7.7 EFFECTIV E MASS AND MOBILITY OF CHARGE CARRIERS
8.1.3 Temperature dependence of conductivity in semiconductors
Can the temperature dependence of conductivity in semiconductors be described by the classical electron theory?
In intrinsic semiconductors, the number density of charge carriers increases with temperature according to the equation
where £ g is the band gap and the above equation assumes that the Fermi level is in the middle of the band gap. This equation shows that there is an increase in the number density of conduction electrons with temperature. In addition, there is a change in mobility of the charge carriers with temperature, but this is less significant than the change in charge carrier density. Therefore in semiconductors the temperature variation of N dominates the temperature dependence of conductivity.
Macroscopic electrical properties 159
8.1.4 Temperature dependence of mobility
How can the temperature dependence of mobility of electrons be explained?
ô ô ô " " > -
- — - >h“ * • '■
ex c
^ m y
. ¡t is the temperature dependence of t which determines and it can be seen that u s ^ can view thjs as the temperature dependence the mobility, o r a . ‘ the equation of motion of the electrons. Therefore
£ ^ T ta r^ n crefses wiih temperature, leading to a reduction in mobility p 3"The"dassicaf m o'd e l” vz s no indication of the temperature dependence of y.
a h i o u g t u is reasonable to suppose that, as the temperature is raised, the increased vibrations of the lattice will cause more collisions with the rec electrons and contribute to a higher resistive coefhcient y or shorter mean free time t.
8.1.5 Different types of mobility
How can we define electron mobility in a material?
There are four different kinds of mobility of electrons in a material which must be distinguished.
1. Microscopic mobility
v
P m i c p*
This is defined for a particular electron moving with drift velocity r in an electric field £ It therefore cannot easily be experimentally verified, and so remains only a concept from which a more practical description of collective mobility of electrons can be developer
160 Electrical and thermal properties o f materials 2. Conductivity mobility
/'con- ex m*
i ms is me macroscopic average m(
measurement of electrical conductivity — niuoimya,
° — N eu“con assuming N and e are both known,
3. Hall mobility
= 0Rh = ^Hall
P o J H
is the mobility of charge carriers as determined from a Hall effect m easurem ent- 4. Drift m obility
. o T , a ™ L dr , r' T ed/ r°,'i; m easure* " ' o f , h e tim e , requ ired for carriers
is an e in e m aterial under (he a c tio n o f a n e le c tr ic field £
8.2 QUANTUM MECHANICAL DESCRIPTION
O F CONDUCTION ELECTRON BEHAVIOUR
D o a ll ■conduction d e c r a m a c tu a lly con tribu te to th e e le c tr ic a l c o n d u c tiv ity ? parhclehr h m h| 7 " .elf rons “ a m aterial b e h a v e n o , lik e c la s s ic a l
[ 1 ™ ,s leads >o Properties w h ich a re d ifferen t fr o m in a m e a ,T T . r “ ° f “ ' t a r i e field, th e v a le n c e e le c t r o n s V e a o n „ lh , I ợ T P w | o e i,y in an y d ir e c tio n . If w e p lo t th e Obrnm a 1 “ ,e el“ ,r o n V n s Pace, then for a f r e e - e l e c o n m e ta l w e aS S Th f t “ ° f Which o o rresp on d s to th e F e r m i v e lo c ity -
■ piTùZS
fid d is a P P " 'd ' ’be Ferm i sph ere is d isp la c e d a s s h o w n u n c o m „ i a maJOr;'y ° f " “ ’to n velocities c a n c e l, b u t n o w s o m e a r e n ô m T h r " 'S efcc,ron s w hich cau se the e le c tr ic cu rren t. W e a°e Closed 1 f im P ° r,aM r e iu " " > ô on ly certain sp e c ific e le c tr o n s w h ic h N o t e d m , 1° , T ? c a " “ ntribute to the c o n d u c tio n m e c h a n is m , S ' 7 m a rF r ° un mr° r h' a ' w h ere o n ly th o s e e le c t r o n s
b of the Fermi level could contribute to the heat capacity.
Conduction electron behaviour 161
Fermi A
surface under zero field
displaced Fermi surface
Applied electric field
Fig. 8.2 Velocity of free electrons within the Fermi sphere under zero electric field and under an applied field 4 along the x direction.
