BAND THEORY OF SOLIDS

Bloch Theorem Blocks theorem was formulated by the German-born US physicist Felix Bloch

(1905ndash83) in 1928Accordind to this theorem in a periodic structure every

Electronic wave function can be represented by a ldquoBloch functionrdquo

A crystalline solid consists of a lattice which is composed of a large number of ionic cores at regular intervals and the conduction electrons can move freely throughout the lattice

A theorem that specifies the form of the wave functions that characterize electron energy levels in a periodic crystal Electrons that move in a constant potential have wave functions that are plane waves having the form exp(i k middot r ) Here k is the wave vector which can assume any value and describes an electron having momentum P= ħk Electrons in a crystal experience a potential that has the periodicity same as the crystal lattice

Now since the lattice is a repetitive structure of the ion arrangement in a crystal the type of variation of V also repeats itself If a is the inter ionic distance then as we move in x-direction the value of V will be same at all points which are separated by a distance equal to a

ie V(x) = V( x + a ) where x is distance of the electron from the core

Such a potential is said to be a periodic potential

Blochs theorem states that ldquothe solution of SWequation has the form of a plane wave multiplied by a function with the period of the Crystal latticerdquostating that the wave function ψ for an electron in a periodic potential has the form

ψ(x) = exp(ik x)Uk(x) in One dimensions BLOCH

FUNCTIONSψ(r) = exp(ik r)U(r) in 3-dimensional motion

where k is the wave vector r is a position vector and U(r) is a periodic function that satisfies

U(r+a) = U(r)

for all vectors lsquoarsquo of the Crystal lattice of the crystal Blocks theorem is interpreted to mean that the wave function for an electron in a periodic potential is a plane wave modulated by a periodic function This explains why a free-electron model has some success in describing the properties of certain metals although it is inadequate to give a quantitative description of the properties of most metals

In order to understand the physical properties of the system it is required to solve the Schroumldingerrsquos equation However it is extremely difficult to solve the Schroumldingerrsquos equation with periodic potential described above Hence the Kronig ndash Penney Model is adopted for simplification

THE KRONIG -PENNEY MODEL

It is assumed in quantum free electron theory of metals that the free electrons in a metal experience a constant potential and is free to move in the metal This theory explains successfully most of the phenomena of solids But it could not explain why some solids are good conductors and some others are insulators and semi conductors It can be understood successfully using the band theory of solids

According to this theory the electrons move in a periodic potential produced by the positive ion cores The potential of electron varies periodically with periodicity of ion core and potential energy of the electrons is zero near nucleus of the positive ion core It is maximum when it is lying between the adjacent nuclei which are separated by interatomic spacing The variation of potential of electrons while it is moving through ion core is shown fig

Fig One dimensional periodic potential

V ( x ) = 0 for the region 0 lt x lt a = V0 for the region -b lt x lt 0

----------------(1)

Applying the time independent Schroumldingerrsquos wave equation for above two regions

d2Ψ dx2 + 2 m E Ψ ħ2 = 0 for region-I 0 lt x lt a ---------------(2)

and d2Ψ dx2 + 2 m ( E ndash V ) Ψ ħ2 = 0 for region-II -b lt x lt a ----------------(3)

Substituting α2 = 2 m E ħ2

---------------(4)

β2 = 2 m ( V ndash E ) ħ2

----------------(5)

d2Ψ dx2 + α2 Ψ = 0 for region 0 lt x lt a ---------------(6)

d2Ψ dx2 - β2 Ψ = 0 for region -b lt x lt a --------------(7)

The solution for the eqnrsquos (6) and (7) can be written as

Ψ ( x ) = Uk ( x ) eikx ---------------(8)

The above solution consists of a plane wave eikx modulated by the periodic function

Uk(x) where this Uk(x) has the periodicity same as that of the ion

ie Uk(x) = Uk(x+a) ---------------(9)

where k is propagating constant along x-direction and is given by k = 2 Π λ This k is also known asPropagation wave vector

Differentiating equation (8) twice with respect to x and substituting in equation (6) and (7) two independent second order linear differential equations can be obtained for the regions-I and II

ie

Substituting these values in equations

+ 2iK + = 0 for 0ltxlta

---------------------(10)

and +2iK - for ndashbltxlt0

------------------------(11)

where u1 represents the value of uk (x) in the interval 0ltxlta and u2 the value of uk (x) in the interval ndashbltxlt0

The solution of the differential equation

U1 = emx

and

Substituting these values in equation we get

m2 emx +2iK memx + emx = 0

m2+2iKm + = 0

m =

m = -ik im1 = -ik +i = i (-K)

m2 = -iK-i = -i (+K)

Thus the general solution is

u1 = Aem1

x + B em2x

u1 = Aei(-K)x + B e-I (+k)x

Similarly we have the solution for equation (12) asu2 = Ce(β-iK)x + D e-i (β +ik)x

Where ABC and D are constants The values of the constants ABC and D can be obtained by applying the boundary conditions

and

Applying the boundary conditions to the solution of above equations four linear equations in terms of ABC and D it can be obtained (where ABCD are constants ) the solution

for these equations can be determined only if the determinant of the coefficients of A B C and D vanishes on solving the determinant Thus we have

(β2 - α2 2 α β)sin hβb sin αa + cos hβb cos αa = cos k ( a + b ) ------------ (12)

The above equation is complicated and Kronig and Penney could conclude with the equation Hence they tried to modify this equation as follows

Let Vo is tending to infinite and b is approaching to zero Such that Vob remains finite Therefore sin hβb rarr βb and cos hβbrarr1

(β2 - α2 2 α β)βb sin αa + cos αa = cos ka ------------ (13)

β2 - α2 = ( 2 m ħ2 ) (Vo ndash E ) ndash ( 2 m E ħ2 )

= ( 2 m ħ2 ) (Vo ndash E - E ) = ( 2 m ħ2 ) (Vo ndash 2 E )

= 2 m Vo ħ2 ( since Vo gtgt E )

Substituting all these values in equation (11) it can be written as

( 2 m Vo 2 ħ2 α β ) β b sin α a + cos α a == cos k a

( m Vo b a ħ2 ) ( sin α a α a ) + cos α a == cos k a

P ( sin α a α a) + cos α a == cos k a -------------------(14)

Where P = [ m Vo b a ħ2 ] and is a measure of scattering power

The left hand side of the equation (12) is plotted as a function of α for the value of P = 3 Π 2 which is shown in fig the right hand side one takes the values between -1 to +1 as indicated by the horizontal lines in fig Therefore the equation (12) is satisfied only for those values of ka for which left hand side between plusmn 1

