The Bloch Theorem

1.5.2 The Bloch Theorem

In the ‘free electron’ theory → → → → → what do we assume ? Actually, we assume that an electron moves in a constant potential V 0 leading to a Schrodinger

equation for a one-dimensional case :

d 2 ψ ⎛ 2m ⎞

2 + ⎜ D ⎟ [E – V 0 ] dx ψ=0 ⎝ ⎠

The solution of this equation is given by the ‘plane waves’ of the type as :

(1.86) Upon substitution, we get the ‘kinetic energy’ of the electron as :

ψ(x) = e ± ikx

E kin =E–V 0 =

2m

The physical meaning of k is that it represents the momentum of electron divided by D . In order to get a complete solution of the above wave function with time, we multiply ψ(x) by exp(– iωt), where

E ω= D , so that the solutions of the type of equation (1.86) represents waves propagating along the x-

axis.

GENERAL INTRODUCTION

The Periodic Potential

Now, let us consider the Schrodinger equation for an electron moving in a one-dimensional periodic potential. Therefore, the potential energy [V(x)] must satisfy the equation :

(1.88) where, a = period of the ‘lattice’. The Schrodinger equation is then expressed as :

V(x) = V(x + a)

⎛ 2m ⎞

dx 2 ⎜ ⎝ D ⎟ [E – V(x)] ψ=0

For a solution of this equation, there is an important ‘theorem’ → which states that there exists solutions of the form : ψ(x) = e ± ikx u k (x) with u k (x) = u k (x + a)

(1.90) In other words, the solutions are ‘plane waves’, which are modulated by the function u k (x) →

which has the same periodicity as the ‘lattice’. This theorem is known as the “Bloch Theorem”. In the theory of differential equations, it is called Floquet’s theorem. The functions of this type (1.90) are called “Bloch Functions”. The Bloch function has the property :

ψ(x + a) = exp[ik(x + a)] u k (x + a) = ψ(x) exp(ika) since u k (x + a) = u k (x). In other words, the Bloch functions have the property : ψ(x + a) = Q ψ(x), with Q = exp(± ika)

(1.91) Now, it is evident that → if we can show that the Schrodinger equation (1.89) has solutions with

the property (1.91), the solutions can be written as Bloch functions, and the Bloch theorem is then proven.

The Proof

Let us suppose A(x) and B(x) are two ‘real’ independent solutions of the Schrodinger equation. We know that a differential equation of the 2nd order has only two independent solutions, and all other solutions can be expressed as a ‘linear combination’ of the independent ones. Hence, since A(x + a) and B(x + a) are also solutions of the Schrodinger equation, we get the following relations :

A(x + a) = α A(x) + β B(x) B(x + a) = γ A(x) + δ B(x)

(1.92) where, α, β, γ and δ are real functions of E. The solution of the Schrodinger equation may be written in

the form : ψ(x) = F A(x) + G B(x)

where, F and G are two arbitrary constants. According to (1.92), we get : ψ(x + a) = (F.α + G.γ) A(x) + (F.β + G.δ) B(x) In view of the ‘property’ of the Bloch function in equation (1.91), let us now choose F and G in

such a manner, so that we get :

F. α + G.γ = Q F

F. β + G.δ = Q G (1.93) where, Q = Constant. In this way, we have obtained a function ψ(x) with the following ‘property’ : ψ(x + a) = Q ψ(x)

NANO MATERIALS

i.e., equation (1.91). Since equation (1.93) have the ‘non-vanishing’ solutions for F and G, only if the “determinant” of their coefficients vanishes, so that we have the equation for Q as :

or, Q 2 –( α + δ)Q + α.δ – β.γ = 0 (1.95) Now, it is possible to show that α.δ – β.γ = 1 in the following manner : we can derive from

A( ′ x + a )B( ′ x + a ) A()B() ′ x ′ x γδ (1.96)

d A( ) x

d B( ) x

where, A ′(x) =

. Now, if we multiply the Schrodinger equation for B(x) by dx

and B ′(x) =

dx

A(x), and the equation for A(x) by B(x), and we do the required subtraction, we get : 0=AB d ′′ – B A′′ = (A B ′ – B A′)

dx

Hence, in this case, the so-called “Wronskian” is a constant :

A( ) x B( ) x

′ (1.97) ′ x This result together with equation (1.96) leads to the conclusion that α.δ – β.γ = 1. Therefore,

W (x) =

= Constant

A()B() x

instead of equation (1.95), we may write it as : Q 2 –( α + δ)Q + 1 = 0

(1.98) Then, in general, there are two ‘roots’ : Q 1 and Q 2 , i.e., there are two functions ψ 1 (x) and ψ 2 (x),

which exhibit the property (1.93). It should be noted that the product Q 1 Q 2 = 1. For a certain range of energy E, i.e., for those corresponding to (

α + δ) 2 <4 → The two roots Q

and Q 2 will be “complex”, and since Q 1 Q 2 = 1, they will be “conjugates”. In those regions of energy, we may then write them as :

Q 1 = exp(ika)

and,

Q 2 = exp(–ika)

(1.99) Then, the corresponding eigenfunctions ψ 1 (x) and ψ 2 (x) have the ‘property’ written as :

ψ 1 (x + a) = exp(ika) ψ 1 (x)

and,

(1.100) Thus, following equation (1.90), they are the “Bloch Functions”. In other regions of energy E, where, if we put ( α + δ) 2 > 4, the two ‘roots’ are real and reciprocals

ψ 2 (x + a) = exp(– ika) ψ 2 (x)

of each other. The roots corresponding to the solution of Schrodinger equation are :

ψ 1 (x) = exp( µx)u(x)

and,

ψ 2 (x) = exp(– µx)u(x)

35 where, µ = a real quantity. Although they are mathematically all right, they can not be accepted as “wave

GENERAL INTRODUCTION

functions” describing the electrons, since they are not bounded. Thus, there are no electronic states in

the energy regions corresponding to the real ‘roots’ Q 1 and Q 2 .

Therefore, we get the “notion” that the energy spectrum of an electron in a periodic potential consists of → ALLOWED and FORBIDDEN energy regions or bands.