approximately constant in size; (They aren't really, but their size varies much slower than n squared.); Therefore, each band's height is approximately the difference of successive squares and a constant; eg: height(n) ~= ( n² - (n-1)² -k ) ~= 2n - 1 - k.

That means, the parabolic shape generally increases in height to width ratio for each band (n) traversed.
Said another way, given a point near zero momentum, the height to with ratio of that point from the bottom of the band is smaller for lower bands. Therefore the parabola is "fatter" looking near the zero momentum point on lower bands than higher ones. This is most easily seen on the very lowest parabola on the third diagram, vs. the one right above it. The same is true between the second and third parabolas, but it's much less of a difference and harder to see.

But: in general -- Holes near the top of a typical valence band are going to be slightly "heavier" than "electrons" near the bottom of a conduction band. This is a fact caused by the "free" electron slope always being larger on average as momentum increases because energy is proportional to the square of momentum.

It is possible for other considerations to upset this general trend as there are interference effects of orbital shapes which give rise to "light" holes, and other carriers; but I wanted to give you a general notion of why at least one of the holes band in a typical semiconducor is usually higher mass (fatter parabola) than the electrons in the band just above it. The "average" trend is that holes are heavier than electrons in a typical semiconductor.