Fig a) P=6pi b) p--gt infinity c) p--gt 0

This plot for a value of P assumed arbitrarily as Indicated in the figure are the permitted values of this function show as solid lines between thin portions This then gives rise to the concept of ranges of permitted values of for a given ion lattice spacing a and since E = permitted bands of energy

are predicted The following interesting conclusions may be drawn

1 The allowed ranges of a which permit a wave-mechanical solution to exist are shown by the shadow portions Thus the motion of electrons in a periodic lattice is characterized by the bands of allowed energy separated by forbidden regions

2 As the value of a increases the width of allowed energy bands also increases and the width of the forbidden bands decreases this is a consequence of the fact that the first term of equation decreases on the average with increasing a

3 Let us now consider the effect of varying P It is known that P is a measure of the potential barrier strength If V0b is large ie if P is large the function described by the left hand side of the equation crosses +1 and -1 region at a steeper angle as shown in fig helliphellip Thus the allowed bands are narrower and the forbidden bands are wider

When a = n cosa = cosKa with Ka = n K = These values of K are points of discontinuity in the (E-K) cure for electrons in the crystal In the limit P the allowed band reduces to one single energy level that is we are back to the case of discrete energy spectrum existing in isolated atoms

P is follows from equation that Sina=0 a = n 2 = Referring equation

2 =

ie E =

E =

Here E is independent of K The energy levels in this case are discrete and the result is similar to the energy levels of a particle in a constant potential box of atomic dimensions This is expected because for large P tunneling through the barrier becomes almost improbable The other extreme case when P0 leads to

Cos a = cos KaThus = Kie 2 = K2

Referring equation (4)

K2 = 2 =

E = E =

E =

E =

Equation is appropriate to completely free particles Hence no energy levels exist that means all energies are allowed to the electrons

Thus by varying P from zero to infinity we cover the whole range from the completely free electron to the completely bound electron

Energy bands

ie V(x) = V( x + a ) where x is distance of the electron from the core

Such a potential is said to be a periodic potential

Blochs theorem states that ldquothe solution of SWequation has the form of a plane wave multiplied by a function with the period of the Crystal latticerdquostating that the wave function ψ for an electron in a periodic potential has the form

ψ(x) = exp(ik x)Uk(x) in One dimensions BLOCH

FUNCTIONSψ(r) = exp(ik r)U(r) in 3-dimensional motion

where k is the wave vector r is a position vector and U(r) is a periodic function that satisfies

U(r+a) = U(r)

for all vectors lsquoarsquo of the Crystal lattice of the crystal Blocks theorem is interpreted to mean that the wave function for an electron in a periodic potential is a plane wave modulated by a periodic function This explains why a free-electron model has some success in describing the properties of certain metals although it is inadequate to give a quantitative description of the properties of most metals

In order to understand the physical properties of the system it is required to solve the Schroumldingerrsquos equation However it is extremely difficult to solve the Schroumldingerrsquos equation with periodic potential described above Hence the Kronig ndash Penney Model is adopted for simplification

THE KRONIG -PENNEY MODEL

It is assumed in quantum free electron theory of metals that the free electrons in a metal experience a constant potential and is free to move in the metal This theory explains successfully most of the phenomena of solids But it could not explain why some solids are good conductors and some others are insulators and semi conductors It can be understood successfully using the band theory of solids

According to this theory the electrons move in a periodic potential produced by the positive ion cores The potential of electron varies periodically with periodicity of ion core and potential energy of the electrons is zero near nucleus of the positive ion core It is maximum when it is lying between the adjacent nuclei which are separated by interatomic spacing The variation of potential of electrons while it is moving through ion core is shown fig

Fig One dimensional periodic potential

V ( x ) = 0 for the region 0 lt x lt a = V0 for the region -b lt x lt 0

----------------(1)

Applying the time independent Schroumldingerrsquos wave equation for above two regions

d2Ψ dx2 + 2 m E Ψ ħ2 = 0 for region-I 0 lt x lt a ---------------(2)

and d2Ψ dx2 + 2 m ( E ndash V ) Ψ ħ2 = 0 for region-II -b lt x lt a ----------------(3)

Substituting α2 = 2 m E ħ2

---------------(4)

β2 = 2 m ( V ndash E ) ħ2

----------------(5)

d2Ψ dx2 + α2 Ψ = 0 for region 0 lt x lt a ---------------(6)

d2Ψ dx2 - β2 Ψ = 0 for region -b lt x lt a --------------(7)

The solution for the eqnrsquos (6) and (7) can be written as

Ψ ( x ) = Uk ( x ) eikx ---------------(8)

The above solution consists of a plane wave eikx modulated by the periodic function

Uk(x) where this Uk(x) has the periodicity same as that of the ion

ie Uk(x) = Uk(x+a) ---------------(9)

where k is propagating constant along x-direction and is given by k = 2 Π λ This k is also known asPropagation wave vector

Differentiating equation (8) twice with respect to x and substituting in equation (6) and (7) two independent second order linear differential equations can be obtained for the regions-I and II

ie

Substituting these values in equations

+ 2iK + = 0 for 0ltxlta

---------------------(10)

and +2iK - for ndashbltxlt0

------------------------(11)

where u1 represents the value of uk (x) in the interval 0ltxlta and u2 the value of uk (x) in the interval ndashbltxlt0

The solution of the differential equation

U1 = emx

and

Substituting these values in equation we get

m2 emx +2iK memx + emx = 0

m2+2iKm + = 0

m =

m = -ik im1 = -ik +i = i (-K)

m2 = -iK-i = -i (+K)

Thus the general solution is

u1 = Aem1

x + B em2x

u1 = Aei(-K)x + B e-I (+k)x

Similarly we have the solution for equation (12) asu2 = Ce(β-iK)x + D e-i (β +ik)x

Where ABC and D are constants The values of the constants ABC and D can be obtained by applying the boundary conditions

and

Applying the boundary conditions to the solution of above equations four linear equations in terms of ABC and D it can be obtained (where ABCD are constants ) the solution

for these equations can be determined only if the determinant of the coefficients of A B C and D vanishes on solving the determinant Thus we have

(β2 - α2 2 α β)sin hβb sin αa + cos hβb cos αa = cos k ( a + b ) ------------ (12)

The above equation is complicated and Kronig and Penney could conclude with the equation Hence they tried to modify this equation as follows

Let Vo is tending to infinite and b is approaching to zero Such that Vob remains finite Therefore sin hβb rarr βb and cos hβbrarr1

