A useful model for understanding how electrons behave inside crystalline materials is the Kronig-Penney model. Although not a realistic potential for crystals, it allows us to calculate the energy of the electrons as a function of the parameter k that appears in Bloch's theorem.
The Kronig-Penney model represents the background periodic potential seen by the electrons in the crystal as a simple potential shown in Fig. 2.24. The one-dimensional potential has the form
U(x) = 0 0 < x < a
= Uo -b<x < 0 (2.124) The potential is repeated periodically as shown in Fig. 2.24 with a periodicity distance d (= a 4- b). Since the potential is periodic, the electron wavefunction satisfies Bloch's theorem and we may write
<*l>(x + d) = ei4>*P(x) (2.125) where the phase <j) is written as
In the region — 6 < x < a, the electron function has the form ( Ae** + Be-**, if - 6 < x < 0 V>(*) = | Deiax + Fe-ia,t i f o < x < a where
. 2mo(E-Uo) p = V ¥
l2m0E
Then, in the following period, a < x < a + d, from Eq. 2.123
^ei(3(x-d) _^_ ge-i(3{x-d) ^ \f a < X < d (
Qeia(x-d) _|_ pe-ia(x-d)^ if d<X<a + d ^
From the continuity conditions for the wavefunction and its derivative at x = 0 and at x — a, the following system of equations is obtained
A+B = D+F (3(A-B) = a(D-F) ei(^(Aei^ + I?e~?^&) = Deiaa + Fe~iaa
(3e^{Ae^h - Be~i(3h) = a(Deiaa - Fe~iaa) (2.129) Non-trivial solutions for the variables A,B,D,F are obtained only if the determinant of their coefficients vanishes, which gives the condition
a2-S2
cos <j) — cos aa cosh bS — sin aa sinh b8, if 0 < E < Uo 2ab
— cos aa cos bf3 —— sin aa sin b/3, if E > Uo (2.130)
Lap
where
2mo(Uo — E)
\ 2 (2.131) The energy E, which appears in Eq. 2.130 through a,/?, and 6, is physically allowed only if
- 1 < cos <f> < + 1
Consider the case where E < Uo- We denote the right-hand side of Eq. 2.130 by f(E)
( /2m0E\ 2mo(Uo-E) j(E) = cos aw j — cosh I by
Uo-2E I l2m0E\ 2mo(Uo - E) , / 0 1 0 0.
=sm a\ 5— smh \b\ ^-= -\ (2.132)
2.6. Electrons in crystalline solids 89
This function must lie between —1 and +1 since it is equal to cos <j> (= cos kxd). We wish to find the relationship between E and <j) or E and kx. In general, we have to write a computer program in which we evaluate f(E) starting from E = 0, and verifies if f(E) lies between —1 and 1. If it does, we get the value of <j> for each allowed value of E.
The approach is shown graphically in Fig. 2.25a. As we can see, f(E) remains between the dzl bounds only for certain regions of energies. These "allowed energies" form the allowed bands and are separated by "bandgaps." We can obtain the E versus k relation or the bandstructure of the electron in the periodic structure, as shown in Fig. 2.25b.
In the figure, the energies between E2 and E\ form the first allowed band, the energies between £4 and E3 form the second bandgap, etc.
We note that the <f> = kxd term on the left-hand side of Eq. 2.130 appears as a cosine. As a result, \ikxd corresponds to a certain allowed electron energy, then kxd+2n7r is also allowed. This simply reflects a periodicity that is present in the problem. It is customary to show the E-k relation for the smallest fc-values. The smallest &-values lie in a region ±7r/d for the simple problem discussed here. In more complex periodic structures the smallest ^-values lie in a more complicated Ar-space. The term Brillouin zone is used to denote the smallest unity cell of fc-space. If a k-value is chosen beyond the Brillouin zone values, the energy values are simply repeated. The concept of allowed bands of energy separated by bandgaps is central to the understanding of crystalline materials. Near the bandedges it is usually possible to define the electron E-k relation as
n [k — k0)
E = 2m* ( 2 1 3 3 )
where k0 is the &-value at the bandedge and m* is the effective mass. The concept of an effective mass is extremely useful, since it represents the response of the electron-crystal system to the outside world.
