ABSTRACT
On comparison with Eq.(40.3), we may infer that ( )Rr GG +ψ and ( )rGψ correspond to the same eigenvalue ,ε that is,
( ) ( ) ( ) ,r R C R rψ + =G GG Gψ where ( ) 2 1C R =G (41.3)
In other words, under any translation R G
that leaves the lattice invariant the wave function is multiplied by a phase factor ( )RC G . Substituting RR ′+ GG for RG yields ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )r R R C R R r C R r R C R C R rψ ψ ψ′ ′ ′ ′+ + = + = + =G G G G G G G GG G G ψ G that is,
( ) ( ) ( )C R R C R C R′+ =G G G G′ or ( ) RkieRC GGG ⋅= (41.4) and hence, Eq.(41.3) reduces to the Bloch condition given by ( ) ( )reRr Rki GGG GG ψψ ⋅=+ (41.5) The significance of may be derived by using periodic boundary conditions on the wave functions (41.5), which read
k G
( ) ( ) ( ) ( )rcSrbQraPr GGGGGGG ψψψψ =+=+=+
and this requires that
2 , 2 , 2Pk a p Qk b q Sk c sπ π π⋅ = ⋅ = ⋅ =G G G GG G where are arbitrary integers. Thus, sqp and, k
G is a vector of k
G -space of the form
∗∗∗ ++= c S sb
Q qa
P pk G
GGG (41.6) where the components / , / and /p P q Q s S are, in general, fractions of an integer. With each -value we may associate a one-electron wave function of the form k
G ( ) ( )ruer krkik GG GG
called a Bloch wave which, upon substitution into the Bloch condition (41.5), leads to ( ) ( ) ( )rueeRrue krkiRkikRrki GGG GGGGGGGGG ⋅⋅+⋅ =+ or ( ) ( )ruRru kk GGG GG =+ (41.8) Equations (41.7) and (41.8) express the Bloch theorem which states that in a periodic potential (41.1) one can always choose the one-electron wave functions to be Bloch waves of the form (41.7), where ( )ruk GG has the periodicity (41.8) of the lattice. Substituting Eq.(41.7) into the one-electron Schrödinger equation (40.3) we find an equation for ( )ruk GG in the form
( ) ( ) ( ) ( )2 2 22 2 k ke
ik k u r U r u r u r m
ε⎡ ⎤− ∇ + ⋅∇ − + =⎣ ⎦ G G kG G G= G G G G (41.9)
which can be rewritten as
( ) ( ) ( )ru m krurUpk
G=GGGG== GG ⎟⎟⎠ ⎞
⎜⎜⎝ ⎛ −=⎥⎦
⎤⎢⎣ ⎡ +⋅+∇−
ε (41.10)
Considering the complex conjugate equation, yields ( ) ( )k ku r u r∗ −=G GG G and this leads to
( ) ( )rr kk GG GG −∗ =ψψ (41.11) and also
( ) ( )kk GG εε =− (41.12) which means that any energy band is symmetric about the origin of -space.k
G
In the interval ,0 ax << where the potential vanishes, to a first approximation, a trial solution is
which can be rewritten as
Figure 41.1. Bloch wave of lowest energy in a one-dimensional periodic potential
xud m
00 2 ∑ −−=+ δε = ) Integrating both sides over a small interval about 0,x = we obtain
The first term on the left-hand side vanishes for 0,ρ → and the remaining terms show that
0 u Vm
dx xdu e
x = −=⎥⎦
⎤⎢⎣ ⎡
As a result of these two conditions on ( )xu0 and its derivative, the lowest energy Bloch wave may be represented as in Figure 41.1. The two homogenous equations for A and B have nonidentically vanishing solutions provided that
/ or tanh
α α ααα += =−
This equation can be graphically solved and provides α in terms of that define the aV and0 periodic potential energy ( ).xU 41.2. THE WEAK-BINDING APPROXIMATION If we think of the potential of the ionic cores as being rather small and acting as a slight perturbation on the free electrons in the solid, we obtain the weak-binding approximation, also called the nearly-free electron model. Such an approximation is valid for electrons in conductors and semiconductors. It is convenient to use the Fourier expansion in terms of reciprocal lattice vectors K
G for the periodic potential energy
( ) ,U rG which again has the translational symmetry of the lattice as ( ) ( ) rKi
K eKUrU
G GG ⋅∑= (41.13)
and also for the periodic part ( )ruk GG of the Bloch waves, which reads
( ) ( ) rKi K
GG ⋅∑= (41.14) From Eq.(41.7) it follows that
r a K eψ + ⋅=∑ G G GG G GG (41.15) Substituting Eqs.(41.13) and (41.15) into the one-electron Schrödinger equation (40.3) yields
( ) ( ) ( ) ( ) ( ) (
k K a K e U K a K e
⎡ ⎤+⎢ ⎥ ′ ′′− =⎢ ⎥⎢ ⎥⎣ ⎦ ∑ ∑G GG G + ⋅GG GG G G
G G= G G G (41.16)
If, in the zeroth order approximation, we consider a constant potential (41.13), where ( ) ( )0 0 and 0 for all 0,U U K≠ =G GK ≠ Eq.(41.16) reduces to ( ) ( ) ( ) ( )KaUKa
m Kk
GGGG= 0 2
=⎥⎥⎦ ⎤
⎢⎢⎣ ⎡ +−ε
and this gives the energy shift of the free electron of wave vector k K+G G as
( ) ( ) ( ) Kk e
U m
KkU GG GG=
++= ++= εε 0
2 0
(41.17)
Thus, the free electron eigenvalues are shifted, for ,0=KG by
( ) ( ) k e
U m kU G G
= εε +=+= 0 2
0 22
(41.18)
In the next approximation, we consider an almost constant potential (41.13), where ( ) ( )0 for all 0,U K U K<< ≠G G so that Eq.(41.16) becomes ( ) ( ) ( ) ( ) ( ) ( )