(β2 - α2 2 α β)βb sin αa + cos αa = cos ka ------------ (13)

β2 - α2 = ( 2 m ħ2 ) (Vo ndash E ) ndash ( 2 m E ħ2 )

= ( 2 m ħ2 ) (Vo ndash E - E ) = ( 2 m ħ2 ) (Vo ndash 2 E )

= 2 m Vo ħ2 ( since Vo gtgt E )

Substituting all these values in equation (11) it can be written as

( 2 m Vo 2 ħ2 α β ) β b sin α a + cos α a == cos k a

( m Vo b a ħ2 ) ( sin α a α a ) + cos α a == cos k a

P ( sin α a α a) + cos α a == cos k a -------------------(14)

Where P = [ m Vo b a ħ2 ] and is a measure of scattering power

The left hand side of the equation (12) is plotted as a function of α for the value of P = 3 Π 2 which is shown in fig the right hand side one takes the values between -1 to +1 as indicated by the horizontal lines in fig Therefore the equation (12) is satisfied only for those values of ka for which left hand side between plusmn 1

Fig a) P=6pi b) p--gt infinity c) p--gt 0

This plot for a value of P assumed arbitrarily as Indicated in the figure are the permitted values of this function show as solid lines between thin portions This then gives rise to the concept of ranges of permitted values of for a given ion lattice spacing a and since E = permitted bands of energy

are predicted The following interesting conclusions may be drawn

1 The allowed ranges of a which permit a wave-mechanical solution to exist are shown by the shadow portions Thus the motion of electrons in a periodic lattice is characterized by the bands of allowed energy separated by forbidden regions

2 As the value of a increases the width of allowed energy bands also increases and the width of the forbidden bands decreases this is a consequence of the fact that the first term of equation decreases on the average with increasing a

3 Let us now consider the effect of varying P It is known that P is a measure of the potential barrier strength If V0b is large ie if P is large the function described by the left hand side of the equation crosses +1 and -1 region at a steeper angle as shown in fig helliphellip Thus the allowed bands are narrower and the forbidden bands are wider

When a = n cosa = cosKa with Ka = n K = These values of K are points of discontinuity in the (E-K) cure for electrons in the crystal In the limit P the allowed band reduces to one single energy level that is we are back to the case of discrete energy spectrum existing in isolated atoms

P is follows from equation that Sina=0 a = n 2 = Referring equation

2 =

ie E =

E =

Here E is independent of K The energy levels in this case are discrete and the result is similar to the energy levels of a particle in a constant potential box of atomic dimensions This is expected because for large P tunneling through the barrier becomes almost improbable The other extreme case when P0 leads to

Cos a = cos KaThus = Kie 2 = K2

Referring equation (4)

K2 = 2 =

E = E =

E =

E =

Equation is appropriate to completely free particles Hence no energy levels exist that means all energies are allowed to the electrons

Thus by varying P from zero to infinity we cover the whole range from the completely free electron to the completely bound electron

Energy bands

In order to understand the physical properties of the system it is required to solve the Schroumldingerrsquos equation However it is extremely difficult to solve the Schroumldingerrsquos equation with periodic potential described above Hence the Kronig ndash Penney Model is adopted for simplification

THE KRONIG -PENNEY MODEL

It is assumed in quantum free electron theory of metals that the free electrons in a metal experience a constant potential and is free to move in the metal This theory explains successfully most of the phenomena of solids But it could not explain why some solids are good conductors and some others are insulators and semi conductors It can be understood successfully using the band theory of solids

According to this theory the electrons move in a periodic potential produced by the positive ion cores The potential of electron varies periodically with periodicity of ion core and potential energy of the electrons is zero near nucleus of the positive ion core It is maximum when it is lying between the adjacent nuclei which are separated by interatomic spacing The variation of potential of electrons while it is moving through ion core is shown fig

Fig One dimensional periodic potential

V ( x ) = 0 for the region 0 lt x lt a = V0 for the region -b lt x lt 0

----------------(1)

Applying the time independent Schroumldingerrsquos wave equation for above two regions

d2Ψ dx2 + 2 m E Ψ ħ2 = 0 for region-I 0 lt x lt a ---------------(2)

and d2Ψ dx2 + 2 m ( E ndash V ) Ψ ħ2 = 0 for region-II -b lt x lt a ----------------(3)

Substituting α2 = 2 m E ħ2

---------------(4)

β2 = 2 m ( V ndash E ) ħ2

----------------(5)

d2Ψ dx2 + α2 Ψ = 0 for region 0 lt x lt a ---------------(6)

d2Ψ dx2 - β2 Ψ = 0 for region -b lt x lt a --------------(7)

The solution for the eqnrsquos (6) and (7) can be written as

Ψ ( x ) = Uk ( x ) eikx ---------------(8)

The above solution consists of a plane wave eikx modulated by the periodic function

Uk(x) where this Uk(x) has the periodicity same as that of the ion

ie Uk(x) = Uk(x+a) ---------------(9)

where k is propagating constant along x-direction and is given by k = 2 Π λ This k is also known asPropagation wave vector

Differentiating equation (8) twice with respect to x and substituting in equation (6) and (7) two independent second order linear differential equations can be obtained for the regions-I and II

ie

Substituting these values in equations

+ 2iK + = 0 for 0ltxlta

---------------------(10)

and +2iK - for ndashbltxlt0

------------------------(11)

where u1 represents the value of uk (x) in the interval 0ltxlta and u2 the value of uk (x) in the interval ndashbltxlt0

The solution of the differential equation

U1 = emx

and

Substituting these values in equation we get

m2 emx +2iK memx + emx = 0

m2+2iKm + = 0

m =

m = -ik im1 = -ik +i = i (-K)

m2 = -iK-i = -i (+K)

Thus the general solution is

u1 = Aem1

x + B em2x

u1 = Aei(-K)x + B e-I (+k)x

Similarly we have the solution for equation (12) asu2 = Ce(β-iK)x + D e-i (β +ik)x

Where ABC and D are constants The values of the constants ABC and D can be obtained by applying the boundary conditions

and

Applying the boundary conditions to the solution of above equations four linear equations in terms of ABC and D it can be obtained (where ABCD are constants ) the solution

for these equations can be determined only if the determinant of the coefficients of A B C and D vanishes on solving the determinant Thus we have

(β2 - α2 2 α β)sin hβb sin αa + cos hβb cos αa = cos k ( a + b ) ------------ (12)

The above equation is complicated and Kronig and Penney could conclude with the equation Hence they tried to modify this equation as follows

