23 Motion of electrons in periodic potential
Prof. P. N. Kotru
TABLE OF CONTENTS
23.1 Motion of electrons in a periodic potential.
23.2 Significance of Translational periodicity.
23. 3 Periodic Boundary Conditions and Wave vector.
23.4 Bloch Theorem.
23.5 Kronig-Penney Model.
23.6 Brillouin Zones.
23.7 Significance of Brillouin Zones.
LEARNING OBJECTIVES
- In this module motion of electrons in a periodic potential is described and discussed.
- Significance of translational periodicity in a crystalline solid is explained.
- Somerfield’s model where electron is assumed to travel in a potential which is constant everywhere inside
- the metal is described.
- The concept of De Broglie wave or Somerfield waves or Bloch waves are given
- The kinetic energy of the electron is shown to be proportional to the square of wave number and the
- model leads to a relationship between energy and wave vector for a free electron that is parabolic.
- The Bloch theorem is explained.
- Kronig Penney model describing the motion of electrons in a periodic potential is explained.
- The model is shown to lead to energy versus wave vector for electron in a periodic potential as being
- discontinuous at k = +nπ/a and k = -nπ/a.
- The concept of Brillioun zones is given and their significance in crystallography is discussed.
23.1 Motion of electrons in a perio dic potential
Atoms in a crystalline solid are positioned at regular intervals because of wh ich it exhib its periodic ity in the arrangement of atoms in all d imensions. On account of this periodicity, it exh ibits various types symmet ries and as such the atoms or molecules composing the solid are always related by some symmet ry elements. Crystalline solids exh ibit various types of physical properties which are structure sensitive. So, the physical behaviour of crystalline solids cannot be explained unless we have a sound background of crystallography. The symmetry elements have already been discussed in the previous sections. It is essential to describe the significance of symmetry in crystalline solids in so far as its relevance to the development of theory in the understanding of their physical properties is concerned. One of the physical properties is that of conduction in crystalline solids. Metals have been considered as being composed of atoms with their valence electrons roaming freely inside the solid. The classical theory considered these electrons as moving freely inside the solids like molecules of a gas in an enclosure. Drude and Lorentz theory assumed that these electrons behave like classical part icles, move in a constant potential field andas such applied Maxwell Bolt zmann statistics tounderstand and exp lain the behaviour of solids. However, it was modified by Sommerfeldby applying quantum mechanical concepts which could explain some of the physical properties of solids. The free elect ron theory, however, could not explain as to why some solids are good conductors like metals, so me semiconductors or insulators. It is in situations of this type that crystallography in general and period icity of crystals in particu lar beco mes significant in the develop ment o f solid state theory.
23.2 Sig nificance of translati onal peri o dici ty.
We may select some function say f(r) in the direct lattice space. It is well known that atoms exhib it translational periodicity in a crystalline solid and because of this fact, this function isalso expected to be periodic in the sense that it should be the same atphysically equivalent points in the crystal latt ice . The period icity of the funct ion f(r) could be expressed as:
f( r + R ) = f ( r )……………….23.1
for all points located at r . The vecto r R by virtue o f t ranslat ional period icity is represented by:
R = l a + m b + n c,
where l, m, n are integers and a,b,c are translational period icit ies along x, y and z directions . Let us start with a one-dimensional case for the sake of convenience and simplicity of mathematical treat ment and represent equat ion 23.1 fo r one-d imensional case as:
f( x + a ) = f(x)……………… 23.2
Since the function of equation 23.2 is periodic, it may be exp ressed in the form o f Fourier series.
The exp ression of equation 23.2, if expressed in the Fourier series form can be written as:
f(x) = ∑ Fn2πinx/a……………..23.3
n
where n is an integer, a is period of the function.
We may exp ress equation 23.3 as:
f(x) = ∑ Aфeiфx , where │ фn │ = n.2π/a …………………23.4
ф
The ф’s represent the reciprocal lattice vectors for the linear lattice in one-dimension:
The Fourier coefficients Aфin equation 23.4 are given by:
Aф = 1/a∫ f(x) e ─iфxd x………………….23.5
In terms of the Fouriercoefficient the L.H.S. of equation 23.2may be expressed as:
f (x+a ) = ∑ Aфeiф(x+a) ………………….23.6
It shows that the function f(x) = ∑ Aфeiфxis a function in the d irect lattice space.
