ELECTRONS IN A PERIODIC POTENTIAL AND …...electric field. •These electrons are not bound to any particular atom in the solid but instead they belong to the whole solid. •In insulators,

ELECTRONS IN A PERIODIC POTENTIAL AND ENERGY

BANDS IN SOLIDS-2

ENERGY BANDS IN A SOLID : A FORMAL APPROACH

* Electrons motion in a crystal will now be considered from a quantum mechanical point of view based on Schroedinger's equation.* The crystal may be viewed as a periodic arrangement of positively-charged ion cores comprising the positively charged nuclei and non-valence electrons that may be assumed stationary in the lattice points.* The potential energy of a valence electron in a single isolated atom is shown in Fig. a. * If several of these atoms are placed in close proximity and equidistant from each other (Fig. b), the potential functions of neighboring atoms overlap so that the net potential function will look like the potential function depicted in Fig. c. * However we can simplify the potential function by using the potential function shape shown in Fig. d (referred to as Kronig-Penny Model).

SCHROEDINGER'S EQUATION FOR A PERIODIC POTENTIAL

positively charged

core

U(r) ~ - 1/r

E = 0

ab

Uo

x

U(x)

(a)

(b)

(c)

(d)

* The time-independent Schroedinger equation for the motion of the electron in the one-dimensional periodic potential of Fig. d takes the form

* It is of course implicit in the equation above that the time-dependent component of the wave-function is of the form

* To obtain the solution to the Schroedinger's equation, we utilize a mathematical theorem by Bloch.

y x( ) = u x( )eikx

* The function u(x) is called Bloch function and it is periodic with the same periodicity as that of the potential function.

* The condition for a well-behaved solution is

* The equation gives a simple relationship between E and k for a simple potential described by a finite product bUo

* We must reiterate here again that the equations above are conditions that ought to be satisfied for the Schroedinger'sequation to have any solution other than the trivial solution. However they provide relationships between the total energy of the electron and its momentum (or wave-number) and their dependence on the shape and magnitude of the barrier.

ALLOWED ENERGY BANDS AND FORBIDDEN ENERGY GAPS

* To understand the nature of the solution, let us begin by considering the special case for which Uo = 0. This is the case where K = 0, and it corresponds to a free electron since there are no potential barriers. In this case the requirement above reduces to

* Which shows that the relationship between E and k is parabolic as depicted in the figure.

E

k

E =2m

22h k

* Let us now explore the relationship between E and kfor the electron in a single crystal lattice. Let us require

-1£ g aa( ) = cos aa( ) + Ksin aa( )

aa£ +1

since -1£ cos ka( ) £ +1

From the figure we can deduce the following information on the motion of an electron in a single crystal :

(i) There are only certain energy values that can be taken by the electron in a crystal. We may compare this to the E versus kplot for the free electron where the energy of the electron can take any value. This energy range is called a band. A band has a quasi (Latin for "almost")-continuous set of energy states which can be occupied by the electron.

(i) The energy range between E1 and E2 is not accessible to electrons in the crystal. That is to say, the energy values between E1 and E2 are all forbidden. This energy zone bounded by E1 and E2 is called the forbidden energy gap or simply energy gap.

First energy

band

First energy

band

Second energy

band

Forbidden energy

gap

Forbidden energy

gap

E

METALS, INSULATORS, AND SEMICONDUCTORS

Full Full Full

Partially

filled

Empty

Empty

conduction bandconduction band

conduction band

valence band valence band valence band

Eg

Eg

Eg

(a) (b) (c)

• Electrical conduction will occur if the highest occupied energy band is partially filled (Fig. (a)). This is the case in metals where the highest energy band is partially filled and there are higher energy states in the band available for the electrons in the band to go to if their energies are somewhat increased by an externally applied electric field.• This highest energy band is called the conduction band. The electrons occupying this band are referred to as the conduction electrons. The reason they are called conduction electrons is that they are the electrons that are able to carry electric current across the solid.• An alternative name to these electrons is "free" electrons as these electrons are delocalized and are free to move throughout the solid if perturbed for instance by an applied electric field.•These electrons are not bound to any particular atom in the solid but instead they belong to the whole solid.