Let Vo is tending to infinite and b is approaching to zero Such that Vob remains finite Therefore sin hβb rarr βb and cos hβbrarr1

(β2 - α2 2 α β)βb sin αa + cos αa = cos ka ------------ (13)

β2 - α2 = ( 2 m ħ2 ) (Vo ndash E ) ndash ( 2 m E ħ2 )

= ( 2 m ħ2 ) (Vo ndash E - E ) = ( 2 m ħ2 ) (Vo ndash 2 E )

= 2 m Vo ħ2 ( since Vo gtgt E )

Substituting all these values in equation (11) it can be written as

( 2 m Vo 2 ħ2 α β ) β b sin α a + cos α a == cos k a

( m Vo b a ħ2 ) ( sin α a α a ) + cos α a == cos k a

P ( sin α a α a) + cos α a == cos k a -------------------(14)

Where P = [ m Vo b a ħ2 ] and is a measure of scattering power

The left hand side of the equation (12) is plotted as a function of α for the value of P = 3 Π 2 which is shown in fig the right hand side one takes the values between -1 to +1 as indicated by the horizontal lines in fig Therefore the equation (12) is satisfied only for those values of ka for which left hand side between plusmn 1

Fig a) P=6pi b) p--gt infinity c) p--gt 0

This plot for a value of P assumed arbitrarily as Indicated in the figure are the permitted values of this function show as solid lines between thin portions This then gives rise to the concept of ranges of permitted values of for a given ion lattice spacing a and since E = permitted bands of energy

are predicted The following interesting conclusions may be drawn

1 The allowed ranges of a which permit a wave-mechanical solution to exist are shown by the shadow portions Thus the motion of electrons in a periodic lattice is characterized by the bands of allowed energy separated by forbidden regions

2 As the value of a increases the width of allowed energy bands also increases and the width of the forbidden bands decreases this is a consequence of the fact that the first term of equation decreases on the average with increasing a

3 Let us now consider the effect of varying P It is known that P is a measure of the potential barrier strength If V0b is large ie if P is large the function described by the left hand side of the equation crosses +1 and -1 region at a steeper angle as shown in fig helliphellip Thus the allowed bands are narrower and the forbidden bands are wider

When a = n cosa = cosKa with Ka = n K = These values of K are points of discontinuity in the (E-K) cure for electrons in the crystal In the limit P the allowed band reduces to one single energy level that is we are back to the case of discrete energy spectrum existing in isolated atoms

P is follows from equation that Sina=0 a = n 2 = Referring equation

2 =

ie E =

E =

Here E is independent of K The energy levels in this case are discrete and the result is similar to the energy levels of a particle in a constant potential box of atomic dimensions This is expected because for large P tunneling through the barrier becomes almost improbable The other extreme case when P0 leads to

Cos a = cos KaThus = Kie 2 = K2

Referring equation (4)

K2 = 2 =

E = E =

E =

E =

Equation is appropriate to completely free particles Hence no energy levels exist that means all energies are allowed to the electrons

Thus by varying P from zero to infinity we cover the whole range from the completely free electron to the completely bound electron

Energy bands

Fig One dimensional periodic potential

V ( x ) = 0 for the region 0 lt x lt a = V0 for the region -b lt x lt 0

----------------(1)

Applying the time independent Schroumldingerrsquos wave equation for above two regions

d2Ψ dx2 + 2 m E Ψ ħ2 = 0 for region-I 0 lt x lt a ---------------(2)

and d2Ψ dx2 + 2 m ( E ndash V ) Ψ ħ2 = 0 for region-II -b lt x lt a ----------------(3)

Substituting α2 = 2 m E ħ2

---------------(4)

β2 = 2 m ( V ndash E ) ħ2

----------------(5)

d2Ψ dx2 + α2 Ψ = 0 for region 0 lt x lt a ---------------(6)

d2Ψ dx2 - β2 Ψ = 0 for region -b lt x lt a --------------(7)

The solution for the eqnrsquos (6) and (7) can be written as

Ψ ( x ) = Uk ( x ) eikx ---------------(8)

The above solution consists of a plane wave eikx modulated by the periodic function

Uk(x) where this Uk(x) has the periodicity same as that of the ion

ie Uk(x) = Uk(x+a) ---------------(9)

where k is propagating constant along x-direction and is given by k = 2 Π λ This k is also known asPropagation wave vector

Differentiating equation (8) twice with respect to x and substituting in equation (6) and (7) two independent second order linear differential equations can be obtained for the regions-I and II

ie

Substituting these values in equations

+ 2iK + = 0 for 0ltxlta

---------------------(10)

and +2iK - for ndashbltxlt0

------------------------(11)

where u1 represents the value of uk (x) in the interval 0ltxlta and u2 the value of uk (x) in the interval ndashbltxlt0

The solution of the differential equation

U1 = emx

and

Substituting these values in equation we get

m2 emx +2iK memx + emx = 0

m2+2iKm + = 0

m =

m = -ik im1 = -ik +i = i (-K)

m2 = -iK-i = -i (+K)

Thus the general solution is

u1 = Aem1

x + B em2x

u1 = Aei(-K)x + B e-I (+k)x

Similarly we have the solution for equation (12) asu2 = Ce(β-iK)x + D e-i (β +ik)x

Where ABC and D are constants The values of the constants ABC and D can be obtained by applying the boundary conditions

and

Applying the boundary conditions to the solution of above equations four linear equations in terms of ABC and D it can be obtained (where ABCD are constants ) the solution

for these equations can be determined only if the determinant of the coefficients of A B C and D vanishes on solving the determinant Thus we have

(β2 - α2 2 α β)sin hβb sin αa + cos hβb cos αa = cos k ( a + b ) ------------ (12)

The above equation is complicated and Kronig and Penney could conclude with the equation Hence they tried to modify this equation as follows

Let Vo is tending to infinite and b is approaching to zero Such that Vob remains finite Therefore sin hβb rarr βb and cos hβbrarr1

(β2 - α2 2 α β)βb sin αa + cos αa = cos ka ------------ (13)

β2 - α2 = ( 2 m ħ2 ) (Vo ndash E ) ndash ( 2 m E ħ2 )

= ( 2 m ħ2 ) (Vo ndash E - E ) = ( 2 m ħ2 ) (Vo ndash 2 E )

= 2 m Vo ħ2 ( since Vo gtgt E )

Substituting all these values in equation (11) it can be written as

( 2 m Vo 2 ħ2 α β ) β b sin α a + cos α a == cos k a

( m Vo b a ħ2 ) ( sin α a α a ) + cos α a == cos k a

P ( sin α a α a) + cos α a == cos k a -------------------(14)