We may extend the expressions 23.2, 23.3&23.4 to the more general case by considering lattice periodicity along the three orthogonal axes which may be represented by │a│ ,│ b │ and │ c │. In that case, we have on using equation 23.1,
f ( r + R ) = f ( r + l a + m b + n c ) = f ( r )……………………23.8
As in the case of one-dimensions, the exp ression for three-dimensions may be written as:
f (r ) = ∑ B (ф 1, ф 2 , ф 3 ) ei (ф 1 x + ф 2 y + ф 3 z ) …………………….23.9
ф 1 ф 2 ф 3
Equation 23.9 may also be written in the form:
f(r) = ∑ B ф ei A r……………………………23.10
where ф is the recip rocal lattice vector with co mponents ф 1 , ф 2 and ф 3 along the x,y and z axes.
As in the one-dimensional case above, here also:
Which is in agreement with equation no. 23.1
23.3 Perio dic Boun dary Con di tions & Wave Vector .
In the So mmerfeld’s model, the free electrons are assumed to travel in a potential wh ich is constant everywhere inside the metal. The electron is considered bound inside a cubical crystal of side say ‘l’.The potential inside the crystal is assumed to be zero and the potential outside the crystal is infinity. The motion of the electrons is calculated by solving the Schrodinger equation :
2Ψ+( 8π2 m/h2 )E Ψ = 0
Instead of assuming that the electron wave function vanishes at x=0 and at x= l, it is considered more realistic to assume that the electron wave function is periodic with a period l. That means there must be integral number o f wavelengths of the electron wave (de Broglie wave) in the distance ‘l’ of the crystal. It can be exp ressed mathematically as:
Ψ [ ( x+l), y , z ] = Ψ (x,y,z) ,
with similar exp ressions for y and z coo rd inates.
This is popularlyknown as periodic boundarycondition, also known as cyclic boundary condition .It is an important concept in solid state physics.
Considering the fact that atoms in a crystal are arranged in an orderly manner consequently leading to periodicity of the lattice, one can express the periodic boundary condition for the electron wave function as:
Ψ ( x + a ) = Ψ (x ) ,
where a is the lattice constant Ψ ( x + a ) = Ψ (x ) ,
The relation between the wave vector k and the energy E is given by : K = 2π (2mE) 1/2 h The De Broglie waves, also known as Sommerfeld waves or Bloch waves in so far as motion of electrons in a crystalline solids is concerned, are expressed in terms of the wave vector. We know
.E =p 2 /2m, Since p = h/λ, E = h 2 /2λ 2m
Substituting this expression for E in the above equation, we get
K = 2π/λ
Therefore, k is a vector whose magnitude is 2π/λand whosedirection is the same as the direction of propagation of electron wave or the De Broglie wave (So mmerfeld waves or Bloch waves). It is also called wave vector or wave number. The kinetic energy of the electron is proportional to the square of the wave number and the relationship between E and k for a free electron is parabolic as shown in the figure 23.1
Figure 23.1: Plot of E Vs. wave vector k fo r a free electron
23.4 Bloch Theorem
What has been discussed above is regarding general functions with the periodicity of the lattice. It is general case of any such function. We shall now consider here a specific function of periodic potential. We know that the atoms orions in a crystalline solid are positioned in regular periodic arrays and assuch theelectrostatic potential exhib ited by them is also periodic which may be exp ressed as:
V ( r + R ) = V ( r ) ………………23.11
for all Bravais lattice vectors r .