8.2.1 Q uantum corrections to the conductivity in Ohm’s law
Hovv is Ohm’s law modified if only the electrons close to the Fermi surface contribute to the conductivity o f a metal?
The highest energy that electrons can take in a metal in its ground state is the Fermi energy £ F. We also know that the density of occupied states is highest around £ F, since for a free-electron model the density of states D(E) has the following form, as shown in section 4.4.7
This means that only a small change of energy A£ is needed to raise a large number of electrons above the Fermi level. We will consider that the velocity of the uncompensated electrons under the action of the field 4 is close to the Fermi velocity. This will be a reasonable simplifying assumption. With this in mind, we can calculate the electric conductivity a, based on quantum mechanical considerations.
O ur Ohm’s law equation of section 8.1.1 needs to be slightly modified to take into account the fact that not all free electrons contribute to the conductivity.
Hence,
where v F is the velocity of electrons at the Fermi level and N* is the number of displaced electrons, that is those in the shaded region of Fig. 8.3 which contribute to the conductivity.
J = N*evh-
162 Electrical and thermal properties of materials
Density of available states
Density of occupied states N(E)
theory Pu*at*on density versus energy for free electrons according to the free-electron
8 2 2 Number of •conduction- electrons contributing to conduction
actuallv ^ ° Ut ^ VV man^ l^e s°-called conduction electrons in a metol actually contribute to electrical conduction?
density o fn ° ^ ta^ c*n exPress^on H*. This wiJI dearly be dependent on the A£, c^upie ^states at the Fermi level N(E) and the displacement energy
and consequently N* = N(E)AE J — N(E)AEevF
= vFe N ( E ) ~ A k . The
for theterm (dE/dk) .s determined from the energy versus wave vector diagram given case. For free electrons, we have E = h2k2/2m and hence,
and this yields
d£ _ h2k d k ~ m ~ hvp
/ = v 2FeN{E)hAk.
8-2 3 Displacement of the Fermi sphere under the action of an electric field How does the displacement o f the electron wave vectors depend on other factor*
such as the mean free time of electrons between collisions?
Now we need to find the displacement of the Fermi sphere Ak under the influe of the electric field £ Since we know that m(dv/dt) = <?<£;, and since p — hk is momentum, it follows that the force on the electrons can be expressed as
the
Dielectric properties 163
it dp , d k So,**
r = m — = h — = e £
dr *
dr
or
d* = ^ d r h
Ak = — At = -- T
h h
exrTrp^Jon f o r ^ r v ^ n ^ tlme ° f the electrons between collisions. With this e arrive at the following expression for the current density
Only the projections of vF along the
vF cos 9, contribute to the current direction of the electric field that is N(E)Çt i +n/2(pFcos6)2'
J -n/2
= $e2N(E)£ xv2.
For a spherical Fermi surface there is a slight correction which gives, J = ^ e2N{E)$ tv2
and finally, the conductivity is given by a = J/£, so that a = F^Af(£).
ô 1S ^ ua,n am mechanical statement of conductivity shows that not all t °*! uction e ec rons can contribute to the conductivity, but only those close of C eT / 7 aCC In ad(*ition’ conductivity is determined by the density Of occupied States near the Fermi level. For metals such as copper, which has one For bivalent metak per atom’ thls dens,ty >s high, leading to high conductivity, atom this density ic SUC i|3S ca*clum has two conduction electrons per Tt ^ h ^ e n s h y of I ! 01311’ T * " 8 ‘° 3 re,ativc|y ,ow conductivity. Therefore total number of c o n d i ^ 31 thf Fermi surface> and not the classically expected Z t Z COnduct,on electrons’ whjch determines the conductivity of a
8-3 D IELECTRIC p r o p e r t i e s
H ° W can we represent the response of a non-conducting material to an electric field?
M ost electronic applications involve the use of alternating electric fields or rrents. In these cases the atoms in insulators oscillate under the action of the
applied electric field, and these oscillations can be expressed in terms of the dielectric constant, This is often expressed in terms of real and imaginary components r 1 and c2
r. = £0£r = + ie2.
This dielectric ‘constant’ is actually dependent on the frequency of the applied electric field. When considering its dependence on the frequency of electro
magnetic radiation it is often represented as e(a>).