Where P = [ m Vo b a ħ2 ] and is a measure of scattering power

The left hand side of the equation (12) is plotted as a function of α for the value of P = 3 Π 2 which is shown in fig the right hand side one takes the values between -1 to +1 as indicated by the horizontal lines in fig Therefore the equation (12) is satisfied only for those values of ka for which left hand side between plusmn 1

Fig a) P=6pi b) p--gt infinity c) p--gt 0

This plot for a value of P assumed arbitrarily as Indicated in the figure are the permitted values of this function show as solid lines between thin portions This then gives rise to the concept of ranges of permitted values of for a given ion lattice spacing a and since E = permitted bands of energy

are predicted The following interesting conclusions may be drawn

1 The allowed ranges of a which permit a wave-mechanical solution to exist are shown by the shadow portions Thus the motion of electrons in a periodic lattice is characterized by the bands of allowed energy separated by forbidden regions

2 As the value of a increases the width of allowed energy bands also increases and the width of the forbidden bands decreases this is a consequence of the fact that the first term of equation decreases on the average with increasing a

3 Let us now consider the effect of varying P It is known that P is a measure of the potential barrier strength If V0b is large ie if P is large the function described by the left hand side of the equation crosses +1 and -1 region at a steeper angle as shown in fig helliphellip Thus the allowed bands are narrower and the forbidden bands are wider

When a = n cosa = cosKa with Ka = n K = These values of K are points of discontinuity in the (E-K) cure for electrons in the crystal In the limit P the allowed band reduces to one single energy level that is we are back to the case of discrete energy spectrum existing in isolated atoms

P is follows from equation that Sina=0 a = n 2 = Referring equation

2 =

ie E =

E =

Here E is independent of K The energy levels in this case are discrete and the result is similar to the energy levels of a particle in a constant potential box of atomic dimensions This is expected because for large P tunneling through the barrier becomes almost improbable The other extreme case when P0 leads to

Cos a = cos KaThus = Kie 2 = K2

Referring equation (4)

K2 = 2 =

E = E =

E =

E =

Equation is appropriate to completely free particles Hence no energy levels exist that means all energies are allowed to the electrons

Thus by varying P from zero to infinity we cover the whole range from the completely free electron to the completely bound electron

Energy bands

d2Ψ dx2 + α2 Ψ = 0 for region 0 lt x lt a ---------------(6)

d2Ψ dx2 - β2 Ψ = 0 for region -b lt x lt a --------------(7)

The solution for the eqnrsquos (6) and (7) can be written as

Ψ ( x ) = Uk ( x ) eikx ---------------(8)

The above solution consists of a plane wave eikx modulated by the periodic function

Uk(x) where this Uk(x) has the periodicity same as that of the ion

ie Uk(x) = Uk(x+a) ---------------(9)

where k is propagating constant along x-direction and is given by k = 2 Π λ This k is also known asPropagation wave vector

Differentiating equation (8) twice with respect to x and substituting in equation (6) and (7) two independent second order linear differential equations can be obtained for the regions-I and II

ie

Substituting these values in equations

+ 2iK + = 0 for 0ltxlta

---------------------(10)

and +2iK - for ndashbltxlt0

------------------------(11)

where u1 represents the value of uk (x) in the interval 0ltxlta and u2 the value of uk (x) in the interval ndashbltxlt0

The solution of the differential equation

U1 = emx

and

Substituting these values in equation we get

m2 emx +2iK memx + emx = 0

m2+2iKm + = 0

m =

m = -ik im1 = -ik +i = i (-K)

m2 = -iK-i = -i (+K)

Thus the general solution is

u1 = Aem1

x + B em2x

u1 = Aei(-K)x + B e-I (+k)x

Similarly we have the solution for equation (12) asu2 = Ce(β-iK)x + D e-i (β +ik)x

Where ABC and D are constants The values of the constants ABC and D can be obtained by applying the boundary conditions

and

Applying the boundary conditions to the solution of above equations four linear equations in terms of ABC and D it can be obtained (where ABCD are constants ) the solution

for these equations can be determined only if the determinant of the coefficients of A B C and D vanishes on solving the determinant Thus we have

(β2 - α2 2 α β)sin hβb sin αa + cos hβb cos αa = cos k ( a + b ) ------------ (12)

The above equation is complicated and Kronig and Penney could conclude with the equation Hence they tried to modify this equation as follows

Let Vo is tending to infinite and b is approaching to zero Such that Vob remains finite Therefore sin hβb rarr βb and cos hβbrarr1

(β2 - α2 2 α β)βb sin αa + cos αa = cos ka ------------ (13)

β2 - α2 = ( 2 m ħ2 ) (Vo ndash E ) ndash ( 2 m E ħ2 )

= ( 2 m ħ2 ) (Vo ndash E - E ) = ( 2 m ħ2 ) (Vo ndash 2 E )

= 2 m Vo ħ2 ( since Vo gtgt E )

Substituting all these values in equation (11) it can be written as

( 2 m Vo 2 ħ2 α β ) β b sin α a + cos α a == cos k a

( m Vo b a ħ2 ) ( sin α a α a ) + cos α a == cos k a

P ( sin α a α a) + cos α a == cos k a -------------------(14)

Where P = [ m Vo b a ħ2 ] and is a measure of scattering power

The left hand side of the equation (12) is plotted as a function of α for the value of P = 3 Π 2 which is shown in fig the right hand side one takes the values between -1 to +1 as indicated by the horizontal lines in fig Therefore the equation (12) is satisfied only for those values of ka for which left hand side between plusmn 1

Fig a) P=6pi b) p--gt infinity c) p--gt 0

This plot for a value of P assumed arbitrarily as Indicated in the figure are the permitted values of this function show as solid lines between thin portions This then gives rise to the concept of ranges of permitted values of for a given ion lattice spacing a and since E = permitted bands of energy

are predicted The following interesting conclusions may be drawn

1 The allowed ranges of a which permit a wave-mechanical solution to exist are shown by the shadow portions Thus the motion of electrons in a periodic lattice is characterized by the bands of allowed energy separated by forbidden regions

2 As the value of a increases the width of allowed energy bands also increases and the width of the forbidden bands decreases this is a consequence of the fact that the first term of equation decreases on the average with increasing a

3 Let us now consider the effect of varying P It is known that P is a measure of the potential barrier strength If V0b is large ie if P is large the function described by the left hand side of the equation crosses +1 and -1 region at a steeper angle as shown in fig helliphellip Thus the allowed bands are narrower and the forbidden bands are wider