This potential assumes significance when we deal with an understanding of motion ofelectrons in a crystalline solid. The periodicity of the electrostatic potential V is of the order of 1Ǻwhich is almost crystal is free fro m any imperfection wh ich is rather an ideal concept. The real crystalline solid is not free fro m any imperfection because of which the periodicity breaks down in the region around imperfection. The imperfections influence the physical properties of crystalline solids. However, it is necessary to assume that we are dealing with an ideal crystalline solid in order to be able to develop the theory regarding physical properties of solids in the background of symmetries and periodicity in crystals. It is in this ideal crystal that we shall consider the motion of electrons as represented by one-electron potential V(r) which fo llo ws the expression no. 23.11 for an ideal crystal, i.e .,
V (r + R) = V (r)
The motion of electrons is described by Schrodinger equation:
HΨ = (- ħ2/2m 2 +V (r) ) Ψ = E Ψ ……………………………23.12
Where H stands for Hamiltonian, m is the effective mass of the electron;ħ= h/2π (where h represents Planck’s constant , Ψ is the wave function , 2is the operator ∂2/∂r2 and E is the total energy). In order to describe the motion of the electrons in crystalline solid , equation 23.12 is required to be solved with the periodic potential of an ideal crystal lattice V (r + R ) = V ( r ). The solution of this equation is given by :
Ψk( r ) = Vk (r) eik. r ,……………….23.13
Where Vk( r + R ) = Vk(r) ………….23.14
Functions of the type of equation23.13 are called Bloch functions,k the wave vector (= λ/2π). It was Bloch who in 1928 showed that the solutions are of the form as given by equation 23.13.
Fro m equations 23.13, 23.14, we have:
Ψk( r + R ) = e ik.R . Ψk (r)………………………….23.15
Bloch function can also be exp ressed by the following equation by selecting
Eigen states of H in such a way that corresponding to each Ψ there is a wave vector k so that:
Ψ ( r + R ) = e ik.R Ψ (r) …………………………..23.16 for all possible Bravais lattice vectors R .
Referring to Schrodinger equation no. 23.12, its solution is given by:
Ψ( r ) = f ( r ) u ( r )………………….23.17
where u ( r ) is a periodic function . Since the potential u(r) is periodic, all the quantities associated with the electron are also periodic. The quantity │ Ψ (r)│2has also to be periodic provided f(r) satisfies the condition :
│ f ( r + R ) │2 = │ f ( r )│2 ………………………………23.18
The solution of this equation is expressed to be of this form:
Ψ (r) = u (r) e ikr ……………………..23.19 This equation represents Bloch theorem.
If one considers the periodicity of atoms in the crystalline solid, it is obvious that the electron moves in the field of periodic potential. For example, figure 23.2 indicates the qualitative form of the electrostatic potential energy of conduction electrons in the field of positive ion cores of monatomic linear lattice. The ion cores left out by conduction electrons are positively charged. The potential energy of an electron in the field of these positive ion cores is negative.
Figure 23.2: Variat ion of electrostatic potential energy of electrons in the field of ions mo no ato mic linear lattice
23.5 Kronig -Penney Mo del
Here, we consider the motion of electrons in a simple one-dimensional periodic potential and use Bloch theorem to understand about the allowed and forbidden electronic energy bands in a onedimensional lattice. For this purpose we shall consider Kronig-Penney model in which the potential energy variation is assumed to be of the form as shown in figure 23.3.It shows an ideal periodic square well potential used in Kronig-Penney Model to illustrate the general characteristics of the quantum behaviour of the electrons in periodic lattices.
The essential features of the behaviour of electrons in a periodic potential may be studied by considering the model first discussed by Kronig and Penney (1931) and so is named after them. In this model, it is assumed that the potential energy of an electron moving in onedimensional perfect crystal lattice is represented in the form of a periodic array of rectangular wells as indicated in the figure; the period of potential being (a+b).In regions ranging 0 < x < a potential energy is assumed to be zero whereas in the regions ranging ─ b < x < 0 , the potential energy is V0 . Each of the potential energy may be treated as a rough approximation for the potential in the vicinity of an atom. With this infinite one-dimensional rectangular well potential, it is possible to obtain an exact solution of the Schrodinger equation.
Figure 23.3: A one-dimensional periodic potential of periodicity (a+b) for the Kronig-Penney
model
Although it is somewhat an idealized periodic potential wh ich in reality is just a crude approximation towhat in reality exists in the real crystal, it is nevertheless very useful because it serves the purpose of illustrating the most explicit way with the help of which several important characteristic features of the quantum behaviour of electrons in a periodic lattice can be described.
And V (x + a + b) = V (x)………………………23.21
So, the periodicity of the potential as represented by the figure is (a + b). The Schrodinger equations for the two regions are given by:
Here, it may be assumed that the energy E of the electrons is smaller than V0 i.e., E <V0 . We also put two real quantities α and β by:
E < V0, because α and β are real quant ities.
If E >V0 then β will be imaginary.