•In insulators, however, all the energy bands are completely full with electrons (Fig. (b)). In this case the first empty band is separated by a forbidden energy gap from the highest filled band as we described in the energy band scheme of solids in the previous section.•The highest completely filled band in an insulator is called the valence band. It is called the valence band because it contains valence electrons ; i. e., the electrons in this valence band are somewhat bound to their individual atoms.•These electrons are , therefore, unable to move in the entire solid and they are consequently unable to carry electric current.•The separation in energy between the top of the valence band and the bottom of the next empty band (the conduction band) is the energy gap and is generally denoted by the symbol Eg.•In an insulator Eg is very large and it is unlikely that electrons in the valence band will be able to make it to the conduction band and, hence, the material remains insulating.

* The band scheme of a semiconductor is essentially similar to that of the insulator ; the highest band is completely full with electrons (Fig. c)).* However in a semiconductor the energy gap Eg is small and electrons in the valence band can potentially acquire enough energy to surmount this energy gap and make it to the conduction band.* These electrons will, therefore, become free and available for conduction in the presence of an electric field across the semiconductor.* Generally, in a semiconductor this energy Eg required to raise the electron to the conduction band can be supplied as thermal or optical energy.* In silicon, the most known and utilized semiconductor, Eg ~ 1 eVand this amount of energy is comparable to the thermal energy of an electron at temperatures below liquid nitrogen temperature (~78K).

Problem (2)The E versus k relation for the conduction band of a certain semiconductor is

given by

Find the maximum and minimum energies and the corresponding values of k for this band, and also find the width, in eV, of this band.Solution (2)

E = 2.8 - Acos ka( ) where E is in eV

A =1.7, a = 2 ´10-8 cm, and -p

a£ k £

p

a

The E versus k relation for the conduction band is

E = 2.8 - Acos ka( ) for -p

a£ k £

p

aand A =1.7

The maximum and minimum values of E are determined by values of k such that

dE

dk= aAsin ka( ) = 0

This is satisfied by

k = 0 and ±p

awithin -

p

a£ k £

p

a

To determine whether these three values of k give maximum or minimum energy values is determined by the sign of the second derivative at these points

d2E

dk 2= a2Acos ka( ) = a2A > 0 for k = 0

and

d2E

dk 2= a2Acos ka( ) = -a2A < 0 for k = ±

p

a

Therefore k = 0 corresponds to the minimum band energy,

Emin = Ec( ) = 2.8 -1.7cos 0( ) a( )[ ] =1.1 eV

k = ±p

acorrespond to maxima

The width of this band DE is

THE EFFECTIVE MASS APPROXIMATION

* Classically, the equation of motion for an electron in a crystal

Ftot = Fint + Fext = Ftot = mdv

dtm is electron's "rest mass"

* Fint is net internal forces in a crystal (~1022 forces/cm3) and Fext is the externally applied force on the electron. The “effective mass”, m*, is an invention which takes into account effects of all the internal forces and reduces the equation of motion to

Fext = m* dv

dt* Once adopting the effective mass approach we can treat the electron in a crystal as a free electron with mass m*

The energy of the electron in the crystal is hence

Differentiation once yields

Differentiation twice yields

Problem (3)Consider the semiconductor in problem (2) above. (a) Find an

expression for the velocity of an electron in the conduction band as a function of k. (b) What are the numerical values of this velocity at k = 0 and k = p/(2a) ? (c) Find an expression for the effective mass of an electron in the conduction band as a function of k, and evaluate this effective mass at k = 0 and k = p/(2a).

Solution (3)The E versus k relation for the conduction band is given by

E = 2.8 -1.7cos ka( )

Upon substitution one gets

v =1.7 2.0X10-8( )sin 2.0X10-8 k( )

6.58X10-14( )or

v = 5.2X105 sin 2.0X10-8 k( ) cm /s for k in cm-1

(a) The velocity is

(b) Substituting k = 0 gives v = 0 at k = 0. At k = p/2a we have

(c) The effective mass for the conduction band electron is given by

Substitution gives

m* =6.58X10-14( )

2

1.7 2.0X10-8( )2

cos 2.0X10-8 k( )[ ]or

m* = 6.37X10-12 sec 2.0X10-8 k( ) gm for k in cm-1

(d) At k = 0 m* = 0. But

And at k = p/2a m* is infinitely large.