When a = n cosa = cosKa with Ka = n K = These values of K are points of discontinuity in the (E-K) cure for electrons in the crystal In the limit P the allowed band reduces to one single energy level that is we are back to the case of discrete energy spectrum existing in isolated atoms

P is follows from equation that Sina=0 a = n 2 = Referring equation

2 =

ie E =

E =

Here E is independent of K The energy levels in this case are discrete and the result is similar to the energy levels of a particle in a constant potential box of atomic dimensions This is expected because for large P tunneling through the barrier becomes almost improbable The other extreme case when P0 leads to

Cos a = cos KaThus = Kie 2 = K2

Referring equation (4)

K2 = 2 =

E = E =

E =

E =

Equation is appropriate to completely free particles Hence no energy levels exist that means all energies are allowed to the electrons

Thus by varying P from zero to infinity we cover the whole range from the completely free electron to the completely bound electron

Energy bands

ie

Substituting these values in equations

+ 2iK + = 0 for 0ltxlta

---------------------(10)

and +2iK - for ndashbltxlt0

------------------------(11)

where u1 represents the value of uk (x) in the interval 0ltxlta and u2 the value of uk (x) in the interval ndashbltxlt0

The solution of the differential equation

U1 = emx

and

Substituting these values in equation we get

m2 emx +2iK memx + emx = 0

m2+2iKm + = 0

m =

m = -ik im1 = -ik +i = i (-K)

m2 = -iK-i = -i (+K)

Thus the general solution is

u1 = Aem1

x + B em2x

u1 = Aei(-K)x + B e-I (+k)x

Similarly we have the solution for equation (12) asu2 = Ce(β-iK)x + D e-i (β +ik)x

Where ABC and D are constants The values of the constants ABC and D can be obtained by applying the boundary conditions

and

Applying the boundary conditions to the solution of above equations four linear equations in terms of ABC and D it can be obtained (where ABCD are constants ) the solution

for these equations can be determined only if the determinant of the coefficients of A B C and D vanishes on solving the determinant Thus we have

(β2 - α2 2 α β)sin hβb sin αa + cos hβb cos αa = cos k ( a + b ) ------------ (12)

The above equation is complicated and Kronig and Penney could conclude with the equation Hence they tried to modify this equation as follows

Let Vo is tending to infinite and b is approaching to zero Such that Vob remains finite Therefore sin hβb rarr βb and cos hβbrarr1

(β2 - α2 2 α β)βb sin αa + cos αa = cos ka ------------ (13)

β2 - α2 = ( 2 m ħ2 ) (Vo ndash E ) ndash ( 2 m E ħ2 )

= ( 2 m ħ2 ) (Vo ndash E - E ) = ( 2 m ħ2 ) (Vo ndash 2 E )

= 2 m Vo ħ2 ( since Vo gtgt E )

Substituting all these values in equation (11) it can be written as

( 2 m Vo 2 ħ2 α β ) β b sin α a + cos α a == cos k a

( m Vo b a ħ2 ) ( sin α a α a ) + cos α a == cos k a

P ( sin α a α a) + cos α a == cos k a -------------------(14)

Where P = [ m Vo b a ħ2 ] and is a measure of scattering power

The left hand side of the equation (12) is plotted as a function of α for the value of P = 3 Π 2 which is shown in fig the right hand side one takes the values between -1 to +1 as indicated by the horizontal lines in fig Therefore the equation (12) is satisfied only for those values of ka for which left hand side between plusmn 1

Fig a) P=6pi b) p--gt infinity c) p--gt 0

This plot for a value of P assumed arbitrarily as Indicated in the figure are the permitted values of this function show as solid lines between thin portions This then gives rise to the concept of ranges of permitted values of for a given ion lattice spacing a and since E = permitted bands of energy

are predicted The following interesting conclusions may be drawn

1 The allowed ranges of a which permit a wave-mechanical solution to exist are shown by the shadow portions Thus the motion of electrons in a periodic lattice is characterized by the bands of allowed energy separated by forbidden regions

2 As the value of a increases the width of allowed energy bands also increases and the width of the forbidden bands decreases this is a consequence of the fact that the first term of equation decreases on the average with increasing a

3 Let us now consider the effect of varying P It is known that P is a measure of the potential barrier strength If V0b is large ie if P is large the function described by the left hand side of the equation crosses +1 and -1 region at a steeper angle as shown in fig helliphellip Thus the allowed bands are narrower and the forbidden bands are wider

When a = n cosa = cosKa with Ka = n K = These values of K are points of discontinuity in the (E-K) cure for electrons in the crystal In the limit P the allowed band reduces to one single energy level that is we are back to the case of discrete energy spectrum existing in isolated atoms

P is follows from equation that Sina=0 a = n 2 = Referring equation

2 =

ie E =

E =

Here E is independent of K The energy levels in this case are discrete and the result is similar to the energy levels of a particle in a constant potential box of atomic dimensions This is expected because for large P tunneling through the barrier becomes almost improbable The other extreme case when P0 leads to

Cos a = cos KaThus = Kie 2 = K2

Referring equation (4)

K2 = 2 =

E = E =

E =

E =

Equation is appropriate to completely free particles Hence no energy levels exist that means all energies are allowed to the electrons

Thus by varying P from zero to infinity we cover the whole range from the completely free electron to the completely bound electron

Energy bands

m2+2iKm + = 0

m =

m = -ik im1 = -ik +i = i (-K)

m2 = -iK-i = -i (+K)

Thus the general solution is

u1 = Aem1

x + B em2x

u1 = Aei(-K)x + B e-I (+k)x

Similarly we have the solution for equation (12) asu2 = Ce(β-iK)x + D e-i (β +ik)x

Where ABC and D are constants The values of the constants ABC and D can be obtained by applying the boundary conditions

and

Applying the boundary conditions to the solution of above equations four linear equations in terms of ABC and D it can be obtained (where ABCD are constants ) the solution

for these equations can be determined only if the determinant of the coefficients of A B C and D vanishes on solving the determinant Thus we have

(β2 - α2 2 α β)sin hβb sin αa + cos hβb cos αa = cos k ( a + b ) ------------ (12)

The above equation is complicated and Kronig and Penney could conclude with the equation Hence they tried to modify this equation as follows

Let Vo is tending to infinite and b is approaching to zero Such that Vob remains finite Therefore sin hβb rarr βb and cos hβbrarr1

(β2 - α2 2 α β)βb sin αa + cos αa = cos ka ------------ (13)