The Schrodinger equation 23.22 and 23.23 may be written as :
Where uk(x) is the periodic function in x with the period (a+b).
Substituting fro m equations 23.27 and 23.29 in equation no. 23.25 we get :
For the sake of convenience let us drop the subscripts and write the above expression as:
Similarly, substituting equations 23.27 and 23.29 in equation 23.26, we have:
Representing the value of u(x) in the interval 0< x < a by u1(x) and in the interval -b < x < 0 by u2(x), equation 23.30 and 23.31 may be written as :
The differential equations 23.32 and 23.33 are easily solved by the standard procedure as follo ws:
In solving these differential equations, it is assumed that the solution is of the follo wing form :
The first two conditions under equations 23.37(a) and 23.37(b) are imposed because of the requirements of continuity of the wave functions. Because of continuity at x=0, the two wave functions and their derivatives must have the same value at x=0. The other two conditions viz., 23.38(a) & 23.38(b) are required because of the periodicity of uk(x). Thus, the function uk(x) should have the same value at x=a and x= ─ b.
Imposition of the first two conditions 23.37(a) & 23.37 (b) in equations 23.34 and 23.36 leads to:
The derivation of equation no.23.40 is given as under:
Fro m equation 23.35, we have:
Similarily, from equation 23.36, we have:
Equation 23.40 is thus derived here. Moving forward we have:
Imposition of the next two conditions 23.38 (a & b ) in equations 23.34 and 23.36, we have:
Thus application of 23.37(a ,b) on equation no. 23.34 and 23.36 leads to four linear homogeneous equations in the constants A, B , C and D. The coefficients A, B, C and D can thus be determined as the solution of a set of these four simu ltaneous linear homogeneous equations in these quantities. There is no solution other than A =B =C =D =0, unless the determinant of the coefficients vanishes. That means these four equations 23.39 to 23.42 have a solution only if the determinant of the coefficients of A,B,C, and D van ishes.
On expansion of this determinant, one can show after rigorous and straight forward algebra that equation 23.43 can be expressed as:
In order to obtain a more convenient equation, Kronig and Penney considers the case for which the potential barriers become delta functions i.e., V0 → ∞ & b → 0 but the product
V0b or β2b remains finite. Such a function is known as delta function. Under these circu mstances equation 23.44 reduces to:
It is because when b → 0 sinh βb → βb and cosh βb → 1 and because E < V0 and V0 → ∞ so E is very small, α2 may be neglected but β cannot be neglected on account of involvement of the term (V0 ─ E).
Equation 23.45 can be further exp ressed as:
Let us put β2ab/2 = P in the above equation, we get:
The quantity P is defined by the exp ression:
potential barrier. The physical significance of this quantity is that if P is increased, the area of the potential barrier is increased and so a given electron is bound more strongly to a particular potential well. When P → 0, the potential barrier beco mes very weak which in other words means that electrons are free electrons
When P → 0 i.e., for free electrons, expression at 23.46 can be written as:
The expression at 23.47 resembles the result as was obtained by considering the So mmerfeld model of
metals.
A plot of the function (Psinαa/αa + cosαa) when drawn against αa for P = 3π/2 appears as shown in figure 23.4.
Figure 23.4: Variation of function (P s in αa/ αa + cosαa) with αa for P = 3π/2.The allowed values of energy are given by those ranges of α =( 2mE/ħ2)1/2 for wh ich the function lies between ─ 1 and + 1
We know that α = (2mE/ħ2)1/2 which suggests that α2 is proportional to energy E. The abscissa is a measure of energy and in order to calculate the energy represented by the function at a point, the value of αa corresponding to that point is determined. The values of αa which satisfy equation 23.46are found out by drawing a line parallel to αa axis at a distance coska from it. If ka is continuously varied from 0 to π, i.e., coska from +1 to ─ 1, one is able to find all possible values of αa and hence that of energy. It is important to realize that the right hand side of equation 23.46 can accept only values between ─ 1 and + 1 (i.e., ka continuously varying from 0 to π and coska from +1 to ─ 1) as indicated by the horizontal lines in the figure. Therefore, the condition as imposed by equation 23.46 can be met only for values of αa for which the L.H.S. lies between ±1.
