18
INTRODUCTION TO PHYSICS OF THE SOLID STATE
where N = 2 for 11-VI and N = 3 for 111-V compounds. The fractional charges all
lie in the range from 0.43 to 0.49 for the compounds under consideration. Using the
e* tabulations in the Boer book and Eq. (2.7), we obtain the fractional covalencies of
a:ov - 0.81 for all the 11-VI compounds, and a:ov - 0.68 for all the 111-V
compounds listed in Tables B.l, B.4, B.5, and so on. These values are consistent
with the better fit of the covalent model to the crystallographic data for these
compounds.
We conclude this section with some observations that will be of use in later
chapters. Table B.l shows that the typical compound GaAs has the lattice constant
a = 0.565 nm, so the volume of its unit cell is 0.180 nm3, corresponding to about 22
of each atom type per cubic nanometer. The distances between atomic layers in the
100, 110, and 11 1 directions are, respectively, 0.565 nm, 0.400 nm, and 0.326 nm for
GaAs. The various 111-V semiconducting compounds under discussion form mixed
crystals over broad concentration ranges, as do the group of 11-VI compounds. In a
mixed crystal of the type In,Ga,-,As it is ordinarily safe to assume that Vegard’s law
is valid, whereby the lattice constant a scales linearly with the concentration
parameter x. As a result, we have the expressions
a(x) = a(GaAs) + [a(InAs) - a(GaAs)]x
= 0.565 + 0 . 0 4 1 ~
(2.8)
where 0 5 x _< 1. In the corresponding expression for the mixed semiconductor
Al,Ga,-,As the term +O.OOlx replaces the term +O.O41x, so the fraction of lattice
mismatch 21aAIAs - u G ~ A ~ ~ / ( u A ~ ~
+ aGaAs) = 0.0018 = 0.18% for this system is
quite minimal compared to that [21aInAs - aGaAsI/(u1ds + aGaAs) = 0.070 = 7.0%]
of the In,Ga,-,As system, as calculated from Eq. (2.8) [see also Eq. (10-3).] Table
B. 1 gives the lattice constants a for various 111-V and 11-VI semiconductors with the
zinc blende structure.
2.1.5. Lattice Vibrations
We have discussed atoms in a crystal as residing at particular lattice sites, but in
reality they undergo continuous fluctuations in the neighborhood of their regular
positions in the lattice. These fluctuations arise from the heat or thermal energy in
the lattice, and become more pronounced at higher temperatures. Since the atoms are
bound together by chemical bonds, the movement of one atom about its site causes
the neighboring atoms to respond to this motion. The chemical bonds act like
springs that stretch and compress repeatedly during oscillatory motion. The result is
that many atoms vibrate in unison, and this collective motion spreads throughout the
crystal. Every type of lattice has its own characteristic modes or frequencies of
vibration called normal modes, and the overall collective vibrational motion of the
lattice is a combination or superposition of many, many normal modes. For a
diatomic lattice such as GaAs, there are low-frequency modes called acoustic modes,
in which the heavy and light atoms tend to vibrate in phase or in unison with each
INTRODUCTION TO PHYSICS OF THE SOLID STATE
where N = 2 for 11-VI and N = 3 for 111-V compounds. The fractional charges all
lie in the range from 0.43 to 0.49 for the compounds under consideration. Using the
e* tabulations in the Boer book and Eq. (2.7), we obtain the fractional covalencies of
a:ov - 0.81 for all the 11-VI compounds, and a:ov - 0.68 for all the 111-V
compounds listed in Tables B.l, B.4, B.5, and so on. These values are consistent
with the better fit of the covalent model to the crystallographic data for these
compounds.
We conclude this section with some observations that will be of use in later
chapters. Table B.l shows that the typical compound GaAs has the lattice constant
a = 0.565 nm, so the volume of its unit cell is 0.180 nm3, corresponding to about 22
of each atom type per cubic nanometer. The distances between atomic layers in the
100, 110, and 11 1 directions are, respectively, 0.565 nm, 0.400 nm, and 0.326 nm for
GaAs. The various 111-V semiconducting compounds under discussion form mixed
crystals over broad concentration ranges, as do the group of 11-VI compounds. In a
mixed crystal of the type In,Ga,-,As it is ordinarily safe to assume that Vegard’s law
is valid, whereby the lattice constant a scales linearly with the concentration
parameter x. As a result, we have the expressions
a(x) = a(GaAs) + [a(InAs) - a(GaAs)]x
= 0.565 + 0 . 0 4 1 ~
(2.8)
where 0 5 x _< 1. In the corresponding expression for the mixed semiconductor
Al,Ga,-,As the term +O.OOlx replaces the term +O.O41x, so the fraction of lattice
mismatch 21aAIAs - u G ~ A ~ ~ / ( u A ~ ~
+ aGaAs) = 0.0018 = 0.18% for this system is
quite minimal compared to that [21aInAs - aGaAsI/(u1ds + aGaAs) = 0.070 = 7.0%]
of the In,Ga,-,As system, as calculated from Eq. (2.8) [see also Eq. (10-3).] Table
B. 1 gives the lattice constants a for various 111-V and 11-VI semiconductors with the
zinc blende structure.
2.1.5. Lattice Vibrations
We have discussed atoms in a crystal as residing at particular lattice sites, but in
reality they undergo continuous fluctuations in the neighborhood of their regular
positions in the lattice. These fluctuations arise from the heat or thermal energy in
the lattice, and become more pronounced at higher temperatures. Since the atoms are
bound together by chemical bonds, the movement of one atom about its site causes
the neighboring atoms to respond to this motion. The chemical bonds act like
springs that stretch and compress repeatedly during oscillatory motion. The result is
that many atoms vibrate in unison, and this collective motion spreads throughout the
crystal. Every type of lattice has its own characteristic modes or frequencies of
vibration called normal modes, and the overall collective vibrational motion of the
lattice is a combination or superposition of many, many normal modes. For a
diatomic lattice such as GaAs, there are low-frequency modes called acoustic modes,
in which the heavy and light atoms tend to vibrate in phase or in unison with each