Deconvolution model 1 : Metallic samples

In this model we assume that the donor concentration is No described by an activation energy

AEo, and that the concentration of deep compensating acceptors is N^. Then the net carrier concentration N t available for current transport is the difference between the shallow donor concentration and the concentration of compensating acceptors {No - Na). At any given temperature, a portion of the net electrons (No - Na) is to be distributed over the conduction band states and the “2No” impurity band states. Electrons that are excited in the CB have a high mobility /// and the unexcited carriers remain in the impurity band but are free to move with a lower mobility jU2. Note that the number of states in the impurity band is now described as 2No,

not just as No (see section 3.3.3). This is because the wavefunctions of the electrons in the EB are not any more localised and are spread throughout the crystal allowing more than one electron to occupy the states of a given impurity atom. Now if the EB states are in the immediate neighborhood of the conduction band energy Ec, the equation expressing neutrality is given by the sum o f electrons in the conduction band and those in the impurity band.

Nj. = N^ - N^ = = const.

f 17 J7 \

1 + exp Ec - E j ,

k j

From Eq. (5.13), it follows that U2 = N t - nj and by defining b = f^ 2 IjJ-i (0 < 6 < 1) we can

reformulate the two-band transport Eqs. (5.11) and (5.12) as follows:

(5.15)

In the above we have assumed that is some fraction of the electron mobility within the conduction band. The number o f unknowns in the above two equations is three, «/, «2 and b and this leads to certain difficulties in solving it as a system. However, at low temperatures, the value of Uh equals Uh = N j and if b is not strongly dependent on temperature we can show that Hh, goes through a minimum, at a given T, for which the conductivities in the two bands are equal,

5. Impurity Band Conduction Processes

(1 + 6)' (5.16)

This minimum occurs when

N^b

" l + 6

(5.17)

In Eq. (5.16) we know the value of Nt from the low temperature measured data and we also have

^Hmin, therefore we can proceed calculating b at the temperature where the minimum in occurs.

Having the value of b at Tmin we are then allowed to extract the carrier concentration nj in the conduction band, at that minimum, from Eq. (5.17).

It should be noted that the de-convoluted data of C063 only are considered in this section since the calculated behaviour is typical of all heavily doped samples. This is a Si-doped specimen above the ~10’* cm'^ range and its nn(T) exhibits a minimum at 140 K equal to riHmw = 2.1x10’^ cm'^. At low temperatures, Uh saturates to a value of «// = 2 .4 x l0 ’^ cm'^ = Nj. By solving Eq. (5.16) we get a quadratic relationship between b and N j! riHmin which has the solution:

V

- 1 (5.18)

From this we obtain, b = 0.459 and from Eq. (5.17) above we find = 7.55xlO’^cm'^ at T = 140 K. In order to calculate the whole temperature dependence of rij, the position of the Fermi level

(Ef-Ec)relative to the conduction band needs to be evaluated at each T.

C alculation o f Ferm i level using Ferm i-D irac statistics

The evaluated parameter Uj at T^m can now be used to initially determine the position of the Fermi level Ep at that temperature. Earlier in chapter 3, an expression for the Fermi-Dirac integral (Eq. 3.50) was presented which reduced Fj/ 2 in a form capable of extending the validity range to

some degree of degeneracy (rj < 1.3). By rearranging Eq. 3.51, for the reduced Fermi energy to read exp(rj) = n J { N ^ - C n ^ ) , with C = 0.27, we can easily find Ep - Ec, at the temperature corresponding to the minimum in «//. That is:

k T

F =_§_isiiLin

\ N c J

5. Impurity Band Conduction Processes

Employing the approximation of Eq. 3.50, we can express the neutrality Eq. (5.13) in terms o f the reduced Fermi level 77;

which yields to a quadratic equation whose solution for exp(77) is [46]:

V"{[Afc + iV„ (3 - C) - N , (1 - C ) ] ' - 8AT„ ( N^ C)} -

Finally the Fermi level variation with temperature can be calculated by solving the above equation iteratively, which has been programmed by Matlab. Using the acquired variation of exp(77), the temperature dependence of n, is obtained by calculating Eq. 3.51. Once the temperature dependence of is established we can obtain the temperature dependence of «2 from Eq. (5.13). Note that Eq. (5.21) is also a function of No and Na, which are normally unknown. However, Na can be determined from the “flat” low-T mobility ijin) data by using the degenerate form of the ionised impurity scattering limited mobility (refer to section 3.4.1). Within this framework there is only one value of No for which Eq. (5.21) is satisfied at Tmi„ for the corresponding exp(7;).

Returning to the two-band transport equations (5.14) and (5.15) it can be seen that the parameters

ri] and «2 {=Nt - «;) are now known, which allows the Independences o f /U] and fj.2 to be found.

Solving Eq. (5.14) with respect to b, we can obtain its temperature dependence. This yields a quadratic equation in b and the solution that corresponds to b ranging from 0 < 6 < 1 is used for the calculations: b = f \ , 72, 72, 72, 1 + - ^ - 72^ --- V ”2 J 722 J (5.22)

Once the ratio b is obtained it easy to find the mobility in the conduction band jui at each temperature from Eq. (5.15), and since /J.2 =bfii, the T-dependence of fj.2 can also be obtained.

5. Impurity Band Conduction Processes