In figure 23.4, the function: f(αa) = Psin αa/αa + cos αa is plotted against αa. Obviously, for k to be real │f (αa) │should be less than 1. Those values of E for which │ (αa) │> 1 will be forbidden.Those values of E for which f(αa) ≤ 1 will correspond to allowed values of energy. It may also be noted that if k is replaced by k+2nπ/a (where n= ±1, ±2, ±3……), the right hand side of equation 23.46 will remain the same. That means one cannot uniquely determine the value of k. It is usually restricted to the domain:
which is known as the first Brillouin zone.
─ π/a≤ k≤ + π/a
The E – k diagram showing the allowed and forbidden energy bands are shown in figures 23.5 & 23.6. At the zone boundaries ( k=± π/a ) the Bloch wave function satisfies the Bragg condition and the group velocity [=( 1/ħ)dE/dk] is zero which corresponds to a standing wave.
Figure 23.5: E Vs. wave vector K for electron in a periodic potential
Figure 23.6: Full curve of E Vs. wave vector k showing d iscontin uit ies at k=±n π/ a (where n=1, 2, 3…)
Let us analyse the details of what may be concluded fro m the figure drawn on the basis of equation 23.46.
(i) There are infinite nu mbers of allo wed energy bands separated by intervals in which there are no energy
levels. Such regions are known as forbidden regions.
The boundaries of the allowed ranges of αa correspond to the values for which coska=±1
That happens if ka = nπ, or k = nπ/a, where n = 1, 2, 3…….
(ii) The first term in the expression for f (αa) i.e ., P.sinαa/αa decreases on an average with increas ing αa. It suggests that the width of the allowed energy bands increases and the forbidden regions get narrower as αa increases or as energy increases.
(iii) As P increases, the width of the allowed band decreases. Increase in the value of P means the increase in “binding energy” of the electrons. In the extreme case, when P is infinite, the allowed energy bands get infinitely narrower, consequently the energy spectrum becomes line spectrum and are independent of k.
If P→ ∞, the allowed energy ranges of αa reduce to points given by:
It shows that E is independent of k.
The energy levels in this case are discrete and the electron is completely bound. It is trapped within the potential wells and moves only in one cell of width ‘a ’.
Fro m the discussion on analysis of equation 23.46, we co me to the conclusion that in the one-dimensional problem for the limit ing case, the spectrum of energy values that are permitted is found to be consisting of continuous regions separated by finite intervals. By vary ing the quantity
P (i.e., V0 b) fro m zero to infinity, we move fro m the case of free electrons to that of bound electrons and so are able to study the changes in the allowed and forbidden ranges of energy and the wave function.
We may also use equation 23.46 to study the variation of energy E with the wave number k.The variation is as shown in figures23.5 & 23.6.Fro m these figures one arrives at the follo wing conclusions:
(i) The energy spectrum of the electrons consists of a number of allowed energy bands separated by forbidden regions.
(ii) The width of the allowed energy bands increases with increase of energy values.
(iii) If V0 is made zero, the E versus k curve becomes a continuous parabola, the same as that of free electrons (see figure 23.1)
(iv) There are d iscontinuities in the E versus k curve which occur for k = nπ/a, where n = ±1, ±2, ±3.The range of allowed values of k between ─ π/a and + π/a constitutes the first Brillouin zone. The
range of values of k between ─π/a and ─ 2π/a and between + π/a and + 2π/a constitutes the second Brillouin zone.
The energy E is represented as function of k for P = 3π/2.Analyzing the curve of E against k further regarding the discontinuities in the E versus k at k = nπ/a, where n= 1, 2, 3…….The k-values define the boundaries of the first (I), second (II), third (III) and so on Brillouin zones. Figure 23.5 shown here provides only half of the E-k curve; the first zone in fact extends from k= ─ π/a to + π/a.
Similarly, the second Brillouin zone is composed of two parts; the one extending from +π/a to + 2π/a as shown in the figure and the other part extending fro m ─ π/a to ─ 2π/a. Figure 23.6 represents the full E-k curve. We may call each position of the curve as a band. The curve of E-k has the follo wing characteristics:
1. The curves are horizontal at the top and bottom.
2. The curves are parabolic near the top and the bottom having curvatures in opposite direct ions.
3. d2E/dk2is positive in the lower portion of the band and negative in the upper port ion of the band.