β2 - α2 = ( 2 m ħ2 ) (Vo ndash E ) ndash ( 2 m E ħ2 )

= ( 2 m ħ2 ) (Vo ndash E - E ) = ( 2 m ħ2 ) (Vo ndash 2 E )

= 2 m Vo ħ2 ( since Vo gtgt E )

Substituting all these values in equation (11) it can be written as

( 2 m Vo 2 ħ2 α β ) β b sin α a + cos α a == cos k a

( m Vo b a ħ2 ) ( sin α a α a ) + cos α a == cos k a

P ( sin α a α a) + cos α a == cos k a -------------------(14)

Where P = [ m Vo b a ħ2 ] and is a measure of scattering power

The left hand side of the equation (12) is plotted as a function of α for the value of P = 3 Π 2 which is shown in fig the right hand side one takes the values between -1 to +1 as indicated by the horizontal lines in fig Therefore the equation (12) is satisfied only for those values of ka for which left hand side between plusmn 1

Fig a) P=6pi b) p--gt infinity c) p--gt 0

This plot for a value of P assumed arbitrarily as Indicated in the figure are the permitted values of this function show as solid lines between thin portions This then gives rise to the concept of ranges of permitted values of for a given ion lattice spacing a and since E = permitted bands of energy

are predicted The following interesting conclusions may be drawn

1 The allowed ranges of a which permit a wave-mechanical solution to exist are shown by the shadow portions Thus the motion of electrons in a periodic lattice is characterized by the bands of allowed energy separated by forbidden regions

2 As the value of a increases the width of allowed energy bands also increases and the width of the forbidden bands decreases this is a consequence of the fact that the first term of equation decreases on the average with increasing a

3 Let us now consider the effect of varying P It is known that P is a measure of the potential barrier strength If V0b is large ie if P is large the function described by the left hand side of the equation crosses +1 and -1 region at a steeper angle as shown in fig helliphellip Thus the allowed bands are narrower and the forbidden bands are wider

When a = n cosa = cosKa with Ka = n K = These values of K are points of discontinuity in the (E-K) cure for electrons in the crystal In the limit P the allowed band reduces to one single energy level that is we are back to the case of discrete energy spectrum existing in isolated atoms

P is follows from equation that Sina=0 a = n 2 = Referring equation

2 =

ie E =

E =

Here E is independent of K The energy levels in this case are discrete and the result is similar to the energy levels of a particle in a constant potential box of atomic dimensions This is expected because for large P tunneling through the barrier becomes almost improbable The other extreme case when P0 leads to

Cos a = cos KaThus = Kie 2 = K2

Referring equation (4)

K2 = 2 =

E = E =

E =

E =

Equation is appropriate to completely free particles Hence no energy levels exist that means all energies are allowed to the electrons

Thus by varying P from zero to infinity we cover the whole range from the completely free electron to the completely bound electron

Energy bands

for these equations can be determined only if the determinant of the coefficients of A B C and D vanishes on solving the determinant Thus we have

(β2 - α2 2 α β)sin hβb sin αa + cos hβb cos αa = cos k ( a + b ) ------------ (12)

The above equation is complicated and Kronig and Penney could conclude with the equation Hence they tried to modify this equation as follows

Let Vo is tending to infinite and b is approaching to zero Such that Vob remains finite Therefore sin hβb rarr βb and cos hβbrarr1

(β2 - α2 2 α β)βb sin αa + cos αa = cos ka ------------ (13)

β2 - α2 = ( 2 m ħ2 ) (Vo ndash E ) ndash ( 2 m E ħ2 )

= ( 2 m ħ2 ) (Vo ndash E - E ) = ( 2 m ħ2 ) (Vo ndash 2 E )

= 2 m Vo ħ2 ( since Vo gtgt E )

Substituting all these values in equation (11) it can be written as

( 2 m Vo 2 ħ2 α β ) β b sin α a + cos α a == cos k a

( m Vo b a ħ2 ) ( sin α a α a ) + cos α a == cos k a

P ( sin α a α a) + cos α a == cos k a -------------------(14)

Where P = [ m Vo b a ħ2 ] and is a measure of scattering power

The left hand side of the equation (12) is plotted as a function of α for the value of P = 3 Π 2 which is shown in fig the right hand side one takes the values between -1 to +1 as indicated by the horizontal lines in fig Therefore the equation (12) is satisfied only for those values of ka for which left hand side between plusmn 1

Fig a) P=6pi b) p--gt infinity c) p--gt 0

This plot for a value of P assumed arbitrarily as Indicated in the figure are the permitted values of this function show as solid lines between thin portions This then gives rise to the concept of ranges of permitted values of for a given ion lattice spacing a and since E = permitted bands of energy

are predicted The following interesting conclusions may be drawn

1 The allowed ranges of a which permit a wave-mechanical solution to exist are shown by the shadow portions Thus the motion of electrons in a periodic lattice is characterized by the bands of allowed energy separated by forbidden regions

2 As the value of a increases the width of allowed energy bands also increases and the width of the forbidden bands decreases this is a consequence of the fact that the first term of equation decreases on the average with increasing a

3 Let us now consider the effect of varying P It is known that P is a measure of the potential barrier strength If V0b is large ie if P is large the function described by the left hand side of the equation crosses +1 and -1 region at a steeper angle as shown in fig helliphellip Thus the allowed bands are narrower and the forbidden bands are wider

When a = n cosa = cosKa with Ka = n K = These values of K are points of discontinuity in the (E-K) cure for electrons in the crystal In the limit P the allowed band reduces to one single energy level that is we are back to the case of discrete energy spectrum existing in isolated atoms

P is follows from equation that Sina=0 a = n 2 = Referring equation

2 =

ie E =

E =

Here E is independent of K The energy levels in this case are discrete and the result is similar to the energy levels of a particle in a constant potential box of atomic dimensions This is expected because for large P tunneling through the barrier becomes almost improbable The other extreme case when P0 leads to

Cos a = cos KaThus = Kie 2 = K2

Referring equation (4)

K2 = 2 =

E = E =

E =

E =

Equation is appropriate to completely free particles Hence no energy levels exist that means all energies are allowed to the electrons

Thus by varying P from zero to infinity we cover the whole range from the completely free electron to the completely bound electron

Energy bands

( 2 m Vo 2 ħ2 α β ) β b sin α a + cos α a == cos k a

( m Vo b a ħ2 ) ( sin α a α a ) + cos α a == cos k a

P ( sin α a α a) + cos α a == cos k a -------------------(14)