K K U a K U K a U K a Kε ε +
≠ ⎡ ⎤ ′− − − = −⎣ ⎦ K ′∑G G G GG G G G G (41.19)
We may drop the sum on the right-hand side of Eq.(41.19), provided that the ( )KKa ′− GG coefficients are negligible for ( ) ( ), that is 0 .K K a K K a′ ′≠ − <<G G G G Since KG is an arbitrary vector of the reciprocal lattice, the validity of this assumption can be understood from the reduced equation
( ) ( ) ( ) ( )0 k KU a K U K aε ε +⎡ ⎤− − − =⎣ ⎦G G 0 0G G (41.20) if is approximated by the free electron shifted eigenvalues (41.18), which gives ε ( ) ( ) ( ) ( )0 0k K a K U K aε ε +− −G G =G G As, in general, we have ( )2 22 / 2 ek K k k K K mε ε+ − = ⋅ + ≠G GG 0,G G G= one obtains
( ) ( ) ( ) ( )( ) ( )2 2 2
0 0 2
U K m U K a K a a
k K Kε ε + = = −− ⋅ +G G G
G G G
G G G= (41.21)
but we have assumed ( )KU G to be negligible, and hence, ( ) ( )0 for all 0.a K a K<< ≠G G Since ( )KU G is a Fourier coefficient of the potential energy expansion (41.13), it can be written as the Fourier transform of ( )rU G , Eqs.(16.61), and this can be regarded as a matrix element of which joins two free-electron states differing in
their wave number by a reciprocal lattice vector
that is,
( ) ( ) 0 00
G G
+∑ −+= ψεε ψψψψ (41.23)
Standard perturbation treatment, developed in Chapter 36, leads to the second-order expression (36.24) for energy, which reads
20 0 2 2 2
U KU r mk kU U m m k K K
ψ ψε ε ε +
+ = + + = + −− ⋅ +∑ ∑
GGG G= = G G G= (41.24)
For each given value of ,K
G the wave vectors k
G which span the boundary of the
corresponding Brillouin zone in k G
GGG εε ==+⋅ +or02 2 (41.25) Thus, if k
G is near a Brillouin zone boundary, defined by a particular value of ,K
G one of
the ( )a KG coefficients becomes large, according to Eq.(41.21), and therefore, all the ( )KKa ′− GG coefficients in Eq.(41.19) become negligible either for ,K K′ ≠G G in which case ( ) ( )0 ,a K K a′− <<G G or for which implies that ,0≠′KG ( ) ( ).a K K a K′− <<G G G In these
conditions, there are two valid approximations only for Eq.(41.19), which form a set of simultaneous homogenous equations for ( ) ( )0and aKa G given by
( ) ( ) ( ) ( )0 0k KU a K U K aε ε +⎡ ⎤− − − =⎣ ⎦G G 0G G (41.26)
( ) ( ) ( ) ( )0 0kU a U K a Kε ε⎡ ⎤− − − − =⎣ ⎦G 0G G where ( ) ( )KUKU GG ∗=− for any Fourier coefficient of the real potential function . ( )rU G The free-electron states andk k Kψ ψ +G G G appear in this case, from Eq.(41.25), to be degenerate, so that the Bloch wave (41.15) should be written in terms of the two coefficients ( ) ( )0 anda a GK as
which may be interpreted to be a linear combination of degenerate states. The matrix eigenvalue equation for the degenerate case is obtained from Eq.(41.26) as
( ) ( )( ) ( ) ( )( ) ( )( )⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠
⎞ ⎜⎜⎝ ⎛
+ +
a Ka
a UKU
KUU
k GGG G
G 00 0
0 εε ε
(41.27)
and the characteristic equation is
( ) ( ) ( ) 20 0k k KU U U Kε ε ε ε +⎡ ⎤ ⎡ ⎤− − − − − =⎣ ⎦ ⎣ ⎦G G G 0G (41.28) This provides two energy eigenvalues of the form
( ) ( )2 20 2 2k k K k k KU Uε ε ε εε + +± + −⎛ ⎞= + ± +⎜ ⎟⎝ ⎠ G G G GG G
K G
(41.29)
For ( ) ,k k KU K ε ε +<< −G G GG Eq.(41.29) reduces to the solutions (41.18) and (41.17) which give the first order energy shifts for the free-electron states and .k k K+G G G At each zone boundary in k
G -space, where ,Kkk GGG += εε there is an energy gap of magnitude
( )2 U Kε ε ε+ −∆ = − = G (41.30)
Figure 41.2. Energy-band structure in the weak binding approximation (a) and its representation reduced to the first Brillouin zone (b)
The energy spectrum of nearly-free electrons exhibits allowed energy bands ( )kGε at successively higher energies, with forbidden energy gaps between adjacent energy bands, as illustrated in Figure 41.2 (a) for the one-dimensional case. The significance of the energy gaps in the ε versus kG spectrum, at -valueskG which are said to define zone boundaries in k
G -space, can be given a simple interpretation in terms of Bragg reflection.
Bloch waves with wave numbers which satisfy Eq.(41.25), and hence, the Laue condition (39.28), are Bragg reflected and do not propagate, giving rise to a gap in energy. In this respect, the effect on the Bloch waves of the potential energy , having the periodicity of the lattice, is similar to that of a diffraction grating on the propagation of plane waves. It follows that we must expect that a "bands and gaps" structure of the energy spectrum will occur for any periodic potential. Since the Bloch wave functions have the periodicity of the reciprocal lattice given by