Let us consider a linear lattice of lattice constant a. According to free electron theory E versus wave vector k is continuous as shown in figure 23.1. The low energy part of the band structure is shown in figure 23.7 for electrons which are nearly free but with an energy gap at wave vector k= ±π/a resulting into creation of forbidden gap as a result of our assumption that electron is moving in a periodic potential (unlike in the case of free electron theory where electron is assumed to be moving in a constant potential field). The curve of E versus wave vector k for an electron in a monato mic linear lattice with lattice constant a appears as shown in figure 23.7. The energy gap Eg is associated with the first Bragg reflection at k = ±π/a.
Figu re 23.7: E as a function of wave vector k fo r an electron in a monatomic linear lattice o f latt ice constant a. The forb idden band associated with the energy gap Egis shown The Band gap is associated with the first Bragg reflection at k=±π/a .
The Bragg condition in terms of the reciprocal lattice imp lies that:
for diffraction of a wave vector k, if taken in one dimension becomes on expanding the dot product and simp lify ing
where G is 2π times a vector fro m the origin to a lattice point of the reciprocal lattice and k is a vector of magnitude 2π/λ along the direction of the incident x=ray beam. Equation 23.49 is the vector form of the Bragg equation.
In one dimensions equation 23.49 beco mes:
k = ± ½ G =± nπ/a, …………………23.50 Where G = ±2nπ/a is the reciprocal lattice vector.
The first reflection and the first energy gap results at k = ±π/a; other energy gaps result for other
values of integer in equation 23.50.
The reflection at the wave vector k =±π/a takes place because the wave reflected fro m one atom in
the linear lattice meets constructive interference with the wave fro m a nearest neighbour atom. The
phase difference between the two reflected waves is ±2π for these two values of k. The region in the k-space between ─π/a and +π/a is called the first Brillouin zone .
23.6 Brillouin Zo nes
Kronig–Penney model has shown that the energy discontinuities in the monatomic one dimensional lattice result when the wave vector satisfies the relation :
k = nπ/a , where n is an integer which may be positive or negative.
In the one-dimensional monatomic lattice a line representing the value of k is divided up by the energy discontinuities into segments of length π/a as shown in figure 23.8
Figure 23.8: Line rep resent ing the value of k for one -d imension al monatomic lattice is sho wn divided into segments of length ±nπ/a (n=1, 2,3…)
These line segments are known as Brillouin zones. The segment ─ π/a < k < π/a represents first Brillouin zone; the two segments ─ 2π/a < k < ─ π/a and π/a < k < 2π/a form the second Brillouin zone and so on. Brillouin zones are characteristic of a particular crystal structure and as such each crystal structure form its own characteristic Brillouin zones.
Let us take the examp le of two dimensional simple square lattice as shown in figure 23.9. The first Brillouin zone for this lattice will be a square ABCD whose boundaries are defined by:
kx = ±π/a; ky =± π/a
Similarly, the boundaries of the second Brillouin zone are defined by :
±kx =±ky = π/a as represented by the diagram EFGH.
Figu re 23.9: Brillouin zones in two d imens ional simple square latt ice
One can apply the same principle for the three d imensional crystal structures. For a simp le cub ic lattice, the first Brillouin zone is a cube of edge 2π/a.
23.7 Significance of Brillouin Zones
(i) Looking at the curve shown in figure 23.6 one finds that electron energy increases continuously from zero until the value of k reaches π/a. Thereafter, it gets stopped as if it meets an obstacle or a wall at that instant and consequently gets reflected.
Treating electrons as waves and the propagation of electrons through a crystal as analogous to the propagation of electromagnetic waves, we arrive at a very important conclusion. We know that x-ray will suffer
reflection if incident normal to a set of planes of interplanar spacing ‘a’ provided this equation is satisfied:
nλ = 2a sin90⁰ ( for x-ray is at right angles to the planes)
nλ = 2a
For electron k = nπ/a and k = 2π/λ
So, 2π/λ = nπ/a, or nλ = 2a, wh ich is the same condition as for Bragg reflect ion. Thus, we may conclude that electron suffers Bragg reflection when n =±1,±2,±3 …… corresponding to first, second, third orderreflections and so on. The zones between the values of k = ─ π/a and + π/a constitutes the first Brillouin zone and so on.
