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Divalent metals 119

7.10 Divalent metals
We may now return to the case of beryllium and magnesium and to their col- The divalent metals, having two
leagues, generally referred to as divalent metals. One-dimensional theory is valency electrons, are found in
unable to explain their electric properties; let us try two dimensions. groups IIA and IIB of the periodic
The E – k x , k y surface may be obtained from eqn (7.34) as follows: table (Fig. 4.5).
E = E 1 –2A x cos k x a –2A y cos k y b. (7.68)
Let us plot now the constant energy curves in the k x –k y plane for the simple
case when
1
E 1 =1, A x = A y = , a = b. (7.69)
4
It may be seen in Fig. 7.15 that the minimum energy E = 0 is at the origin
and for higher values of k x and k y the energy increases. Note well that the
boundaries k x = ±π/a and k y = ±π/a represent a discontinuity in energy.
(This is something we have proved only for the one-dimensional case, but the
generalization to two dimensions is fairly obvious.) There is an energy gap
there. If the wave vector changes from point B just inside the rectangle, to
point C, just outside the rectangle, the corresponding energy may jump from
one unit to (say) 1.5 units. The usual notation is to call the
Let us now follow what happens at T = 0 as we fill up the available states rectangle the ‘first Brillouin zone’,
with electrons. There is nothing particularly interesting until all the states up and as we step out of it (say at
to E = 1 are filled, as shown in Fig. 7.16(a). The next electron coming has an point C) we reach the ‘second
itch to leave the rectangle; it looks out, sees that the energy outside is 1.5 units, Brillouin zone’. The shape of the
and therefore stays inside. This will go on until all the states are filled up to higher Brillouin zones can be de-
an energy E = 1.5, as shown in Fig. 7.16(b). The remaining states inside our termined with not too much effort
rectangle have energies in excess of 1.5, and so the next electron in its search but it is beyond the scope of the
for lowest energy will go outside. It will go into a higher band because there present book.
are lower energy states in that higher band (in spite of the energy gap) than
inside the rectangle.

C
π/a
B
k 1.85
y
1.5
1.0
0.5

0 E=0.15

Fig. 7.15
Constant energy contours for a
two-dimensional crystal in the k x – k y
–π/a plane on the basis of the Feynman
–π/a 0 k π/a
x model.