Significance of the k-vector
In our discussion of free electrons the quantity /ik represents the momentum of the elec- tron. We have now seen that when electrons are in crystalline systems their properties are described by a wave vector k. What is the significance of k?
For free electrons moving in space two important laws are used to describe their properties: (i) Newton's second law of motion tells us how the electron's trajectory evolves in the presence of an external force; (ii) the law of conservation of momentum allows us to determine the electron's trajectory when there is a collision. As noted in Section 2.3, the Ehrenfest theorem tells us that these laws are applicable to particles in quantum mechanics as well. We are obviously interested in finding out what the analogous laws are when an electron is inside a crystal and not in free space.
An extremely important implication of the Bloch theorem is that in the perfectly periodic background potential that the crystal presents, the electron propagates without scattering. The electronic state (~ exp(ik • r)) is an extended wave which occupies the entire crystal. To complete our understanding, we need to derive an equation of motion for the electrons which tells us how electrons will respond to external forces.
cos kyd = 1
r~
3r a allowed band Bandgap
} 2n d allowed band
> Bandgap
> 1s* allowed band
Numerical approach to calculate E - kx relation
Start with energy E equal to the lowest potential energy value,
Ef ~
f Ei + ±E
^E) >^J~
^No
3S Ef is in forbidden gap Et = Ef
E* is in an allowed band
Use cos <|> = cos kyd =f(E) to obtain fcr-values
(b)
Figure 2.25: (a) The graphical solution to obtain the allowed energy levels. The function f(E) is plotted as a function of E. Only energies for which f(E) lies between +1 and —1 are allowed.
(b) The allowed and forbidden bands are plotted in the E versus k relation using the results from (a). The inset shows a flow chart of how we can obtain the E-kx relation.
2.6. Electrons in crystalline solids 91 The equation of motion
— = Fext+Fint (2.134)
| , •"- C A l I lilt \ /
is not very useful for a meaningful description of the electron because it includes the internal forces on the electron. We need a description which does not include the eval- uation of the internal forces.
As in classical wave theory, associated with any wave phenomena is the wave group velocity that represents the propagation of wave energy. In the case of a particle wave the group velocity represents the particle velocity. We can define the group velocity of this wavepacket as
V9 = ^ r (2.135)
where u> is the frequency associated with the electron of energy E; i.e., LJ — E/Ti:
idE
If we have an electric field F present, the work done on the electron during a time interval 6t is
6E = -eF -vg6t (2.136)
We may also write, in general
= n-vg-Sk (2.137)
Comparing the two equations for 6E, we get
giving us the relation
nft = -eF (2.138)
In general, we may write
h^- = Fe x t (2.139)
Eq. 2.139 looks identical to Newton's second law of motion
F
It ~ F e x t
in free space if we associate the quantity ftk with the momentum of the electron in the crystal. The term ftk responds to the external forces as if it is the momentum of the electron, although, as can be seen by comparing the true Newtons equation of motion, it
Electron in a periodic potential
<£•
<+ *
E versus k relation effective mass Equation of motion ^ ~Jf -
Electron behaves as if it is in free space, but with a different effective mass
Figure 2.26: A physical description of electrons in a periodic potential. As shown the electrons can be treated as if they are in free space except that their energy-momentum relation is modified because of the potential. Near the bandedges the electrons respond to the outside world as if they have an effective mass m*. The effective mass can have a positive or negative value.
is clear that hh contains the effects of the internal crystal potentials and is therefore not the true electron momentum. The quantity ftk is called the crystal momentum. Once the E versus k relation is established, we can, for all practical purposes, forget about the background potential U(r) and treat the electrons as if they are free and obey the effective Newtons equation of motion. This physical picture is summarized in Fig. 2.26.