(ii) The zone boundaries represent the maximu m energies that the electron can have without developing any discontinuity.
(iii) The energy gap at the zero boundary is called the Forbidden zone or band; electrons cannot have those energies.
(iv) Considering velocity of electrons in a periodic potential (given vg = 2π.dν/dk; E = hν and so vg
=2π/h.d E/dk)
Slope d E/dk = 0 when k = 0,
& d E/ dk = 0 when k= π/a.
Therefore, the velocity of electron is zero both at the bottom and at the top of the first Brillou in zone or band
At intermediate regions in the zone the electron velocity reaches the free electron velocity h k/ 2π m.
(v) Let us now consider the motion of the electron in the first Brillouin zone, under the continued application of a force F (either due to electric field or any other agency). Velocity of the electron at k = 0 will
increase but when it approaches the value of k close to + π/a its velocity begins to decrease and at k = + π/a
the velocity becomes zero, indicating that the electron wave packet suffers a Bragg reflection and begins to
travel in a direction opposite to the applied force. The propagation in the negative direction continues until
it becomes equal to ─ π/a. Once again the electron suffers Bragg reflection and the forward propagation
starts until value of k reaches +π/a. In other words, the electron is shunted back and forth in the first
Brillouin zone under the application of a constant unidirectional force. It means that a stationary electron
wave is set up in the first Brillouin zone instead of a travelling wave
vi) An electron remain ing in one zone or band cannot cross over to another zone by continuous application of force.Two consecutive Brillouin zones are separated by a forbidden energy gap. So, unless and until the electronin the first zone absorbs an energy equal to that of the forbidden gap in a single dose, the electron
cannot cross over to the next Brillouin zone irrespective of the duration of the applied force
vii) If the electrons continue to move back and forth in their respective Brillouin zones there is no change in the sum of the mo menta of all the electrons and, as a result, no net current is conducted by the crystal. It, obviously, means that the material is an insulator. Now suppose the forbidden gap is small enough or if impurity atoms provide some localized electronic states within the forb idden energy gap, one will find a semi conductor behaviour. The third situation is that there is no forbidden energy gap. In that case there is no restriction for increase in the value of k and so electric conduction can take place. The material in this case is a good conductor.
SUMMAR Y
- We have discussed motion of electrons, both in a constant potential field (as assumed by Somerfield) as well as in a period ic potential field as assumed in Kronig Penney model.
- Significance of t ranslational period icity in a crystalline solid is explained.
- The concept of De Broglie wave or So merfield waves or Bloch waves are given.
- It is shown that Somerfield’s model of motion of a free electrons in a constant potential field leads to a relation between energy and wave vector for a free electron that is parabolic.
- Bloch theorem is explained.
- It is shown that if electron is assumed to travel in a periodic potential as proposed by Kronig Penney model it leads to energy versus wave vector tha shows discontinuities at
- The concept of brillioun zones and their significance in crystallography is exp lained
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References.
1.Verma,A.R. &Srivastava,O.N.” Crystallography for Solid State Physics”,Wiley Eastern Ltd., N.Delhi,1982.
2. Seitz ,F.: “ Modern Theory of Solids”, McGraw-Hill, N.Y.,1940.
3. Omar,M.A. “ Elementary Solid State Physics”, Addison-Wesley, Readin,1975.
4. Kittel, C.:”Introduction to Solid State Physics”,Wiley-Eastern Ltd.,N.Delhi,1985.
5. Wannier, G.H.: “ Elements of Solid State Theory”,Cambridge Univ. Press,1959.
6. Ziman,J.M.: “ Principles of the Theory of Solids”, Cambridge Univ. Press,1964.
7. Clark,H.: “ Solid State Physics”,Macmillan,London,1968.
8. Pillai,S.O. : “Solid State Physics”, New Age Int.(P) Ltd.Publishers, N.Delhi,1997.
Suggested Reading.
- Madelung,O :” Introduction to Solid State Theory”, Springer-Verlag,N.Y.,1978.
- Ghatak, A.K. &Kothari,L.S.: “ Introduction to Lattice Dynamics”,Addison Wesley,Reading,1971.
- Ashcroft,N.W. &Mermin,N.D.: “Solid State Physics”,New York: Holt,Rinchart and Winston,1976.