Where P = [ m Vo b a ħ2 ] and is a measure of scattering power

The left hand side of the equation (12) is plotted as a function of α for the value of P = 3 Π 2 which is shown in fig the right hand side one takes the values between -1 to +1 as indicated by the horizontal lines in fig Therefore the equation (12) is satisfied only for those values of ka for which left hand side between plusmn 1

Fig a) P=6pi b) p--gt infinity c) p--gt 0

This plot for a value of P assumed arbitrarily as Indicated in the figure are the permitted values of this function show as solid lines between thin portions This then gives rise to the concept of ranges of permitted values of for a given ion lattice spacing a and since E = permitted bands of energy

are predicted The following interesting conclusions may be drawn

1 The allowed ranges of a which permit a wave-mechanical solution to exist are shown by the shadow portions Thus the motion of electrons in a periodic lattice is characterized by the bands of allowed energy separated by forbidden regions

2 As the value of a increases the width of allowed energy bands also increases and the width of the forbidden bands decreases this is a consequence of the fact that the first term of equation decreases on the average with increasing a

3 Let us now consider the effect of varying P It is known that P is a measure of the potential barrier strength If V0b is large ie if P is large the function described by the left hand side of the equation crosses +1 and -1 region at a steeper angle as shown in fig helliphellip Thus the allowed bands are narrower and the forbidden bands are wider

When a = n cosa = cosKa with Ka = n K = These values of K are points of discontinuity in the (E-K) cure for electrons in the crystal In the limit P the allowed band reduces to one single energy level that is we are back to the case of discrete energy spectrum existing in isolated atoms

P is follows from equation that Sina=0 a = n 2 = Referring equation

2 =

ie E =

E =

Here E is independent of K The energy levels in this case are discrete and the result is similar to the energy levels of a particle in a constant potential box of atomic dimensions This is expected because for large P tunneling through the barrier becomes almost improbable The other extreme case when P0 leads to

Cos a = cos KaThus = Kie 2 = K2

Referring equation (4)

K2 = 2 =

E = E =

E =

E =

Equation is appropriate to completely free particles Hence no energy levels exist that means all energies are allowed to the electrons

Thus by varying P from zero to infinity we cover the whole range from the completely free electron to the completely bound electron

Energy bands

Fig a) P=6pi b) p--gt infinity c) p--gt 0

This plot for a value of P assumed arbitrarily as Indicated in the figure are the permitted values of this function show as solid lines between thin portions This then gives rise to the concept of ranges of permitted values of for a given ion lattice spacing a and since E = permitted bands of energy

are predicted The following interesting conclusions may be drawn

1 The allowed ranges of a which permit a wave-mechanical solution to exist are shown by the shadow portions Thus the motion of electrons in a periodic lattice is characterized by the bands of allowed energy separated by forbidden regions

2 As the value of a increases the width of allowed energy bands also increases and the width of the forbidden bands decreases this is a consequence of the fact that the first term of equation decreases on the average with increasing a

3 Let us now consider the effect of varying P It is known that P is a measure of the potential barrier strength If V0b is large ie if P is large the function described by the left hand side of the equation crosses +1 and -1 region at a steeper angle as shown in fig helliphellip Thus the allowed bands are narrower and the forbidden bands are wider

When a = n cosa = cosKa with Ka = n K = These values of K are points of discontinuity in the (E-K) cure for electrons in the crystal In the limit P the allowed band reduces to one single energy level that is we are back to the case of discrete energy spectrum existing in isolated atoms

P is follows from equation that Sina=0 a = n 2 = Referring equation

2 =

ie E =

E =

Here E is independent of K The energy levels in this case are discrete and the result is similar to the energy levels of a particle in a constant potential box of atomic dimensions This is expected because for large P tunneling through the barrier becomes almost improbable The other extreme case when P0 leads to

Cos a = cos KaThus = Kie 2 = K2

Referring equation (4)

K2 = 2 =

E = E =

E =

E =

Equation is appropriate to completely free particles Hence no energy levels exist that means all energies are allowed to the electrons

Thus by varying P from zero to infinity we cover the whole range from the completely free electron to the completely bound electron

Energy bands

are predicted The following interesting conclusions may be drawn

1 The allowed ranges of a which permit a wave-mechanical solution to exist are shown by the shadow portions Thus the motion of electrons in a periodic lattice is characterized by the bands of allowed energy separated by forbidden regions

2 As the value of a increases the width of allowed energy bands also increases and the width of the forbidden bands decreases this is a consequence of the fact that the first term of equation decreases on the average with increasing a

3 Let us now consider the effect of varying P It is known that P is a measure of the potential barrier strength If V0b is large ie if P is large the function described by the left hand side of the equation crosses +1 and -1 region at a steeper angle as shown in fig helliphellip Thus the allowed bands are narrower and the forbidden bands are wider

When a = n cosa = cosKa with Ka = n K = These values of K are points of discontinuity in the (E-K) cure for electrons in the crystal In the limit P the allowed band reduces to one single energy level that is we are back to the case of discrete energy spectrum existing in isolated atoms

P is follows from equation that Sina=0 a = n 2 = Referring equation

2 =

ie E =

E =

Here E is independent of K The energy levels in this case are discrete and the result is similar to the energy levels of a particle in a constant potential box of atomic dimensions This is expected because for large P tunneling through the barrier becomes almost improbable The other extreme case when P0 leads to

Cos a = cos KaThus = Kie 2 = K2

Referring equation (4)

K2 = 2 =

E = E =

E =

E =

Equation is appropriate to completely free particles Hence no energy levels exist that means all energies are allowed to the electrons

Thus by varying P from zero to infinity we cover the whole range from the completely free electron to the completely bound electron

Energy bands

P is follows from equation that Sina=0 a = n 2 = Referring equation

2 =

ie E =

E =

Here E is independent of K The energy levels in this case are discrete and the result is similar to the energy levels of a particle in a constant potential box of atomic dimensions This is expected because for large P tunneling through the barrier becomes almost improbable The other extreme case when P0 leads to

Cos a = cos KaThus = Kie 2 = K2

Referring equation (4)

K2 = 2 =

E = E =

E =

E =

Equation is appropriate to completely free particles Hence no energy levels exist that means all energies are allowed to the electrons

Thus by varying P from zero to infinity we cover the whole range from the completely free electron to the completely bound electron

Energy bands

E =

Equation is appropriate to completely free particles Hence no energy levels exist that means all energies are allowed to the electrons

Thus by varying P from zero to infinity we cover the whole range from the completely free electron to the completely bound electron

Energy bands